id	sid	tid	token	lemma	pos
ejde-320	1	1	electronic	electronic	ADJ
ejde-320	1	2	journal	journal	NOUN
ejde-320	1	3	of	of	ADP
ejde-320	1	4	differential	differential	ADJ
ejde-320	1	5	equations	equation	NOUN
ejde-320	1	6	,	,	PUNCT
ejde-320	1	7	vol	vol	NOUN
ejde-320	1	8	.	.	PROPN
ejde-320	1	9	2021	2021	NUM
ejde-320	1	10	(	(	PUNCT
ejde-320	1	11	2021	2021	NUM
ejde-320	1	12	)	)	PUNCT
ejde-320	1	13	,	,	PUNCT
ejde-320	1	14	no	no	INTJ
ejde-320	1	15	.	.	NOUN
ejde-320	1	16	82	82	NUM
ejde-320	1	17	,	,	PUNCT
ejde-320	1	18	pp	pp	ADJ
ejde-320	1	19	.	.	PUNCT
ejde-320	2	1	1–19	1–19	PROPN
ejde-320	2	2	.	.	PUNCT
ejde-320	3	1	issn	issn	PROPN
ejde-320	3	2	:	:	PUNCT
ejde-320	3	3	1072	1072	NUM
ejde-320	3	4	-	-	SYM
ejde-320	3	5	6691	6691	NUM
ejde-320	3	6	.	.	PUNCT
ejde-320	4	1	url	url	PROPN
ejde-320	4	2	:	:	PUNCT
ejde-320	4	3	http://ejde.math.txstate.edu	http://ejde.math.txstate.edu	PROPN
ejde-320	4	4	or	or	CCONJ
ejde-320	4	5	http://ejde.math.unt.edu	http://ejde.math.unt.edu	PROPN
ejde-320	4	6	orlicz	orlicz	PROPN
ejde-320	4	7	-	-	PUNCT
ejde-320	4	8	sobolev	sobolev	NOUN
ejde-320	4	9	inequalities	inequality	NOUN
ejde-320	4	10	and	and	CCONJ
ejde-320	4	11	the	the	DET
ejde-320	4	12	dirichlet	dirichlet	PROPN
ejde-320	4	13	problem	problem	NOUN
ejde-320	4	14	for	for	ADP
ejde-320	4	15	infinitely	infinitely	ADV
ejde-320	4	16	degenerate	degenerate	ADJ
ejde-320	4	17	elliptic	elliptic	ADJ
ejde-320	4	18	operators	operator	NOUN
ejde-320	4	19	usman	usman	PROPN
ejde-320	4	20	hafeez	hafeez	PROPN
ejde-320	4	21	,	,	PUNCT
ejde-320	4	22	théo	théo	PROPN
ejde-320	4	23	lavier	lavier	NOUN
ejde-320	4	24	,	,	PUNCT
ejde-320	4	25	lucas	lucas	PROPN
ejde-320	4	26	williams	williams	PROPN
ejde-320	4	27	,	,	PUNCT
ejde-320	4	28	lyudmila	lyudmila	NOUN
ejde-320	4	29	korobenko	korobenko	ADJ
ejde-320	4	30	abstract	abstract	ADJ
ejde-320	4	31	.	.	PUNCT
ejde-320	5	1	we	we	PRON
ejde-320	5	2	investigate	investigate	VERB
ejde-320	5	3	a	a	DET
ejde-320	5	4	connection	connection	NOUN
ejde-320	5	5	between	between	ADP
ejde-320	5	6	solvability	solvability	NOUN
ejde-320	5	7	of	of	ADP
ejde-320	5	8	the	the	DET
ejde-320	5	9	dirichlet	dirichlet	PROPN
ejde-320	5	10	problem	problem	NOUN
ejde-320	5	11	for	for	ADP
ejde-320	5	12	an	an	DET
ejde-320	5	13	infinitely	infinitely	ADV
ejde-320	5	14	degenerate	degenerate	ADJ
ejde-320	5	15	elliptic	elliptic	ADJ
ejde-320	5	16	operator	operator	NOUN
ejde-320	5	17	and	and	CCONJ
ejde-320	5	18	the	the	DET
ejde-320	5	19	validity	validity	NOUN
ejde-320	5	20	of	of	ADP
ejde-320	5	21	an	an	DET
ejde-320	5	22	orlicz	orlicz	ADJ
ejde-320	5	23	-	-	PUNCT
ejde-320	5	24	sobolev	sobolev	NOUN
ejde-320	5	25	inequality	inequality	NOUN
ejde-320	5	26	in	in	ADP
ejde-320	5	27	the	the	DET
ejde-320	5	28	associated	associated	ADJ
ejde-320	5	29	subunit	subunit	NOUN
ejde-320	5	30	metric	metric	ADJ
ejde-320	5	31	space	space	NOUN
ejde-320	5	32	.	.	PUNCT
ejde-320	6	1	for	for	ADP
ejde-320	6	2	subelliptic	subelliptic	ADJ
ejde-320	6	3	operators	operator	NOUN
ejde-320	6	4	it	it	PRON
ejde-320	6	5	is	be	AUX
ejde-320	6	6	known	know	VERB
ejde-320	6	7	that	that	SCONJ
ejde-320	6	8	the	the	DET
ejde-320	6	9	classical	classical	ADJ
ejde-320	6	10	sobolev	sobolev	NOUN
ejde-320	6	11	inequality	inequality	NOUN
ejde-320	6	12	is	be	AUX
ejde-320	6	13	sufficient	sufficient	ADJ
ejde-320	6	14	and	and	CCONJ
ejde-320	6	15	almost	almost	ADV
ejde-320	6	16	necessary	necessary	ADJ
ejde-320	6	17	for	for	SCONJ
ejde-320	6	18	the	the	DET
ejde-320	6	19	dirichlet	dirichlet	PROPN
ejde-320	6	20	problem	problem	NOUN
ejde-320	6	21	to	to	PART
ejde-320	6	22	be	be	AUX
ejde-320	6	23	solvable	solvable	ADJ
ejde-320	6	24	with	with	ADP
ejde-320	6	25	a	a	DET
ejde-320	6	26	quantitative	quantitative	NOUN
ejde-320	6	27	bound	bind	VERB
ejde-320	6	28	on	on	ADP
ejde-320	6	29	the	the	DET
ejde-320	6	30	solution	solution	NOUN
ejde-320	6	31	[	[	X
ejde-320	6	32	11	11	NUM
ejde-320	6	33	]	]	PUNCT
ejde-320	6	34	.	.	PUNCT
ejde-320	7	1	when	when	SCONJ
ejde-320	7	2	the	the	DET
ejde-320	7	3	degeneracy	degeneracy	NOUN
ejde-320	7	4	is	be	AUX
ejde-320	7	5	of	of	ADP
ejde-320	7	6	infinite	infinite	ADJ
ejde-320	7	7	type	type	NOUN
ejde-320	7	8	,	,	PUNCT
ejde-320	7	9	a	a	DET
ejde-320	7	10	weaker	weak	ADJ
ejde-320	7	11	orlicz	orlicz	NOUN
ejde-320	7	12	-	-	PUNCT
ejde-320	7	13	sobolev	sobolev	NOUN
ejde-320	7	14	inequality	inequality	NOUN
ejde-320	7	15	seems	seem	VERB
ejde-320	7	16	to	to	PART
ejde-320	7	17	be	be	AUX
ejde-320	7	18	the	the	DET
ejde-320	7	19	right	right	ADJ
ejde-320	7	20	substitute	substitute	NOUN
ejde-320	8	1	[	[	X
ejde-320	8	2	7	7	NUM
ejde-320	8	3	]	]	PUNCT
ejde-320	8	4	.	.	PUNCT
ejde-320	9	1	in	in	ADP
ejde-320	9	2	this	this	DET
ejde-320	9	3	paper	paper	NOUN
ejde-320	9	4	we	we	PRON
ejde-320	9	5	investigate	investigate	VERB
ejde-320	9	6	this	this	DET
ejde-320	9	7	connection	connection	NOUN
ejde-320	9	8	further	far	ADV
ejde-320	9	9	and	and	CCONJ
ejde-320	9	10	reduce	reduce	VERB
ejde-320	9	11	the	the	DET
ejde-320	9	12	gap	gap	NOUN
ejde-320	9	13	between	between	ADP
ejde-320	9	14	necessary	necessary	ADJ
ejde-320	9	15	and	and	CCONJ
ejde-320	9	16	sufficient	sufficient	ADJ
ejde-320	9	17	conditions	condition	NOUN
ejde-320	9	18	for	for	ADP
ejde-320	9	19	solvability	solvability	NOUN
ejde-320	9	20	of	of	ADP
ejde-320	9	21	the	the	DET
ejde-320	9	22	dirichlet	dirichlet	PROPN
ejde-320	9	23	problem	problem	NOUN
ejde-320	9	24	.	.	PUNCT
ejde-320	10	1	1	1	X
ejde-320	10	2	.	.	X
ejde-320	10	3	introduction	introduction	NOUN
ejde-320	10	4	consider	consider	VERB
ejde-320	10	5	the	the	DET
ejde-320	10	6	dirichlet	dirichlet	PROPN
ejde-320	10	7	problem	problem	NOUN
ejde-320	10	8	with	with	ADP
ejde-320	10	9	a	a	DET
ejde-320	10	10	divergence	divergence	NOUN
ejde-320	10	11	form	form	NOUN
ejde-320	10	12	(	(	PUNCT
ejde-320	10	13	degenerate	degenerate	ADJ
ejde-320	10	14	)	)	PUNCT
ejde-320	10	15	elliptic	elliptic	ADJ
ejde-320	10	16	operator	operator	NOUN
ejde-320	10	17	∇	∇	X
ejde-320	10	18	·	·	PUNCT
ejde-320	10	19	a∇u	a∇u	X
ejde-320	10	20	=	=	SYM
ejde-320	10	21	f	f	PROPN
ejde-320	10	22	in	in	ADP
ejde-320	10	23	ω	ω	PROPN
ejde-320	10	24	,	,	PUNCT
ejde-320	10	25	u|∂ω	u|∂ω	PROPN
ejde-320	10	26	=	=	SYM
ejde-320	10	27	0	0	NUM
ejde-320	10	28	,	,	PUNCT
ejde-320	10	29	(	(	PUNCT
ejde-320	10	30	1.1	1.1	NUM
ejde-320	10	31	)	)	PUNCT
ejde-320	10	32	where	where	SCONJ
ejde-320	10	33	a	a	PRON
ejde-320	10	34	is	be	AUX
ejde-320	10	35	nonnegative	nonnegative	ADJ
ejde-320	10	36	semidefinite	semidefinite	NOUN
ejde-320	10	37	and	and	CCONJ
ejde-320	10	38	has	have	AUX
ejde-320	10	39	bounded	bound	VERB
ejde-320	10	40	measurable	measurable	ADJ
ejde-320	10	41	coefficients	coefficient	NOUN
ejde-320	10	42	,	,	PUNCT
ejde-320	10	43	and	and	CCONJ
ejde-320	10	44	ω	ω	PROPN
ejde-320	10	45	is	be	AUX
ejde-320	10	46	a	a	DET
ejde-320	10	47	bounded	bounded	ADJ
ejde-320	10	48	domain	domain	NOUN
ejde-320	10	49	in	in	ADP
ejde-320	10	50	rn	rn	PROPN
ejde-320	10	51	with	with	ADP
ejde-320	10	52	sufficiently	sufficiently	ADV
ejde-320	10	53	smooth	smooth	ADJ
ejde-320	10	54	boundary	boundary	NOUN
ejde-320	10	55	.	.	PUNCT
ejde-320	11	1	we	we	PRON
ejde-320	11	2	are	be	AUX
ejde-320	11	3	interested	interested	ADJ
ejde-320	11	4	in	in	ADP
ejde-320	11	5	establishing	establish	VERB
ejde-320	11	6	sharp	sharp	ADJ
ejde-320	11	7	conditions	condition	NOUN
ejde-320	11	8	on	on	ADP
ejde-320	11	9	the	the	DET
ejde-320	11	10	matrix	matrix	NOUN
ejde-320	11	11	a	a	DET
ejde-320	11	12	that	that	PRON
ejde-320	11	13	guarantee	guarantee	NOUN
ejde-320	11	14	existence	existence	NOUN
ejde-320	11	15	of	of	ADP
ejde-320	11	16	bounded	bound	VERB
ejde-320	11	17	weak	weak	ADJ
ejde-320	11	18	solutions	solution	NOUN
ejde-320	11	19	.	.	PUNCT
ejde-320	12	1	more	more	ADV
ejde-320	12	2	precisely	precisely	ADV
ejde-320	12	3	,	,	PUNCT
ejde-320	12	4	we	we	PRON
ejde-320	12	5	are	be	AUX
ejde-320	12	6	looking	look	VERB
ejde-320	12	7	for	for	ADP
ejde-320	12	8	a	a	DET
ejde-320	12	9	function	function	NOUN
ejde-320	12	10	u	u	NOUN
ejde-320	12	11	from	from	ADP
ejde-320	12	12	the	the	DET
ejde-320	12	13	degenerate	degenerate	ADJ
ejde-320	12	14	sobolev	sobolev	NOUN
ejde-320	12	15	space	space	NOUN
ejde-320	12	16	(	(	PUNCT
ejde-320	12	17	w	w	PROPN
ejde-320	12	18	1,2	1,2	NUM
ejde-320	12	19	a	a	PRON
ejde-320	12	20	)	)	PUNCT
ejde-320	12	21	0	0	NUM
ejde-320	13	1	(	(	PUNCT
ejde-320	13	2	ω	ω	NOUN
ejde-320	13	3	)	)	PUNCT
ejde-320	13	4	satisfying∫	satisfying∫	NOUN
ejde-320	13	5	∇u	∇u	PROPN
ejde-320	13	6	·	·	PUNCT
ejde-320	13	7	a∇ϕ	a∇ϕ	PROPN
ejde-320	13	8	=	=	SYM
ejde-320	14	1	−	−	PROPN
ejde-320	14	2	∫	∫	PROPN
ejde-320	14	3	fϕ	fϕ	X
ejde-320	14	4	for	for	ADP
ejde-320	14	5	every	every	DET
ejde-320	14	6	test	test	NOUN
ejde-320	14	7	function	function	NOUN
ejde-320	14	8	ϕ	ϕ	PROPN
ejde-320	14	9	∈	∈	PROPN
ejde-320	14	10	c1	c1	PROPN
ejde-320	14	11	0	0	NUM
ejde-320	15	1	(	(	PUNCT
ejde-320	15	2	ω	ω	NOUN
ejde-320	15	3	)	)	PUNCT
ejde-320	15	4	(	(	PUNCT
ejde-320	15	5	in	in	ADP
ejde-320	15	6	which	which	DET
ejde-320	15	7	case	case	NOUN
ejde-320	15	8	we	we	PRON
ejde-320	15	9	say	say	VERB
ejde-320	15	10	that	that	SCONJ
ejde-320	15	11	u	u	PROPN
ejde-320	15	12	is	be	AUX
ejde-320	15	13	a	a	DET
ejde-320	15	14	weak	weak	ADJ
ejde-320	15	15	solution	solution	NOUN
ejde-320	15	16	of	of	ADP
ejde-320	15	17	(	(	PUNCT
ejde-320	15	18	1.1	1.1	NUM
ejde-320	15	19	)	)	PUNCT
ejde-320	15	20	)	)	PUNCT
ejde-320	15	21	,	,	PUNCT
ejde-320	15	22	as	as	ADV
ejde-320	15	23	well	well	ADV
ejde-320	15	24	as	as	ADP
ejde-320	15	25	the	the	DET
ejde-320	15	26	qualitative	qualitative	ADJ
ejde-320	15	27	estimate	estimate	NOUN
ejde-320	15	28	‖u‖l∞(ω	‖u‖l∞(ω	NOUN
ejde-320	15	29	)	)	PUNCT
ejde-320	15	30	≤	≤	NOUN
ejde-320	15	31	c‖f‖x	c‖f‖x	NOUN
ejde-320	15	32	for	for	ADP
ejde-320	15	33	some	some	DET
ejde-320	15	34	appropriate	appropriate	ADJ
ejde-320	15	35	normed	normed	ADJ
ejde-320	15	36	space	space	NOUN
ejde-320	15	37	x.	x.	NOUN
ejde-320	16	1	the	the	DET
ejde-320	16	2	case	case	NOUN
ejde-320	16	3	when	when	SCONJ
ejde-320	16	4	a	a	PRON
ejde-320	16	5	is	be	AUX
ejde-320	16	6	uniformly	uniformly	ADV
ejde-320	16	7	elliptic	elliptic	ADJ
ejde-320	16	8	has	have	AUX
ejde-320	16	9	been	be	AUX
ejde-320	16	10	completely	completely	ADV
ejde-320	16	11	settled	settle	VERB
ejde-320	16	12	by	by	ADP
ejde-320	16	13	nash	nash	NOUN
ejde-320	16	14	[	[	X
ejde-320	16	15	10	10	NUM
ejde-320	16	16	]	]	PUNCT
ejde-320	16	17	,	,	PUNCT
ejde-320	16	18	moser	moser	PROPN
ejde-320	17	1	[	[	X
ejde-320	17	2	9	9	NUM
ejde-320	17	3	]	]	PUNCT
ejde-320	17	4	,	,	PUNCT
ejde-320	17	5	and	and	CCONJ
ejde-320	17	6	degiorgi	degiorgi	NOUN
ejde-320	18	1	[	[	X
ejde-320	18	2	1	1	NUM
ejde-320	18	3	]	]	PUNCT
ejde-320	18	4	,	,	PUNCT
ejde-320	18	5	and	and	CCONJ
ejde-320	18	6	is	be	AUX
ejde-320	18	7	now	now	ADV
ejde-320	18	8	considered	consider	VERB
ejde-320	18	9	a	a	DET
ejde-320	18	10	classical	classical	ADJ
ejde-320	18	11	theory	theory	NOUN
ejde-320	18	12	[	[	X
ejde-320	18	13	4	4	NUM
ejde-320	18	14	]	]	PUNCT
ejde-320	18	15	.	.	PUNCT
ejde-320	19	1	when	when	SCONJ
ejde-320	19	2	the	the	DET
ejde-320	19	3	eigenvalues	eigenvalue	NOUN
ejde-320	19	4	of	of	ADP
ejde-320	19	5	the	the	DET
ejde-320	19	6	matrix	matrix	NOUN
ejde-320	19	7	a	a	PRON
ejde-320	19	8	are	be	AUX
ejde-320	19	9	allowed	allow	VERB
ejde-320	19	10	to	to	PART
ejde-320	19	11	vanish	vanish	VERB
ejde-320	19	12	,	,	PUNCT
ejde-320	19	13	i.e.	i.e.	X
ejde-320	19	14	the	the	DET
ejde-320	19	15	operator	operator	NOUN
ejde-320	19	16	is	be	AUX
ejde-320	19	17	degenerate	degenerate	ADJ
ejde-320	19	18	elliptic	elliptic	ADJ
ejde-320	19	19	,	,	PUNCT
ejde-320	19	20	the	the	DET
ejde-320	19	21	theory	theory	NOUN
ejde-320	19	22	is	be	AUX
ejde-320	19	23	far	far	ADV
ejde-320	19	24	from	from	ADP
ejde-320	19	25	complete	complete	ADJ
ejde-320	19	26	.	.	PUNCT
ejde-320	20	1	there	there	PRON
ejde-320	20	2	are	be	VERB
ejde-320	20	3	generally	generally	ADV
ejde-320	20	4	two	two	NUM
ejde-320	20	5	cases	case	NOUN
ejde-320	20	6	considered	consider	VERB
ejde-320	20	7	in	in	ADP
ejde-320	20	8	the	the	DET
ejde-320	20	9	literature	literature	NOUN
ejde-320	20	10	:	:	PUNCT
ejde-320	20	11	finite	finite	PROPN
ejde-320	20	12	vanishing	vanish	VERB
ejde-320	20	13	with	with	ADP
ejde-320	20	14	2010	2010	NUM
ejde-320	20	15	mathematics	mathematic	NOUN
ejde-320	20	16	subject	subject	NOUN
ejde-320	20	17	classification	classification	NOUN
ejde-320	20	18	.	.	PUNCT
ejde-320	21	1	35a01	35a01	NOUN
ejde-320	21	2	,	,	PUNCT
ejde-320	21	3	35b65	35b65	NUM
ejde-320	21	4	,	,	PUNCT
ejde-320	21	5	35d30	35d30	NUM
ejde-320	21	6	,	,	PUNCT
ejde-320	21	7	35h99	35h99	NUM
ejde-320	21	8	,	,	PUNCT
ejde-320	21	9	35j25	35j25	NUM
ejde-320	21	10	,	,	PUNCT
ejde-320	21	11	46e36	46e36	NUM
ejde-320	21	12	.	.	PUNCT
ejde-320	22	1	key	key	ADJ
ejde-320	22	2	words	word	NOUN
ejde-320	22	3	and	and	CCONJ
ejde-320	22	4	phrases	phrase	NOUN
ejde-320	22	5	.	.	PUNCT
ejde-320	23	1	elliptic	elliptic	ADJ
ejde-320	23	2	equations	equation	NOUN
ejde-320	23	3	;	;	PUNCT
ejde-320	23	4	infinite	infinite	ADJ
ejde-320	23	5	degeneracy	degeneracy	NOUN
ejde-320	23	6	;	;	PUNCT
ejde-320	23	7	rough	rough	ADJ
ejde-320	23	8	coefficients	coefficient	NOUN
ejde-320	23	9	;	;	PUNCT
ejde-320	23	10	dirichlet	dirichlet	PROPN
ejde-320	23	11	problem	problem	NOUN
ejde-320	23	12	;	;	PUNCT
ejde-320	23	13	solvability	solvability	NOUN
ejde-320	23	14	;	;	PUNCT
ejde-320	23	15	global	global	ADJ
ejde-320	23	16	boundedness	boundedness	NOUN
ejde-320	23	17	;	;	PUNCT
ejde-320	23	18	orlicz	orlicz	PROPN
ejde-320	23	19	-	-	PUNCT
ejde-320	23	20	sobolev	sobolev	NOUN
ejde-320	23	21	inequality	inequality	NOUN
ejde-320	23	22	.	.	PUNCT
ejde-320	24	1	c	c	X
ejde-320	24	2	©	©	PROPN
ejde-320	24	3	2021	2021	NUM
ejde-320	24	4	.	.	PUNCT
ejde-320	25	1	this	this	DET
ejde-320	25	2	work	work	NOUN
ejde-320	25	3	is	be	AUX
ejde-320	25	4	licensed	license	VERB
ejde-320	25	5	under	under	ADP
ejde-320	25	6	a	a	DET
ejde-320	25	7	cc	cc	NOUN
ejde-320	25	8	by	by	ADP
ejde-320	25	9	4.0	4.0	NUM
ejde-320	25	10	license	license	NOUN
ejde-320	25	11	.	.	PUNCT
ejde-320	26	1	submitted	submit	VERB
ejde-320	26	2	july	july	PROPN
ejde-320	26	3	30	30	NUM
ejde-320	26	4	,	,	PUNCT
ejde-320	26	5	2021	2021	NUM
ejde-320	26	6	.	.	PUNCT
ejde-320	27	1	published	publish	VERB
ejde-320	27	2	september	september	PROPN
ejde-320	27	3	23	23	NUM
ejde-320	27	4	,	,	PUNCT
ejde-320	27	5	2021	2021	NUM
ejde-320	27	6	.	.	PUNCT
ejde-320	28	1	1	1	NUM
ejde-320	28	2	2	2	NUM
ejde-320	28	3	u.	u.	PROPN
ejde-320	28	4	hafeez	hafeez	PROPN
ejde-320	28	5	,	,	PUNCT
ejde-320	28	6	t.	t.	PROPN
ejde-320	28	7	lavier	lavier	PROPN
ejde-320	28	8	,	,	PUNCT
ejde-320	28	9	l.	l.	PROPN
ejde-320	28	10	williams	williams	PROPN
ejde-320	28	11	,	,	PUNCT
ejde-320	28	12	l.	l.	PROPN
ejde-320	28	13	korobenko	korobenko	PROPN
ejde-320	28	14	ejde-2021/82	ejde-2021/82	ADJ
ejde-320	28	15	rough	rough	ADJ
ejde-320	28	16	coefficients	coefficient	NOUN
ejde-320	28	17	,	,	PUNCT
ejde-320	28	18	and	and	CCONJ
ejde-320	28	19	infinite	infinite	VERB
ejde-320	28	20	vanishing	vanish	VERB
ejde-320	28	21	with	with	ADP
ejde-320	28	22	smooth	smooth	ADJ
ejde-320	28	23	coefficients	coefficient	NOUN
ejde-320	28	24	.	.	PUNCT
ejde-320	29	1	in	in	ADP
ejde-320	29	2	the	the	DET
ejde-320	29	3	case	case	NOUN
ejde-320	29	4	of	of	ADP
ejde-320	29	5	finite	finite	ADJ
ejde-320	29	6	vanishing	vanishing	NOUN
ejde-320	29	7	,	,	PUNCT
ejde-320	29	8	the	the	DET
ejde-320	29	9	first	first	ADJ
ejde-320	29	10	generalizations	generalization	NOUN
ejde-320	29	11	of	of	ADP
ejde-320	29	12	the	the	DET
ejde-320	29	13	moser	moser	PROPN
ejde-320	29	14	-	-	PUNCT
ejde-320	29	15	degiorgi	degiorgi	PROPN
ejde-320	29	16	theory	theory	NOUN
ejde-320	29	17	are	be	AUX
ejde-320	29	18	due	due	ADJ
ejde-320	29	19	to	to	ADP
ejde-320	29	20	fabes	fabe	NOUN
ejde-320	29	21	,	,	PUNCT
ejde-320	29	22	kenig	kenig	ADJ
ejde-320	29	23	,	,	PUNCT
ejde-320	29	24	and	and	CCONJ
ejde-320	29	25	serapioni	serapioni	VERB
ejde-320	29	26	[	[	X
ejde-320	29	27	2	2	NUM
ejde-320	29	28	]	]	PUNCT
ejde-320	29	29	,	,	PUNCT
ejde-320	29	30	and	and	CCONJ
ejde-320	29	31	franchi	franchi	PROPN
ejde-320	29	32	and	and	CCONJ
ejde-320	29	33	lanconelli	lanconelli	NOUN
ejde-320	30	1	[	[	X
ejde-320	30	2	3	3	NUM
ejde-320	30	3	]	]	PUNCT
ejde-320	30	4	.	.	PUNCT
ejde-320	31	1	the	the	DET
ejde-320	31	2	latter	latter	ADJ
ejde-320	31	3	deals	deal	NOUN
ejde-320	31	4	with	with	ADP
ejde-320	31	5	the	the	DET
ejde-320	31	6	case	case	NOUN
ejde-320	31	7	when	when	SCONJ
ejde-320	31	8	one	one	NUM
ejde-320	31	9	of	of	ADP
ejde-320	31	10	the	the	DET
ejde-320	31	11	eigenvalues	eigenvalue	NOUN
ejde-320	31	12	of	of	ADP
ejde-320	31	13	a	a	PRON
ejde-320	31	14	is	be	AUX
ejde-320	31	15	constant	constant	ADJ
ejde-320	31	16	,	,	PUNCT
ejde-320	31	17	while	while	SCONJ
ejde-320	31	18	others	other	NOUN
ejde-320	31	19	may	may	AUX
ejde-320	31	20	vanish	vanish	VERB
ejde-320	31	21	to	to	ADP
ejde-320	31	22	finite	finite	ADJ
ejde-320	31	23	order	order	NOUN
ejde-320	31	24	.	.	PUNCT
ejde-320	32	1	franchi	franchi	PROPN
ejde-320	32	2	and	and	CCONJ
ejde-320	32	3	lanconelli	lanconelli	PROPN
ejde-320	32	4	’s	’s	PART
ejde-320	32	5	big	big	ADJ
ejde-320	32	6	idea	idea	NOUN
ejde-320	32	7	was	be	AUX
ejde-320	32	8	to	to	PART
ejde-320	32	9	use	use	VERB
ejde-320	32	10	the	the	DET
ejde-320	32	11	subunit	subunit	NOUN
ejde-320	32	12	metric	metric	ADJ
ejde-320	32	13	space	space	NOUN
ejde-320	32	14	associated	associate	VERB
ejde-320	32	15	with	with	ADP
ejde-320	32	16	the	the	DET
ejde-320	32	17	operator	operator	NOUN
ejde-320	32	18	,	,	PUNCT
ejde-320	32	19	and	and	CCONJ
ejde-320	32	20	adapt	adapt	VERB
ejde-320	32	21	the	the	DET
ejde-320	32	22	classical	classical	ADJ
ejde-320	32	23	moser	moser	PROPN
ejde-320	32	24	iteration	iteration	NOUN
ejde-320	32	25	to	to	ADP
ejde-320	32	26	that	that	DET
ejde-320	32	27	setting	setting	NOUN
ejde-320	32	28	.	.	PUNCT
ejde-320	33	1	using	use	VERB
ejde-320	33	2	this	this	DET
ejde-320	33	3	approach	approach	NOUN
ejde-320	33	4	,	,	PUNCT
ejde-320	33	5	sawyer	sawyer	NOUN
ejde-320	33	6	and	and	CCONJ
ejde-320	33	7	wheeden	wheeden	ADJ
ejde-320	33	8	[	[	X
ejde-320	33	9	11	11	NUM
ejde-320	33	10	]	]	PUNCT
ejde-320	33	11	built	build	VERB
ejde-320	33	12	on	on	ADP
ejde-320	33	13	the	the	DET
ejde-320	33	14	work	work	NOUN
ejde-320	33	15	of	of	ADP
ejde-320	33	16	franchi	franchi	PROPN
ejde-320	33	17	and	and	CCONJ
ejde-320	33	18	lanconelli	lanconelli	PROPN
ejde-320	33	19	,	,	PUNCT
ejde-320	33	20	among	among	ADP
ejde-320	33	21	others	other	NOUN
ejde-320	33	22	,	,	PUNCT
ejde-320	33	23	to	to	PART
ejde-320	33	24	further	far	ADV
ejde-320	33	25	investigate	investigate	VERB
ejde-320	33	26	regularity	regularity	NOUN
ejde-320	33	27	questions	question	NOUN
ejde-320	33	28	for	for	ADP
ejde-320	33	29	subelliptic	subelliptic	ADJ
ejde-320	33	30	operators	operator	NOUN
ejde-320	33	31	with	with	ADP
ejde-320	33	32	rough	rough	ADJ
ejde-320	33	33	coefficients	coefficient	NOUN
ejde-320	33	34	.	.	PUNCT
ejde-320	34	1	in	in	ADP
ejde-320	34	2	particular	particular	ADJ
ejde-320	34	3	,	,	PUNCT
ejde-320	34	4	they	they	PRON
ejde-320	34	5	showed	show	VERB
ejde-320	34	6	that	that	SCONJ
ejde-320	34	7	the	the	DET
ejde-320	34	8	(	(	PUNCT
ejde-320	34	9	2σ	2σ	NUM
ejde-320	34	10	,	,	PUNCT
ejde-320	34	11	2	2	NUM
ejde-320	34	12	)	)	PUNCT
ejde-320	34	13	weak	weak	ADJ
ejde-320	34	14	sobolev	sobolev	NOUN
ejde-320	34	15	inequality	inequality	NOUN
ejde-320	34	16	with	with	ADP
ejde-320	34	17	σ	σ	PROPN
ejde-320	34	18	>	>	X
ejde-320	34	19	1	1	NUM
ejde-320	34	20	in	in	ADP
ejde-320	34	21	the	the	DET
ejde-320	34	22	subunit	subunit	NOUN
ejde-320	34	23	metric	metric	ADJ
ejde-320	34	24	space	space	NOUN
ejde-320	34	25	(	(	PUNCT
ejde-320	34	26	1	1	NUM
ejde-320	34	27	|b|	|b|	PROPN
ejde-320	34	28	∫	∫	PROPN
ejde-320	34	29	b	b	PROPN
ejde-320	34	30	|w|2σ	|w|2σ	NOUN
ejde-320	34	31	)	)	PUNCT
ejde-320	34	32	1	1	NUM
ejde-320	34	33	2σ	2σ	NOUN
ejde-320	34	34	≤	≤	NUM
ejde-320	34	35	cr	cr	NOUN
ejde-320	34	36	(	(	PUNCT
ejde-320	34	37	1	1	NUM
ejde-320	34	38	|b|	|b|	PROPN
ejde-320	34	39	∫	∫	PROPN
ejde-320	34	40	b	b	X
ejde-320	34	41	|∇aw|2	|∇aw|2	NOUN
ejde-320	34	42	)	)	PUNCT
ejde-320	34	43	1/2	1/2	NUM
ejde-320	34	44	+	+	NUM
ejde-320	34	45	c	c	NOUN
ejde-320	34	46	(	(	PUNCT
ejde-320	34	47	1	1	NUM
ejde-320	34	48	|b|	|b|	PROPN
ejde-320	34	49	∫	∫	PROPN
ejde-320	34	50	b	b	PROPN
ejde-320	34	51	|w|2	|w|2	PROPN
ejde-320	34	52	)	)	PUNCT
ejde-320	34	53	1/2	1/2	NUM
ejde-320	34	54	(	(	PUNCT
ejde-320	34	55	1.2	1.2	NUM
ejde-320	34	56	)	)	PUNCT
ejde-320	34	57	for	for	ADP
ejde-320	34	58	all	all	DET
ejde-320	34	59	w	w	NOUN
ejde-320	34	60	∈w	∈w	PROPN
ejde-320	34	61	1,2	1,2	NUM
ejde-320	34	62	0	0	NUM
ejde-320	34	63	(	(	PUNCT
ejde-320	34	64	b	b	NOUN
ejde-320	34	65	)	)	PUNCT
ejde-320	34	66	,	,	PUNCT
ejde-320	34	67	is	be	AUX
ejde-320	34	68	sufficient	sufficient	ADJ
ejde-320	34	69	for	for	ADP
ejde-320	34	70	solvability	solvability	NOUN
ejde-320	34	71	of	of	ADP
ejde-320	34	72	the	the	DET
ejde-320	34	73	dirichlet	dirichlet	PROPN
ejde-320	34	74	problem	problem	NOUN
ejde-320	34	75	(	(	PUNCT
ejde-320	34	76	1.1	1.1	NUM
ejde-320	34	77	)	)	PUNCT
ejde-320	34	78	when	when	SCONJ
ejde-320	34	79	ω	ω	PROPN
ejde-320	34	80	=	=	SYM
ejde-320	34	81	b	b	PROPN
ejde-320	34	82	,	,	PUNCT
ejde-320	34	83	a	a	DET
ejde-320	34	84	subunit	subunit	NOUN
ejde-320	34	85	metric	metric	PROPN
ejde-320	34	86	ball	ball	PROPN
ejde-320	34	87	,	,	PUNCT
ejde-320	34	88	with	with	ADP
ejde-320	34	89	the	the	DET
ejde-320	34	90	quantitative	quantitative	ADJ
ejde-320	34	91	estimate	estimate	NOUN
ejde-320	34	92	‖u‖l∞(b	‖u‖l∞(b	NOUN
ejde-320	34	93	)	)	PUNCT
ejde-320	34	94	≤	≤	NOUN
ejde-320	34	95	c‖f‖lq(b	c‖f‖lq(b	ADJ
ejde-320	34	96	)	)	PUNCT
ejde-320	34	97	,	,	PUNCT
ejde-320	34	98	where	where	SCONJ
ejde-320	34	99	q	q	X
ejde-320	34	100	>	>	X
ejde-320	34	101	σ′	σ′	PROPN
ejde-320	34	102	,	,	PUNCT
ejde-320	34	103	and	and	CCONJ
ejde-320	34	104	σ′	σ′	PROPN
ejde-320	34	105	is	be	AUX
ejde-320	34	106	the	the	DET
ejde-320	34	107	dual	dual	ADJ
ejde-320	34	108	of	of	ADP
ejde-320	34	109	σ	σ	PROPN
ejde-320	34	110	.	.	PUNCT
ejde-320	35	1	moreover	moreover	ADV
ejde-320	35	2	,	,	PUNCT
ejde-320	35	3	if	if	SCONJ
ejde-320	35	4	the	the	DET
ejde-320	35	5	above	above	ADJ
ejde-320	35	6	estimate	estimate	NOUN
ejde-320	35	7	holds	hold	VERB
ejde-320	35	8	for	for	ADP
ejde-320	35	9	q	q	NOUN
ejde-320	35	10	=	=	PUNCT
ejde-320	35	11	σ′	σ′	PROPN
ejde-320	35	12	then	then	ADV
ejde-320	35	13	sobolev	sobolev	PROPN
ejde-320	35	14	inequality	inequality	NOUN
ejde-320	35	15	(	(	PUNCT
ejde-320	35	16	1.2	1.2	NUM
ejde-320	35	17	)	)	PUNCT
ejde-320	35	18	holds	hold	VERB
ejde-320	35	19	(	(	PUNCT
ejde-320	35	20	almost	almost	ADV
ejde-320	35	21	necessity	necessity	NOUN
ejde-320	35	22	)	)	PUNCT
ejde-320	35	23	.	.	PUNCT
ejde-320	36	1	in	in	ADP
ejde-320	36	2	this	this	DET
ejde-320	36	3	paper	paper	NOUN
ejde-320	36	4	we	we	PRON
ejde-320	36	5	investigate	investigate	VERB
ejde-320	36	6	the	the	DET
ejde-320	36	7	same	same	ADJ
ejde-320	36	8	question	question	NOUN
ejde-320	36	9	for	for	ADP
ejde-320	36	10	the	the	DET
ejde-320	36	11	case	case	NOUN
ejde-320	36	12	of	of	ADP
ejde-320	36	13	the	the	DET
ejde-320	36	14	infinitely	infinitely	ADV
ejde-320	36	15	degenerate	degenerate	ADJ
ejde-320	36	16	operator	operator	NOUN
ejde-320	36	17	l	l	NOUN
ejde-320	36	18	=	=	X
ejde-320	36	19	∇	∇	X
ejde-320	36	20	·	·	PUNCT
ejde-320	36	21	a∇.	a∇.	X
ejde-320	37	1	more	more	ADV
ejde-320	37	2	precisely	precisely	ADV
ejde-320	37	3	,	,	PUNCT
ejde-320	37	4	we	we	PRON
ejde-320	37	5	make	make	VERB
ejde-320	37	6	use	use	NOUN
ejde-320	37	7	of	of	ADP
ejde-320	37	8	an	an	DET
ejde-320	37	9	analogue	analogue	NOUN
ejde-320	37	10	of	of	ADP
ejde-320	37	11	(	(	PUNCT
ejde-320	37	12	1.2	1.2	NUM
ejde-320	37	13	)	)	PUNCT
ejde-320	37	14	,	,	PUNCT
ejde-320	37	15	considering	consider	VERB
ejde-320	37	16	the	the	DET
ejde-320	37	17	more	more	ADV
ejde-320	37	18	general	general	ADJ
ejde-320	37	19	orlicz	orlicz	NOUN
ejde-320	37	20	spaces	space	NOUN
ejde-320	37	21	,	,	PUNCT
ejde-320	37	22	lφ	lφ	ADV
ejde-320	37	23	,	,	PUNCT
ejde-320	37	24	instead	instead	ADV
ejde-320	37	25	of	of	ADP
ejde-320	37	26	the	the	DET
ejde-320	37	27	traditional	traditional	ADJ
ejde-320	37	28	lebesgue	lebesgue	NOUN
ejde-320	37	29	spaces	space	NOUN
ejde-320	37	30	.	.	PUNCT
ejde-320	38	1	by	by	ADP
ejde-320	38	2	a	a	DET
ejde-320	38	3	(	(	PUNCT
ejde-320	38	4	φ	φ	PROPN
ejde-320	38	5	,	,	PUNCT
ejde-320	38	6	2	2	NUM
ejde-320	38	7	)	)	PUNCT
ejde-320	38	8	orlicz	orlicz	ADJ
ejde-320	38	9	-	-	PUNCT
ejde-320	38	10	sobolev	sobolev	NOUN
ejde-320	38	11	inequality	inequality	NOUN
ejde-320	38	12	we	we	PRON
ejde-320	38	13	mean	mean	VERB
ejde-320	38	14	‖w‖lφ(b	‖w‖lφ(b	VERB
ejde-320	38	15	,	,	PUNCT
ejde-320	38	16	dµ	dµ	ADJ
ejde-320	38	17	)	)	PUNCT
ejde-320	38	18	≤	≤	NUM
ejde-320	38	19	c(r	c(r	NOUN
ejde-320	38	20	)	)	PUNCT
ejde-320	38	21	(	(	PUNCT
ejde-320	38	22	∫	∫	PROPN
ejde-320	38	23	b	b	PROPN
ejde-320	39	1	|∇aw|2dµ	|∇aw|2dµ	PROPN
ejde-320	39	2	)	)	PUNCT
ejde-320	39	3	1/2	1/2	NUM
ejde-320	39	4	(	(	PUNCT
ejde-320	39	5	1.3	1.3	NUM
ejde-320	39	6	)	)	PUNCT
ejde-320	39	7	for	for	ADP
ejde-320	39	8	all	all	DET
ejde-320	39	9	w	w	PROPN
ejde-320	39	10	∈	∈	PROPN
ejde-320	39	11	(	(	PUNCT
ejde-320	39	12	w	w	PROPN
ejde-320	39	13	1,2	1,2	NUM
ejde-320	39	14	a	a	PRON
ejde-320	39	15	)	)	PUNCT
ejde-320	39	16	0	0	PUNCT
ejde-320	40	1	(	(	PUNCT
ejde-320	40	2	b	b	NOUN
ejde-320	40	3	)	)	PUNCT
ejde-320	40	4	and	and	CCONJ
ejde-320	40	5	some	some	DET
ejde-320	40	6	young	young	ADJ
ejde-320	40	7	function	function	NOUN
ejde-320	40	8	φ	φ	PROPN
ejde-320	40	9	(	(	PUNCT
ejde-320	40	10	typically	typically	ADV
ejde-320	40	11	satisfying	satisfy	VERB
ejde-320	40	12	φ(t	φ(t	PROPN
ejde-320	40	13	)	)	PUNCT
ejde-320	40	14	>	>	X
ejde-320	40	15	t2	t2	PROPN
ejde-320	40	16	for	for	ADP
ejde-320	40	17	all	all	DET
ejde-320	40	18	t	t	PROPN
ejde-320	40	19	>	>	X
ejde-320	40	20	1	1	NUM
ejde-320	40	21	)	)	PUNCT
ejde-320	40	22	,	,	PUNCT
ejde-320	40	23	see	see	VERB
ejde-320	40	24	section	section	NOUN
ejde-320	40	25	2	2	NUM
ejde-320	40	26	for	for	ADP
ejde-320	40	27	precise	precise	ADJ
ejde-320	40	28	definitions	definition	NOUN
ejde-320	40	29	,	,	PUNCT
ejde-320	40	30	and	and	CCONJ
ejde-320	40	31	the	the	DET
ejde-320	40	32	measure	measure	NOUN
ejde-320	40	33	dµ	dµ	VERB
ejde-320	40	34	=	=	PUNCT
ejde-320	40	35	dx/|b|	dx/|b|	PROPN
ejde-320	40	36	.	.	PUNCT
ejde-320	41	1	there	there	PRON
ejde-320	41	2	are	be	VERB
ejde-320	41	3	a	a	DET
ejde-320	41	4	few	few	ADJ
ejde-320	41	5	recent	recent	ADJ
ejde-320	41	6	results	result	NOUN
ejde-320	41	7	indicating	indicate	VERB
ejde-320	41	8	that	that	SCONJ
ejde-320	41	9	orlicz	orlicz	PROPN
ejde-320	41	10	-	-	PUNCT
ejde-320	41	11	sobolev	sobolev	NOUN
ejde-320	41	12	inequalities	inequality	NOUN
ejde-320	41	13	of	of	ADP
ejde-320	41	14	the	the	DET
ejde-320	41	15	type	type	NOUN
ejde-320	41	16	(	(	PUNCT
ejde-320	41	17	1.3	1.3	NUM
ejde-320	41	18	)	)	PUNCT
ejde-320	41	19	are	be	AUX
ejde-320	41	20	the	the	DET
ejde-320	41	21	correct	correct	ADJ
ejde-320	41	22	substitute	substitute	NOUN
ejde-320	41	23	for	for	ADP
ejde-320	41	24	(	(	PUNCT
ejde-320	41	25	1.2	1.2	NUM
ejde-320	41	26	)	)	PUNCT
ejde-320	41	27	when	when	SCONJ
ejde-320	41	28	the	the	DET
ejde-320	41	29	operator	operator	NOUN
ejde-320	41	30	is	be	AUX
ejde-320	41	31	infinitely	infinitely	ADV
ejde-320	41	32	degenerate	degenerate	ADJ
ejde-320	41	33	.	.	PUNCT
ejde-320	42	1	first	first	ADV
ejde-320	42	2	,	,	PUNCT
ejde-320	42	3	as	as	SCONJ
ejde-320	42	4	has	have	AUX
ejde-320	42	5	been	be	AUX
ejde-320	42	6	shown	show	VERB
ejde-320	42	7	in	in	ADP
ejde-320	42	8	[	[	X
ejde-320	42	9	5	5	NUM
ejde-320	42	10	]	]	PUNCT
ejde-320	42	11	,	,	PUNCT
ejde-320	42	12	a	a	DET
ejde-320	42	13	classical	classical	ADJ
ejde-320	42	14	weak	weak	ADJ
ejde-320	42	15	sobolev	sobolev	NOUN
ejde-320	42	16	inequality	inequality	NOUN
ejde-320	42	17	(	(	PUNCT
ejde-320	42	18	1.2	1.2	NUM
ejde-320	42	19	)	)	PUNCT
ejde-320	42	20	implies	imply	VERB
ejde-320	42	21	the	the	DET
ejde-320	42	22	doubling	double	VERB
ejde-320	42	23	property	property	NOUN
ejde-320	42	24	of	of	ADP
ejde-320	42	25	the	the	DET
ejde-320	42	26	underlying	underlying	ADJ
ejde-320	42	27	metric	metric	ADJ
ejde-320	42	28	measure	measure	NOUN
ejde-320	42	29	space	space	NOUN
ejde-320	42	30	,	,	PUNCT
ejde-320	42	31	and	and	CCONJ
ejde-320	42	32	hence	hence	ADV
ejde-320	42	33	the	the	DET
ejde-320	42	34	degeneracy	degeneracy	NOUN
ejde-320	42	35	must	must	AUX
ejde-320	42	36	be	be	AUX
ejde-320	42	37	of	of	ADP
ejde-320	42	38	finite	finite	ADJ
ejde-320	42	39	type	type	NOUN
ejde-320	42	40	.	.	PUNCT
ejde-320	43	1	on	on	ADP
ejde-320	43	2	the	the	DET
ejde-320	43	3	other	other	ADJ
ejde-320	43	4	hand	hand	NOUN
ejde-320	43	5	,	,	PUNCT
ejde-320	43	6	in	in	ADP
ejde-320	43	7	[	[	X
ejde-320	43	8	6	6	NUM
ejde-320	43	9	,	,	PUNCT
ejde-320	43	10	7	7	NUM
ejde-320	43	11	]	]	PUNCT
ejde-320	43	12	an	an	DET
ejde-320	43	13	abstract	abstract	ADJ
ejde-320	43	14	regularity	regularity	NOUN
ejde-320	43	15	theory	theory	NOUN
ejde-320	43	16	for	for	ADP
ejde-320	43	17	degenerate	degenerate	ADJ
ejde-320	43	18	operators	operator	NOUN
ejde-320	43	19	has	have	AUX
ejde-320	43	20	been	be	AUX
ejde-320	43	21	developed	develop	VERB
ejde-320	43	22	under	under	ADP
ejde-320	43	23	the	the	DET
ejde-320	43	24	assumption	assumption	NOUN
ejde-320	43	25	of	of	ADP
ejde-320	43	26	appropriate	appropriate	ADJ
ejde-320	43	27	orlicz	orlicz	ADJ
ejde-320	43	28	-	-	PUNCT
ejde-320	43	29	sobolev	sobolev	NOUN
ejde-320	43	30	inequalities	inequality	NOUN
ejde-320	43	31	(	(	PUNCT
ejde-320	43	32	stronger	strong	ADJ
ejde-320	43	33	versions	version	NOUN
ejde-320	43	34	of	of	ADP
ejde-320	43	35	(	(	PUNCT
ejde-320	43	36	1.3	1.3	NUM
ejde-320	43	37	)	)	PUNCT
ejde-320	43	38	)	)	PUNCT
ejde-320	43	39	.	.	PUNCT
ejde-320	44	1	moreover	moreover	ADV
ejde-320	44	2	,	,	PUNCT
ejde-320	44	3	for	for	ADP
ejde-320	44	4	particular	particular	ADJ
ejde-320	44	5	classes	class	NOUN
ejde-320	44	6	of	of	ADP
ejde-320	44	7	infinitely	infinitely	ADV
ejde-320	44	8	degenerate	degenerate	ADJ
ejde-320	44	9	operators	operator	NOUN
ejde-320	44	10	these	these	DET
ejde-320	44	11	inequalities	inequality	NOUN
ejde-320	44	12	were	be	AUX
ejde-320	44	13	proved	prove	VERB
ejde-320	44	14	to	to	PART
ejde-320	44	15	hold	hold	VERB
ejde-320	44	16	in	in	ADP
ejde-320	44	17	the	the	DET
ejde-320	44	18	degenerate	degenerate	ADJ
ejde-320	44	19	sobolev	sobolev	NOUN
ejde-320	44	20	spaces	space	NOUN
ejde-320	44	21	associated	associate	VERB
ejde-320	44	22	with	with	ADP
ejde-320	44	23	the	the	DET
ejde-320	44	24	operator	operator	NOUN
ejde-320	44	25	.	.	PUNCT
ejde-320	45	1	in	in	ADP
ejde-320	45	2	this	this	DET
ejde-320	45	3	paper	paper	NOUN
ejde-320	45	4	we	we	PRON
ejde-320	45	5	investigate	investigate	VERB
ejde-320	45	6	the	the	DET
ejde-320	45	7	connection	connection	NOUN
ejde-320	45	8	between	between	ADP
ejde-320	45	9	the	the	DET
ejde-320	45	10	dirichlet	dirichlet	PROPN
ejde-320	45	11	problem	problem	NOUN
ejde-320	45	12	(	(	PUNCT
ejde-320	45	13	1.1	1.1	NUM
ejde-320	45	14	)	)	PUNCT
ejde-320	45	15	and	and	CCONJ
ejde-320	45	16	the	the	DET
ejde-320	45	17	validity	validity	NOUN
ejde-320	45	18	of	of	ADP
ejde-320	45	19	(	(	PUNCT
ejde-320	45	20	1.3	1.3	NUM
ejde-320	45	21	)	)	PUNCT
ejde-320	45	22	.	.	PUNCT
ejde-320	46	1	in	in	ADP
ejde-320	46	2	particular	particular	ADJ
ejde-320	46	3	,	,	PUNCT
ejde-320	46	4	we	we	PRON
ejde-320	46	5	prove	prove	VERB
ejde-320	46	6	sufficiency	sufficiency	NOUN
ejde-320	46	7	and	and	CCONJ
ejde-320	46	8	almost	almost	ADV
ejde-320	46	9	necessity	necessity	NOUN
ejde-320	46	10	of	of	ADP
ejde-320	46	11	an	an	DET
ejde-320	46	12	orlicz	orlicz	ADJ
ejde-320	46	13	-	-	PUNCT
ejde-320	46	14	sobolev	sobolev	NOUN
ejde-320	46	15	type	type	NOUN
ejde-320	46	16	inequality	inequality	NOUN
ejde-320	46	17	for	for	ADP
ejde-320	46	18	the	the	DET
ejde-320	46	19	existence	existence	NOUN
ejde-320	46	20	,	,	PUNCT
ejde-320	46	21	uniqueness	uniqueness	NOUN
ejde-320	46	22	,	,	PUNCT
ejde-320	46	23	and	and	CCONJ
ejde-320	46	24	boundedness	boundedness	NOUN
ejde-320	46	25	of	of	ADP
ejde-320	46	26	weak	weak	ADJ
ejde-320	46	27	solutions	solution	NOUN
ejde-320	46	28	to	to	PART
ejde-320	46	29	infinitely	infinitely	ADV
ejde-320	46	30	degenerate	degenerate	ADJ
ejde-320	46	31	elliptic	elliptic	ADJ
ejde-320	46	32	partial	partial	ADJ
ejde-320	46	33	differential	differential	ADJ
ejde-320	46	34	equations	equation	NOUN
ejde-320	46	35	with	with	ADP
ejde-320	46	36	homogeneous	homogeneous	ADJ
ejde-320	46	37	dirichlet	dirichlet	PROPN
ejde-320	46	38	boundary	boundary	PROPN
ejde-320	46	39	conditions	condition	NOUN
ejde-320	46	40	.	.	PUNCT
ejde-320	47	1	our	our	PRON
ejde-320	47	2	main	main	ADJ
ejde-320	47	3	results	result	NOUN
ejde-320	47	4	are	be	AUX
ejde-320	47	5	as	as	SCONJ
ejde-320	47	6	follows	follow	NOUN
ejde-320	47	7	theorem	theorem	VERB
ejde-320	47	8	1.1	1.1	NUM
ejde-320	47	9	.	.	PUNCT
ejde-320	48	1	let	let	VERB
ejde-320	48	2	l	l	NOUN
ejde-320	48	3	=	=	X
ejde-320	48	4	∇	∇	X
ejde-320	48	5	·	·	PUNCT
ejde-320	49	1	a∇	a∇	NOUN
ejde-320	49	2	with	with	ADP
ejde-320	49	3	bounded	bounded	ADJ
ejde-320	49	4	measurable	measurable	ADJ
ejde-320	49	5	non	non	ADJ
ejde-320	49	6	-	-	ADJ
ejde-320	49	7	negative	negative	ADJ
ejde-320	49	8	semidefinite	semidefinite	NOUN
ejde-320	49	9	matrix	matrix	NOUN
ejde-320	49	10	a	a	PRON
ejde-320	49	11	,	,	PUNCT
ejde-320	49	12	and	and	CCONJ
ejde-320	49	13	d	d	ADP
ejde-320	49	14	a	a	DET
ejde-320	49	15	metric	metric	NOUN
ejde-320	49	16	on	on	ADP
ejde-320	49	17	rn	rn	PROPN
ejde-320	49	18	.	.	PROPN
ejde-320	49	19	suppose	suppose	VERB
ejde-320	49	20	also	also	ADV
ejde-320	49	21	that	that	SCONJ
ejde-320	49	22	(	(	PUNCT
ejde-320	49	23	1.3	1.3	NUM
ejde-320	49	24	)	)	PUNCT
ejde-320	49	25	holds	hold	VERB
ejde-320	49	26	for	for	ADP
ejde-320	49	27	all	all	DET
ejde-320	49	28	w	w	NOUN
ejde-320	49	29	∈	∈	PROPN
ejde-320	49	30	(	(	PUNCT
ejde-320	49	31	w	w	PROPN
ejde-320	49	32	1,2	1,2	NUM
ejde-320	49	33	a	a	PRON
ejde-320	49	34	)	)	PUNCT
ejde-320	49	35	0	0	PUNCT
ejde-320	50	1	(	(	PUNCT
ejde-320	50	2	b	b	NOUN
ejde-320	50	3	)	)	PUNCT
ejde-320	50	4	and	and	CCONJ
ejde-320	50	5	the	the	DET
ejde-320	50	6	metric	metric	ADJ
ejde-320	50	7	ball	ball	PROPN
ejde-320	50	8	b	b	PROPN
ejde-320	50	9	=	=	SYM
ejde-320	50	10	ω	ω	PROPN
ejde-320	51	1	⊂	⊂	PROPN
ejde-320	51	2	rn	rn	PROPN
ejde-320	51	3	with	with	ADP
ejde-320	51	4	φ	φ	PROPN
ejde-320	51	5	satisfying	satisfying	PROPN
ejde-320	51	6	φ(t	φ(t	PROPN
ejde-320	51	7	)	)	PUNCT
ejde-320	51	8	≥	≥	NOUN
ejde-320	51	9	t2	t2	NOUN
ejde-320	51	10	for	for	ADP
ejde-320	51	11	all	all	DET
ejde-320	51	12	t	t	PROPN
ejde-320	51	13	≥	≥	NOUN
ejde-320	51	14	0	0	NUM
ejde-320	51	15	,	,	PUNCT
ejde-320	51	16	and	and	CCONJ
ejde-320	51	17	φ(t	φ(t	PROPN
ejde-320	51	18	)	)	PUNCT
ejde-320	51	19	≥	≥	NOUN
ejde-320	51	20	t2(ln	t2(ln	PROPN
ejde-320	51	21	t)n	t)n	NOUN
ejde-320	51	22	,	,	PUNCT
ejde-320	51	23	n	n	CCONJ
ejde-320	51	24	>	>	X
ejde-320	51	25	1	1	NUM
ejde-320	51	26	,	,	PUNCT
ejde-320	51	27	for	for	ADP
ejde-320	51	28	all	all	DET
ejde-320	51	29	t	t	PROPN
ejde-320	51	30	≥	≥	NOUN
ejde-320	51	31	1	1	NUM
ejde-320	51	32	.	.	PUNCT
ejde-320	52	1	if	if	SCONJ
ejde-320	52	2	f	f	PROPN
ejde-320	52	3	∈	∈	PROPN
ejde-320	52	4	l∞(b	l∞(b	VERB
ejde-320	52	5	)	)	PUNCT
ejde-320	52	6	,	,	PUNCT
ejde-320	52	7	then	then	ADV
ejde-320	52	8	there	there	PRON
ejde-320	52	9	exists	exist	VERB
ejde-320	52	10	a	a	DET
ejde-320	52	11	unique	unique	ADJ
ejde-320	52	12	weak	weak	ADJ
ejde-320	52	13	solution	solution	NOUN
ejde-320	52	14	u	u	NOUN
ejde-320	52	15	∈	∈	PROPN
ejde-320	52	16	(	(	PUNCT
ejde-320	52	17	w	w	PROPN
ejde-320	52	18	1,2	1,2	NUM
ejde-320	52	19	a	a	PRON
ejde-320	52	20	)	)	PUNCT
ejde-320	52	21	0	0	PUNCT
ejde-320	53	1	(	(	PUNCT
ejde-320	53	2	b	b	NOUN
ejde-320	53	3	)	)	PUNCT
ejde-320	53	4	of	of	ADP
ejde-320	53	5	(	(	PUNCT
ejde-320	53	6	1.1	1.1	NUM
ejde-320	53	7	)	)	PUNCT
ejde-320	53	8	in	in	ADP
ejde-320	53	9	the	the	DET
ejde-320	53	10	ball	ball	NOUN
ejde-320	53	11	ω	ω	PROPN
ejde-320	53	12	=	=	SYM
ejde-320	53	13	b	b	PROPN
ejde-320	53	14	and	and	CCONJ
ejde-320	53	15	it	it	PRON
ejde-320	53	16	ejde-2021/82	ejde-2021/82	VERB
ejde-320	53	17	orlicz	orlicz	ADJ
ejde-320	53	18	-	-	PUNCT
ejde-320	53	19	sobolev	sobolev	NOUN
ejde-320	53	20	inequalities	inequality	NOUN
ejde-320	53	21	and	and	CCONJ
ejde-320	53	22	the	the	DET
ejde-320	53	23	dirichlet	dirichlet	PROPN
ejde-320	53	24	problem	problem	NOUN
ejde-320	53	25	3	3	NUM
ejde-320	53	26	satisfies	satisfie	NOUN
ejde-320	53	27	‖u‖l∞(b	‖u‖l∞(b	NOUN
ejde-320	53	28	)	)	PUNCT
ejde-320	53	29	≤	≤	NUM
ejde-320	53	30	c‖f‖l∞(b	c‖f‖l∞(b	NOUN
ejde-320	53	31	)	)	PUNCT
ejde-320	53	32	.	.	PUNCT
ejde-320	54	1	theorem	theorem	VERB
ejde-320	54	2	1.2	1.2	NUM
ejde-320	54	3	.	.	PUNCT
ejde-320	55	1	let	let	VERB
ejde-320	55	2	ϕ	ϕ	NOUN
ejde-320	55	3	be	be	AUX
ejde-320	55	4	a	a	DET
ejde-320	55	5	young	young	ADJ
ejde-320	55	6	function	function	NOUN
ejde-320	55	7	with	with	ADP
ejde-320	55	8	ϕ̃	ϕ̃	PROPN
ejde-320	55	9	being	be	AUX
ejde-320	55	10	its	its	PRON
ejde-320	55	11	dual	dual	ADJ
ejde-320	55	12	,	,	PUNCT
ejde-320	55	13	and	and	CCONJ
ejde-320	55	14	define	define	VERB
ejde-320	55	15	φ	φ	NUM
ejde-320	55	16	by	by	ADP
ejde-320	55	17	φ(t	φ(t	PROPN
ejde-320	55	18	)	)	PUNCT
ejde-320	55	19	=	=	SYM
ejde-320	55	20	ϕ(t2	ϕ(t2	NOUN
ejde-320	55	21	)	)	PUNCT
ejde-320	55	22	for	for	ADP
ejde-320	55	23	all	all	DET
ejde-320	55	24	t	t	PROPN
ejde-320	55	25	∈	∈	PROPN
ejde-320	55	26	r.	r.	PROPN
ejde-320	55	27	suppose	suppose	VERB
ejde-320	55	28	that	that	SCONJ
ejde-320	55	29	for	for	ADP
ejde-320	55	30	every	every	DET
ejde-320	55	31	f	f	PROPN
ejde-320	55	32	∈	∈	PROPN
ejde-320	55	33	lϕ̃(b	lϕ̃(b	NOUN
ejde-320	55	34	)	)	PUNCT
ejde-320	55	35	there	there	PRON
ejde-320	55	36	exists	exist	VERB
ejde-320	55	37	a	a	DET
ejde-320	55	38	unique	unique	ADJ
ejde-320	55	39	weak	weak	ADJ
ejde-320	55	40	solution	solution	NOUN
ejde-320	55	41	u	u	NOUN
ejde-320	55	42	∈	∈	PROPN
ejde-320	55	43	(	(	PUNCT
ejde-320	55	44	w	w	PROPN
ejde-320	55	45	1,2	1,2	NUM
ejde-320	55	46	a	a	PRON
ejde-320	55	47	)	)	PUNCT
ejde-320	55	48	0	0	PUNCT
ejde-320	56	1	(	(	PUNCT
ejde-320	56	2	b	b	NOUN
ejde-320	56	3	)	)	PUNCT
ejde-320	56	4	of	of	ADP
ejde-320	56	5	(	(	PUNCT
ejde-320	56	6	1.1	1.1	NUM
ejde-320	56	7	)	)	PUNCT
ejde-320	56	8	in	in	ADP
ejde-320	56	9	the	the	DET
ejde-320	56	10	ball	ball	NOUN
ejde-320	56	11	ω	ω	PROPN
ejde-320	56	12	=	=	SYM
ejde-320	56	13	b	b	PROPN
ejde-320	56	14	which	which	PRON
ejde-320	56	15	satisfies	satisfy	VERB
ejde-320	56	16	‖u‖l∞(b	‖u‖l∞(b	NOUN
ejde-320	56	17	,	,	PUNCT
ejde-320	56	18	dµ	dµ	PROPN
ejde-320	56	19	)	)	PUNCT
ejde-320	56	20	≤	≤	NOUN
ejde-320	56	21	c‖f‖lϕ̃(b	c‖f‖lϕ̃(b	PROPN
ejde-320	56	22	,	,	PUNCT
ejde-320	56	23	dµ	dµ	PROPN
ejde-320	56	24	)	)	PUNCT
ejde-320	56	25	,	,	PUNCT
ejde-320	56	26	with	with	ADP
ejde-320	56	27	dµ	dµ	PROPN
ejde-320	56	28	=	=	SYM
ejde-320	56	29	dx	dx	PROPN
ejde-320	56	30	|b|	|b|	PROPN
ejde-320	56	31	.	.	PUNCT
ejde-320	57	1	then	then	ADV
ejde-320	57	2	orlicz	orlicz	PROPN
ejde-320	57	3	-	-	PUNCT
ejde-320	57	4	sobolev	sobolev	NOUN
ejde-320	57	5	inequality	inequality	NOUN
ejde-320	57	6	(	(	PUNCT
ejde-320	57	7	1.3	1.3	NUM
ejde-320	57	8	)	)	PUNCT
ejde-320	57	9	holds	hold	VERB
ejde-320	57	10	for	for	ADP
ejde-320	57	11	all	all	DET
ejde-320	57	12	w	w	NOUN
ejde-320	57	13	∈	∈	PROPN
ejde-320	57	14	(	(	PUNCT
ejde-320	57	15	w	w	PROPN
ejde-320	57	16	1,2	1,2	NUM
ejde-320	57	17	a	a	PRON
ejde-320	57	18	)	)	PUNCT
ejde-320	57	19	0	0	PUNCT
ejde-320	58	1	(	(	PUNCT
ejde-320	58	2	b	b	NOUN
ejde-320	58	3	)	)	PUNCT
ejde-320	58	4	.	.	PUNCT
ejde-320	59	1	remark	remark	VERB
ejde-320	59	2	1.3	1.3	NUM
ejde-320	59	3	.	.	PUNCT
ejde-320	60	1	note	note	VERB
ejde-320	60	2	that	that	SCONJ
ejde-320	60	3	in	in	ADP
ejde-320	60	4	the	the	DET
ejde-320	60	5	above	above	ADJ
ejde-320	60	6	theorems	theorem	NOUN
ejde-320	60	7	we	we	PRON
ejde-320	60	8	do	do	AUX
ejde-320	60	9	not	not	PART
ejde-320	60	10	assume	assume	VERB
ejde-320	60	11	that	that	SCONJ
ejde-320	60	12	the	the	DET
ejde-320	60	13	metric	metric	NOUN
ejde-320	60	14	d	d	PROPN
ejde-320	60	15	is	be	AUX
ejde-320	60	16	the	the	DET
ejde-320	60	17	subunit	subunit	NOUN
ejde-320	60	18	metric	metric	NOUN
ejde-320	60	19	associated	associate	VERB
ejde-320	60	20	with	with	ADP
ejde-320	60	21	a.	a.	NOUN
ejde-320	60	22	in	in	ADP
ejde-320	60	23	practice	practice	NOUN
ejde-320	60	24	,	,	PUNCT
ejde-320	60	25	to	to	PART
ejde-320	60	26	prove	prove	VERB
ejde-320	60	27	orlicz	orlicz	ADJ
ejde-320	60	28	-	-	PUNCT
ejde-320	60	29	sobolev	sobolev	NOUN
ejde-320	60	30	inequality	inequality	NOUN
ejde-320	60	31	(	(	PUNCT
ejde-320	60	32	1.3	1.3	NUM
ejde-320	60	33	)	)	PUNCT
ejde-320	60	34	one	one	PRON
ejde-320	60	35	would	would	AUX
ejde-320	60	36	need	need	VERB
ejde-320	60	37	to	to	PART
ejde-320	60	38	work	work	VERB
ejde-320	60	39	in	in	ADP
ejde-320	60	40	a	a	DET
ejde-320	60	41	subunit	subunit	NOUN
ejde-320	60	42	metric	metric	ADJ
ejde-320	60	43	space	space	NOUN
ejde-320	61	1	[	[	X
ejde-320	61	2	3	3	NUM
ejde-320	61	3	]	]	PUNCT
ejde-320	61	4	,	,	PUNCT
ejde-320	61	5	or	or	CCONJ
ejde-320	61	6	a	a	DET
ejde-320	61	7	measure	measure	NOUN
ejde-320	61	8	space	space	NOUN
ejde-320	61	9	associated	associate	VERB
ejde-320	61	10	with	with	ADP
ejde-320	61	11	the	the	DET
ejde-320	61	12	operator	operator	NOUN
ejde-320	61	13	[	[	X
ejde-320	61	14	2	2	NUM
ejde-320	61	15	]	]	PUNCT
ejde-320	61	16	.	.	PUNCT
ejde-320	62	1	a	a	DET
ejde-320	62	2	version	version	NOUN
ejde-320	62	3	of	of	ADP
ejde-320	62	4	the	the	DET
ejde-320	62	5	result	result	NOUN
ejde-320	62	6	in	in	ADP
ejde-320	62	7	theorem	theorem	ADJ
ejde-320	62	8	1.2	1.2	NUM
ejde-320	62	9	and	and	CCONJ
ejde-320	62	10	a	a	DET
ejde-320	62	11	sketch	sketch	NOUN
ejde-320	62	12	of	of	ADP
ejde-320	62	13	the	the	DET
ejde-320	62	14	proof	proof	NOUN
ejde-320	62	15	appears	appear	VERB
ejde-320	62	16	in	in	ADP
ejde-320	62	17	[	[	X
ejde-320	62	18	6	6	NUM
ejde-320	62	19	,	,	PUNCT
ejde-320	62	20	sections	section	NOUN
ejde-320	62	21	1	1	NUM
ejde-320	62	22	and	and	CCONJ
ejde-320	62	23	2	2	NUM
ejde-320	62	24	of	of	ADP
ejde-320	62	25	chapter	chapter	NOUN
ejde-320	62	26	9	9	NUM
ejde-320	62	27	]	]	PUNCT
ejde-320	62	28	.	.	PUNCT
ejde-320	63	1	it	it	PRON
ejde-320	63	2	can	can	AUX
ejde-320	63	3	be	be	AUX
ejde-320	63	4	seen	see	VERB
ejde-320	63	5	as	as	ADP
ejde-320	63	6	a	a	DET
ejde-320	63	7	generalization	generalization	NOUN
ejde-320	63	8	of	of	ADP
ejde-320	63	9	the	the	DET
ejde-320	63	10	subelliptic	subelliptic	ADJ
ejde-320	63	11	result	result	NOUN
ejde-320	63	12	[	[	X
ejde-320	63	13	11	11	NUM
ejde-320	63	14	,	,	PUNCT
ejde-320	63	15	lemma	lemma	PROPN
ejde-320	63	16	102	102	NUM
ejde-320	63	17	]	]	PUNCT
ejde-320	63	18	with	with	ADP
ejde-320	63	19	lϕ	lϕ	NOUN
ejde-320	63	20	replacing	replace	VERB
ejde-320	63	21	lσ	lσ	PRON
ejde-320	63	22	and	and	CCONJ
ejde-320	63	23	lφ	lφ	AUX
ejde-320	63	24	replacing	replace	VERB
ejde-320	63	25	l2σ	l2σ	PROPN
ejde-320	63	26	.	.	PUNCT
ejde-320	64	1	in	in	ADP
ejde-320	64	2	the	the	DET
ejde-320	64	3	subelliptic	subelliptic	ADJ
ejde-320	64	4	case	case	NOUN
ejde-320	64	5	,	,	PUNCT
ejde-320	64	6	the	the	DET
ejde-320	64	7	requirement	requirement	NOUN
ejde-320	64	8	on	on	ADP
ejde-320	64	9	the	the	DET
ejde-320	64	10	right	right	ADJ
ejde-320	64	11	hand	hand	NOUN
ejde-320	64	12	side	side	NOUN
ejde-320	64	13	is	be	AUX
ejde-320	64	14	f	f	PROPN
ejde-320	64	15	∈	∈	PROPN
ejde-320	64	16	lq	lq	VERB
ejde-320	64	17	with	with	ADP
ejde-320	64	18	q	q	PROPN
ejde-320	64	19	>	>	X
ejde-320	64	20	σ′.	σ′.	X
ejde-320	64	21	in	in	ADP
ejde-320	64	22	theorem	theorem	ADJ
ejde-320	64	23	1.2	1.2	NUM
ejde-320	64	24	we	we	PRON
ejde-320	64	25	require	require	VERB
ejde-320	64	26	f	f	PROPN
ejde-320	64	27	∈	∈	PROPN
ejde-320	64	28	l∞	l∞	PROPN
ejde-320	64	29	,	,	PUNCT
ejde-320	64	30	a	a	DET
ejde-320	64	31	strengthening	strengthening	NOUN
ejde-320	64	32	of	of	ADP
ejde-320	64	33	lϕ̃.	lϕ̃.	ADJ
ejde-320	64	34	note	note	VERB
ejde-320	64	35	that	that	SCONJ
ejde-320	64	36	just	just	ADV
ejde-320	64	37	like	like	INTJ
ejde-320	64	38	in	in	ADP
ejde-320	64	39	the	the	DET
ejde-320	64	40	subelliptic	subelliptic	ADJ
ejde-320	64	41	case	case	NOUN
ejde-320	64	42	,	,	PUNCT
ejde-320	64	43	there	there	PRON
ejde-320	64	44	is	be	VERB
ejde-320	64	45	a	a	DET
ejde-320	64	46	gap	gap	NOUN
ejde-320	64	47	between	between	ADP
ejde-320	64	48	necessary	necessary	ADJ
ejde-320	64	49	and	and	CCONJ
ejde-320	64	50	sufficient	sufficient	ADJ
ejde-320	64	51	conditions	condition	NOUN
ejde-320	64	52	.	.	PUNCT
ejde-320	65	1	we	we	PRON
ejde-320	65	2	suspect	suspect	VERB
ejde-320	65	3	that	that	SCONJ
ejde-320	65	4	the	the	DET
ejde-320	65	5	sufficient	sufficient	ADJ
ejde-320	65	6	condition	condition	NOUN
ejde-320	65	7	in	in	ADP
ejde-320	65	8	theorem	theorem	ADJ
ejde-320	65	9	1.1	1.1	NUM
ejde-320	65	10	can	can	AUX
ejde-320	65	11	be	be	AUX
ejde-320	65	12	sharpened	sharpen	VERB
ejde-320	65	13	,	,	PUNCT
ejde-320	65	14	but	but	CCONJ
ejde-320	65	15	not	not	PART
ejde-320	65	16	with	with	ADP
ejde-320	65	17	our	our	PRON
ejde-320	65	18	current	current	ADJ
ejde-320	65	19	method	method	NOUN
ejde-320	65	20	of	of	ADP
ejde-320	65	21	proof	proof	NOUN
ejde-320	65	22	.	.	PUNCT
ejde-320	66	1	at	at	ADP
ejde-320	66	2	this	this	DET
ejde-320	66	3	point	point	NOUN
ejde-320	66	4	we	we	PRON
ejde-320	66	5	do	do	AUX
ejde-320	66	6	not	not	PART
ejde-320	66	7	know	know	VERB
ejde-320	66	8	if	if	SCONJ
ejde-320	66	9	the	the	DET
ejde-320	66	10	gap	gap	NOUN
ejde-320	66	11	can	can	AUX
ejde-320	66	12	be	be	AUX
ejde-320	66	13	closed	close	VERB
ejde-320	66	14	completely	completely	ADV
ejde-320	66	15	.	.	PUNCT
ejde-320	67	1	the	the	DET
ejde-320	67	2	article	article	NOUN
ejde-320	67	3	is	be	AUX
ejde-320	67	4	organized	organize	VERB
ejde-320	67	5	as	as	SCONJ
ejde-320	67	6	follows	follow	VERB
ejde-320	67	7	.	.	PUNCT
ejde-320	68	1	after	after	ADP
ejde-320	68	2	giving	give	VERB
ejde-320	68	3	some	some	DET
ejde-320	68	4	background	background	NOUN
ejde-320	68	5	and	and	CCONJ
ejde-320	68	6	preliminaries	preliminary	NOUN
ejde-320	68	7	in	in	ADP
ejde-320	68	8	section	section	NOUN
ejde-320	68	9	2	2	NUM
ejde-320	68	10	,	,	PUNCT
ejde-320	68	11	we	we	PRON
ejde-320	68	12	prove	prove	VERB
ejde-320	68	13	the	the	DET
ejde-320	68	14	existence	existence	NOUN
ejde-320	68	15	and	and	CCONJ
ejde-320	68	16	global	global	ADJ
ejde-320	68	17	boundedness	boundedness	NOUN
ejde-320	68	18	of	of	ADP
ejde-320	68	19	weak	weak	ADJ
ejde-320	68	20	solutions	solution	NOUN
ejde-320	68	21	,	,	PUNCT
ejde-320	68	22	theorem	theorem	VERB
ejde-320	68	23	1.1	1.1	NUM
ejde-320	68	24	,	,	PUNCT
ejde-320	68	25	in	in	ADP
ejde-320	68	26	section	section	NOUN
ejde-320	68	27	3	3	NUM
ejde-320	68	28	.	.	PUNCT
ejde-320	68	29	section	section	NOUN
ejde-320	68	30	4	4	NUM
ejde-320	68	31	is	be	AUX
ejde-320	68	32	devoted	devote	VERB
ejde-320	68	33	to	to	ADP
ejde-320	68	34	the	the	DET
ejde-320	68	35	proof	proof	NOUN
ejde-320	68	36	of	of	ADP
ejde-320	68	37	theorem	theorem	ADJ
ejde-320	68	38	1.2	1.2	NUM
ejde-320	68	39	,	,	PUNCT
ejde-320	68	40	the	the	DET
ejde-320	68	41	necessity	necessity	NOUN
ejde-320	68	42	of	of	ADP
ejde-320	68	43	orlicz	orlicz	NOUN
ejde-320	68	44	-	-	PUNCT
ejde-320	68	45	sobolev	sobolev	NOUN
ejde-320	68	46	for	for	ADP
ejde-320	68	47	solvability	solvability	NOUN
ejde-320	68	48	of	of	ADP
ejde-320	68	49	the	the	DET
ejde-320	68	50	dirichlet	dirichlet	PROPN
ejde-320	68	51	problem	problem	NOUN
ejde-320	68	52	with	with	ADP
ejde-320	68	53	a	a	DET
ejde-320	68	54	quantitative	quantitative	ADJ
ejde-320	68	55	bound	bind	VERB
ejde-320	68	56	.	.	PUNCT
ejde-320	69	1	the	the	DET
ejde-320	69	2	proof	proof	NOUN
ejde-320	69	3	follows	follow	VERB
ejde-320	69	4	closely	closely	ADV
ejde-320	69	5	the	the	DET
ejde-320	69	6	proof	proof	NOUN
ejde-320	69	7	of	of	ADP
ejde-320	69	8	lemma	lemma	PROPN
ejde-320	69	9	102	102	NUM
ejde-320	69	10	in	in	ADP
ejde-320	69	11	[	[	X
ejde-320	69	12	11	11	NUM
ejde-320	69	13	]	]	PUNCT
ejde-320	69	14	,	,	PUNCT
ejde-320	69	15	and	and	CCONJ
ejde-320	69	16	it	it	PRON
ejde-320	69	17	also	also	ADV
ejde-320	69	18	appears	appear	VERB
ejde-320	69	19	in	in	ADP
ejde-320	69	20	[	[	X
ejde-320	69	21	6	6	NUM
ejde-320	69	22	,	,	PUNCT
ejde-320	69	23	sections	section	NOUN
ejde-320	69	24	1	1	NUM
ejde-320	69	25	and	and	CCONJ
ejde-320	69	26	2	2	NUM
ejde-320	69	27	of	of	ADP
ejde-320	69	28	chapter	chapter	NOUN
ejde-320	69	29	9	9	NUM
ejde-320	69	30	]	]	PUNCT
ejde-320	69	31	.	.	PUNCT
ejde-320	70	1	however	however	ADV
ejde-320	70	2	,	,	PUNCT
ejde-320	70	3	the	the	DET
ejde-320	70	4	case	case	NOUN
ejde-320	70	5	of	of	ADP
ejde-320	70	6	orlicz	orlicz	NOUN
ejde-320	70	7	-	-	PUNCT
ejde-320	70	8	sobolev	sobolev	NOUN
ejde-320	70	9	spaces	space	NOUN
ejde-320	70	10	is	be	AUX
ejde-320	70	11	more	more	ADV
ejde-320	70	12	delicate	delicate	ADJ
ejde-320	70	13	,	,	PUNCT
ejde-320	70	14	so	so	SCONJ
ejde-320	70	15	we	we	PRON
ejde-320	70	16	fill	fill	VERB
ejde-320	70	17	in	in	ADP
ejde-320	70	18	the	the	DET
ejde-320	70	19	gaps	gap	NOUN
ejde-320	70	20	and	and	CCONJ
ejde-320	70	21	provide	provide	VERB
ejde-320	70	22	all	all	DET
ejde-320	70	23	the	the	DET
ejde-320	70	24	details	detail	NOUN
ejde-320	70	25	.	.	PUNCT
ejde-320	71	1	finally	finally	ADV
ejde-320	71	2	,	,	PUNCT
ejde-320	71	3	section	section	NOUN
ejde-320	71	4	5	5	NUM
ejde-320	71	5	provides	provide	VERB
ejde-320	71	6	some	some	DET
ejde-320	71	7	counterexamples	counterexample	NOUN
ejde-320	71	8	demonstrating	demonstrate	VERB
ejde-320	71	9	that	that	SCONJ
ejde-320	71	10	the	the	DET
ejde-320	71	11	requirement	requirement	NOUN
ejde-320	71	12	on	on	ADP
ejde-320	71	13	the	the	DET
ejde-320	71	14	right	right	ADJ
ejde-320	71	15	hand	hand	NOUN
ejde-320	71	16	side	side	NOUN
ejde-320	71	17	in	in	ADP
ejde-320	71	18	theorem	theorem	ADJ
ejde-320	71	19	1.1	1.1	NUM
ejde-320	71	20	can	can	AUX
ejde-320	71	21	not	not	PART
ejde-320	71	22	be	be	AUX
ejde-320	71	23	significantly	significantly	ADV
ejde-320	71	24	relaxed	relax	VERB
ejde-320	71	25	.	.	PUNCT
ejde-320	72	1	more	more	ADV
ejde-320	72	2	precisely	precisely	ADV
ejde-320	72	3	,	,	PUNCT
ejde-320	72	4	we	we	PRON
ejde-320	72	5	give	give	VERB
ejde-320	72	6	examples	example	NOUN
ejde-320	72	7	of	of	ADP
ejde-320	72	8	equations	equation	NOUN
ejde-320	72	9	admitting	admit	VERB
ejde-320	72	10	unbounded	unbounded	ADJ
ejde-320	72	11	weak	weak	ADJ
ejde-320	72	12	solutions	solution	NOUN
ejde-320	72	13	in	in	ADP
ejde-320	72	14	the	the	DET
ejde-320	72	15	case	case	NOUN
ejde-320	72	16	of	of	ADP
ejde-320	72	17	laplacian	laplacian	ADJ
ejde-320	72	18	,	,	PUNCT
ejde-320	72	19	subelliptic	subelliptic	ADJ
ejde-320	72	20	,	,	PUNCT
ejde-320	72	21	and	and	CCONJ
ejde-320	72	22	infinitely	infinitely	ADV
ejde-320	72	23	degenerate	degenerate	ADJ
ejde-320	72	24	elliptic	elliptic	ADJ
ejde-320	72	25	operators	operator	NOUN
ejde-320	72	26	.	.	PUNCT
ejde-320	73	1	2	2	X
ejde-320	73	2	.	.	NUM
ejde-320	73	3	preliminaries	preliminary	NOUN
ejde-320	73	4	2.1	2.1	NUM
ejde-320	73	5	.	.	PUNCT
ejde-320	74	1	subunit	subunit	NOUN
ejde-320	74	2	metric	metric	ADJ
ejde-320	74	3	spaces	space	NOUN
ejde-320	74	4	.	.	PUNCT
ejde-320	75	1	we	we	PRON
ejde-320	75	2	start	start	VERB
ejde-320	75	3	this	this	DET
ejde-320	75	4	section	section	NOUN
ejde-320	75	5	with	with	ADP
ejde-320	75	6	some	some	DET
ejde-320	75	7	background	background	NOUN
ejde-320	75	8	material	material	NOUN
ejde-320	75	9	on	on	ADP
ejde-320	75	10	subunit	subunit	NOUN
ejde-320	75	11	metric	metric	ADJ
ejde-320	75	12	spaces	space	NOUN
ejde-320	75	13	associated	associate	VERB
ejde-320	75	14	with	with	ADP
ejde-320	75	15	degenerate	degenerate	ADJ
ejde-320	75	16	operators	operator	NOUN
ejde-320	75	17	,	,	PUNCT
ejde-320	75	18	all	all	PRON
ejde-320	75	19	of	of	ADP
ejde-320	75	20	which	which	PRON
ejde-320	75	21	can	can	AUX
ejde-320	75	22	be	be	AUX
ejde-320	75	23	found	find	VERB
ejde-320	75	24	in	in	ADP
ejde-320	75	25	[	[	X
ejde-320	75	26	7	7	NUM
ejde-320	75	27	,	,	PUNCT
ejde-320	75	28	chapter	chapter	NOUN
ejde-320	75	29	7	7	NUM
ejde-320	75	30	]	]	PUNCT
ejde-320	75	31	.	.	PUNCT
ejde-320	76	1	as	as	SCONJ
ejde-320	76	2	mentioned	mention	VERB
ejde-320	76	3	in	in	ADP
ejde-320	76	4	the	the	DET
ejde-320	76	5	introduction	introduction	NOUN
ejde-320	76	6	,	,	PUNCT
ejde-320	76	7	we	we	PRON
ejde-320	76	8	do	do	AUX
ejde-320	76	9	not	not	PART
ejde-320	76	10	assume	assume	VERB
ejde-320	76	11	the	the	DET
ejde-320	76	12	underlying	underlie	VERB
ejde-320	76	13	metric	metric	ADJ
ejde-320	76	14	space	space	NOUN
ejde-320	76	15	is	be	AUX
ejde-320	76	16	the	the	DET
ejde-320	76	17	subunit	subunit	NOUN
ejde-320	76	18	metric	metric	ADJ
ejde-320	76	19	space	space	NOUN
ejde-320	76	20	,	,	PUNCT
ejde-320	76	21	however	however	ADV
ejde-320	76	22	,	,	PUNCT
ejde-320	76	23	it	it	PRON
ejde-320	76	24	will	will	AUX
ejde-320	76	25	be	be	AUX
ejde-320	76	26	used	use	VERB
ejde-320	76	27	to	to	PART
ejde-320	76	28	construct	construct	VERB
ejde-320	76	29	counterexamples	counterexample	NOUN
ejde-320	76	30	in	in	ADP
ejde-320	76	31	section	section	NOUN
ejde-320	76	32	5	5	NUM
ejde-320	76	33	.	.	NOUN
ejde-320	77	1	2.1.1	2.1.1	NUM
ejde-320	77	2	.	.	PUNCT
ejde-320	77	3	degenerate	degenerate	ADJ
ejde-320	77	4	sobolev	sobolev	NOUN
ejde-320	77	5	spaces	space	VERB
ejde-320	77	6	.	.	PUNCT
ejde-320	78	1	let	let	VERB
ejde-320	78	2	a	a	PRON
ejde-320	78	3	be	be	AUX
ejde-320	78	4	a	a	DET
ejde-320	78	5	nonnegative	nonnegative	ADJ
ejde-320	78	6	semidefinite	semidefinite	NOUN
ejde-320	78	7	bounded	bound	VERB
ejde-320	78	8	measurable	measurable	ADJ
ejde-320	78	9	matrix	matrix	NOUN
ejde-320	78	10	,	,	PUNCT
ejde-320	78	11	and	and	CCONJ
ejde-320	78	12	assume	assume	VERB
ejde-320	78	13	that	that	SCONJ
ejde-320	78	14	a(x	a(x	NOUN
ejde-320	78	15	)	)	PUNCT
ejde-320	78	16	≈	≈	PROPN
ejde-320	78	17	b(x)trb(x	b(x)trb(x	NOUN
ejde-320	78	18	)	)	PUNCT
ejde-320	78	19	,	,	PUNCT
ejde-320	78	20	i.e.	i.e.	X
ejde-320	78	21	,	,	PUNCT
ejde-320	78	22	there	there	PRON
ejde-320	78	23	exist	exist	VERB
ejde-320	78	24	positive	positive	ADJ
ejde-320	78	25	constants	constant	NOUN
ejde-320	78	26	c1	c1	PROPN
ejde-320	78	27	and	and	CCONJ
ejde-320	78	28	c2	c2	PROPN
ejde-320	78	29	such	such	ADJ
ejde-320	78	30	that	that	PRON
ejde-320	78	31	for	for	ADP
ejde-320	78	32	a.e	a.e	PROPN
ejde-320	78	33	.	.	PUNCT
ejde-320	78	34	x	x	PUNCT
ejde-320	78	35	∈	∈	PROPN
ejde-320	78	36	ω	ω	NOUN
ejde-320	78	37	and	and	CCONJ
ejde-320	78	38	all	all	DET
ejde-320	78	39	ξ	ξ	PROPN
ejde-320	78	40	∈	∈	PROPN
ejde-320	78	41	rn	rn	PROPN
ejde-320	78	42	,	,	PUNCT
ejde-320	78	43	c1|b(x)ξ|2	c1|b(x)ξ|2	PROPN
ejde-320	78	44	≤	≤	X
ejde-320	78	45	ξ	ξ	X
ejde-320	78	46	·	·	PUNCT
ejde-320	78	47	a(x)ξ	a(x)ξ	X
ejde-320	78	48	≤	≤	ADJ
ejde-320	78	49	c2|b(x)ξ|2	c2|b(x)ξ|2	NOUN
ejde-320	78	50	,	,	PUNCT
ejde-320	78	51	where	where	SCONJ
ejde-320	78	52	b(x	b(x	NOUN
ejde-320	78	53	)	)	PUNCT
ejde-320	78	54	is	be	AUX
ejde-320	78	55	a	a	DET
ejde-320	78	56	lipschitz	lipschitz	NOUN
ejde-320	78	57	continuous	continuous	ADJ
ejde-320	78	58	n	n	NUM
ejde-320	78	59	×	×	NOUN
ejde-320	78	60	n	n	CCONJ
ejde-320	78	61	real	real	ADV
ejde-320	78	62	-	-	PUNCT
ejde-320	78	63	valued	value	VERB
ejde-320	78	64	matrix	matrix	NOUN
ejde-320	78	65	defined	define	VERB
ejde-320	78	66	for	for	ADP
ejde-320	78	67	x	x	PROPN
ejde-320	78	68	∈	∈	PROPN
ejde-320	78	69	ω	ω	PROPN
ejde-320	78	70	.	.	PUNCT
ejde-320	79	1	we	we	PRON
ejde-320	79	2	define	define	VERB
ejde-320	79	3	the	the	DET
ejde-320	79	4	a	a	DET
ejde-320	79	5	-	-	PUNCT
ejde-320	79	6	gradient	gradient	NOUN
ejde-320	79	7	by	by	ADP
ejde-320	79	8	∇a	∇a	PROPN
ejde-320	79	9	=	=	PUNCT
ejde-320	79	10	b(x)∇	b(x)∇	NOUN
ejde-320	79	11	,	,	PUNCT
ejde-320	79	12	(	(	PUNCT
ejde-320	79	13	2.1	2.1	NUM
ejde-320	79	14	)	)	PUNCT
ejde-320	79	15	4	4	NUM
ejde-320	79	16	u.	u.	PROPN
ejde-320	79	17	hafeez	hafeez	PROPN
ejde-320	79	18	,	,	PUNCT
ejde-320	79	19	t.	t.	PROPN
ejde-320	79	20	lavier	lavier	PROPN
ejde-320	79	21	,	,	PUNCT
ejde-320	79	22	l.	l.	PROPN
ejde-320	79	23	williams	williams	PROPN
ejde-320	79	24	,	,	PUNCT
ejde-320	79	25	l.	l.	PROPN
ejde-320	79	26	korobenko	korobenko	PROPN
ejde-320	79	27	ejde-2021/82	ejde-2021/82	PROPN
ejde-320	79	28	and	and	CCONJ
ejde-320	79	29	the	the	DET
ejde-320	79	30	associated	associated	ADJ
ejde-320	79	31	degenerate	degenerate	ADJ
ejde-320	79	32	sobolev	sobolev	NOUN
ejde-320	79	33	space	space	NOUN
ejde-320	79	34	w	w	PROPN
ejde-320	79	35	1,2	1,2	NUM
ejde-320	79	36	a	a	DET
ejde-320	79	37	(	(	PUNCT
ejde-320	79	38	ω	ω	NOUN
ejde-320	79	39	)	)	PUNCT
ejde-320	79	40	to	to	PART
ejde-320	79	41	have	have	VERB
ejde-320	79	42	norm	norm	NOUN
ejde-320	79	43	‖v‖w	‖v‖w	NOUN
ejde-320	79	44	1,2	1,2	NUM
ejde-320	79	45	a	a	DET
ejde-320	79	46	≡	≡	PROPN
ejde-320	79	47	(	(	PUNCT
ejde-320	79	48	∫	∫	PROPN
ejde-320	79	49	ω	ω	PROPN
ejde-320	79	50	(	(	PUNCT
ejde-320	79	51	|v|2	|v|2	PROPN
ejde-320	79	52	+	+	PROPN
ejde-320	79	53	∇v	∇v	PROPN
ejde-320	79	54	·	·	SYM
ejde-320	79	55	a∇v	a∇v	NOUN
ejde-320	79	56	)	)	PUNCT
ejde-320	79	57	)	)	PUNCT
ejde-320	79	58	1/2	1/2	NUM
ejde-320	79	59	=	=	SYM
ejde-320	79	60	(	(	PUNCT
ejde-320	79	61	∫	∫	PROPN
ejde-320	79	62	ω	ω	PROPN
ejde-320	79	63	(	(	PUNCT
ejde-320	79	64	|v|2	|v|2	PROPN
ejde-320	79	65	+	+	CCONJ
ejde-320	79	66	|∇av|2	|∇av|2	NOUN
ejde-320	79	67	)	)	PUNCT
ejde-320	79	68	)	)	PUNCT
ejde-320	79	69	1/2	1/2	NUM
ejde-320	79	70	.	.	PUNCT
ejde-320	80	1	the	the	DET
ejde-320	80	2	space	space	NOUN
ejde-320	80	3	(	(	PUNCT
ejde-320	80	4	w	w	PROPN
ejde-320	80	5	1,2	1,2	NUM
ejde-320	80	6	a	a	PRON
ejde-320	80	7	)	)	PUNCT
ejde-320	80	8	0	0	NUM
ejde-320	80	9	(	(	PUNCT
ejde-320	80	10	ω	ω	NOUN
ejde-320	80	11	)	)	PUNCT
ejde-320	80	12	is	be	AUX
ejde-320	80	13	defined	define	VERB
ejde-320	80	14	as	as	ADP
ejde-320	80	15	the	the	DET
ejde-320	80	16	closure	closure	NOUN
ejde-320	80	17	in	in	ADP
ejde-320	80	18	w	w	PROPN
ejde-320	80	19	1,2	1,2	NUM
ejde-320	80	20	a	a	DET
ejde-320	80	21	(	(	PUNCT
ejde-320	80	22	ω	ω	NOUN
ejde-320	80	23	)	)	PUNCT
ejde-320	80	24	of	of	ADP
ejde-320	80	25	the	the	DET
ejde-320	80	26	subspace	subspace	NOUN
ejde-320	80	27	of	of	ADP
ejde-320	80	28	lipschitz	lipschitz	VERB
ejde-320	80	29	continuous	continuous	ADJ
ejde-320	80	30	functions	function	NOUN
ejde-320	80	31	with	with	ADP
ejde-320	80	32	compact	compact	ADJ
ejde-320	80	33	support	support	NOUN
ejde-320	80	34	in	in	ADP
ejde-320	80	35	ω	ω	PROPN
ejde-320	80	36	.	.	PUNCT
ejde-320	81	1	note	note	VERB
ejde-320	81	2	that	that	SCONJ
ejde-320	81	3	even	even	ADV
ejde-320	81	4	though	though	SCONJ
ejde-320	81	5	the	the	DET
ejde-320	81	6	definition	definition	NOUN
ejde-320	81	7	of	of	ADP
ejde-320	81	8	the	the	DET
ejde-320	81	9	a	a	DET
ejde-320	81	10	-	-	PUNCT
ejde-320	81	11	gradient	gradient	NOUN
ejde-320	81	12	depends	depend	VERB
ejde-320	81	13	on	on	ADP
ejde-320	81	14	the	the	DET
ejde-320	81	15	choice	choice	NOUN
ejde-320	81	16	of	of	ADP
ejde-320	81	17	the	the	DET
ejde-320	81	18	matrix	matrix	NOUN
ejde-320	81	19	b	b	NOUN
ejde-320	81	20	,	,	PUNCT
ejde-320	81	21	all	all	DET
ejde-320	81	22	these	these	DET
ejde-320	81	23	definitions	definition	NOUN
ejde-320	81	24	are	be	AUX
ejde-320	81	25	equivalent	equivalent	ADJ
ejde-320	81	26	,	,	PUNCT
ejde-320	81	27	and	and	CCONJ
ejde-320	81	28	associated	associated	ADJ
ejde-320	81	29	sobolev	sobolev	NOUN
ejde-320	81	30	spaces	space	NOUN
ejde-320	81	31	are	be	AUX
ejde-320	81	32	the	the	DET
ejde-320	81	33	same	same	ADJ
ejde-320	81	34	.	.	PUNCT
ejde-320	82	1	definition	definition	NOUN
ejde-320	82	2	2.1	2.1	NUM
ejde-320	82	3	.	.	PUNCT
ejde-320	83	1	given	give	VERB
ejde-320	83	2	u	u	NOUN
ejde-320	83	3	,	,	PUNCT
ejde-320	83	4	v	v	ADP
ejde-320	83	5	∈w	∈w	VERB
ejde-320	83	6	1,2	1,2	NUM
ejde-320	83	7	a	a	PRON
ejde-320	83	8	,	,	PUNCT
ejde-320	83	9	define	define	VERB
ejde-320	83	10	the	the	DET
ejde-320	83	11	inner	inner	ADJ
ejde-320	83	12	product	product	NOUN
ejde-320	83	13	on	on	ADP
ejde-320	83	14	the	the	DET
ejde-320	83	15	gradients	gradient	NOUN
ejde-320	83	16	of	of	ADP
ejde-320	83	17	u	u	NOUN
ejde-320	83	18	and	and	CCONJ
ejde-320	83	19	v	v	NOUN
ejde-320	83	20	to	to	PART
ejde-320	83	21	be	be	AUX
ejde-320	83	22	〈	〈	PROPN
ejde-320	83	23	∇u,∇v	∇u,∇v	PROPN
ejde-320	83	24	〉	〉	NOUN
ejde-320	83	25	:	:	PUNCT
ejde-320	84	1	=	=	PUNCT
ejde-320	84	2	∇u	∇u	ADJ
ejde-320	84	3	·	·	SYM
ejde-320	84	4	a∇v	a∇v	NOUN
ejde-320	84	5	=	=	SYM
ejde-320	84	6	∇utra∇v	∇utra∇v	PROPN
ejde-320	84	7	.	.	PUNCT
ejde-320	85	1	furthermore	furthermore	ADV
ejde-320	85	2	,	,	PUNCT
ejde-320	85	3	define	define	VERB
ejde-320	85	4	the	the	DET
ejde-320	85	5	a	a	DET
ejde-320	85	6	semi	semi	NOUN
ejde-320	85	7	-	-	NOUN
ejde-320	85	8	norm	norm	NOUN
ejde-320	85	9	of	of	ADP
ejde-320	85	10	∇u	∇u	PROPN
ejde-320	85	11	to	to	PART
ejde-320	85	12	be	be	AUX
ejde-320	85	13	[	[	X
ejde-320	85	14	∇u]2a	∇u]2a	NOUN
ejde-320	85	15	:	:	PUNCT
ejde-320	85	16	=	=	PUNCT
ejde-320	85	17	〈	〈	PROPN
ejde-320	85	18	∇u,∇u	∇u,∇u	PROPN
ejde-320	85	19	〉	〉	PROPN
ejde-320	85	20	.	.	PUNCT
ejde-320	86	1	2.1.2	2.1.2	NUM
ejde-320	86	2	.	.	NOUN
ejde-320	86	3	subunit	subunit	NOUN
ejde-320	86	4	metrics	metric	NOUN
ejde-320	86	5	.	.	PUNCT
ejde-320	87	1	we	we	PRON
ejde-320	87	2	now	now	ADV
ejde-320	87	3	define	define	VERB
ejde-320	87	4	subunit	subunit	NOUN
ejde-320	87	5	(	(	PUNCT
ejde-320	87	6	or	or	CCONJ
ejde-320	87	7	control	control	NOUN
ejde-320	87	8	,	,	PUNCT
ejde-320	87	9	or	or	CCONJ
ejde-320	87	10	carnot	carnot	NOUN
ejde-320	87	11	-	-	PUNCT
ejde-320	87	12	carathéodory	carathéodory	NOUN
ejde-320	87	13	)	)	PUNCT
ejde-320	87	14	metric	metric	NOUN
ejde-320	87	15	associated	associate	VERB
ejde-320	87	16	with	with	ADP
ejde-320	87	17	the	the	DET
ejde-320	87	18	operator	operator	NOUN
ejde-320	87	19	l	l	NOUN
ejde-320	87	20	=	=	SYM
ejde-320	87	21	∇	∇	X
ejde-320	87	22	·	·	PUNCT
ejde-320	87	23	a∇	a∇	PROPN
ejde-320	87	24	,	,	PUNCT
ejde-320	87	25	see	see	VERB
ejde-320	87	26	[	[	X
ejde-320	87	27	3	3	NUM
ejde-320	87	28	]	]	PUNCT
ejde-320	87	29	.	.	PUNCT
ejde-320	88	1	definition	definition	NOUN
ejde-320	88	2	2.2	2.2	NUM
ejde-320	88	3	.	.	PUNCT
ejde-320	89	1	a	a	DET
ejde-320	89	2	subunit	subunit	NOUN
ejde-320	89	3	curve	curve	NOUN
ejde-320	89	4	is	be	AUX
ejde-320	89	5	a	a	DET
ejde-320	89	6	lipschitz	lipschitz	NOUN
ejde-320	89	7	curve	curve	NOUN
ejde-320	89	8	γ	γ	NOUN
ejde-320	89	9	:	:	PUNCT
ejde-320	90	1	[	[	X
ejde-320	90	2	0	0	NUM
ejde-320	90	3	,	,	PUNCT
ejde-320	90	4	r]→	r]→	X
ejde-320	90	5	ω	ω	NUM
ejde-320	90	6	such	such	ADJ
ejde-320	90	7	that	that	SCONJ
ejde-320	90	8	(	(	PUNCT
ejde-320	90	9	γ′(t)ξ)2	γ′(t)ξ)2	ADJ
ejde-320	90	10	≤	≤	NUM
ejde-320	90	11	ξ′a(γ(t))ξ	ξ′a(γ(t))ξ	NOUN
ejde-320	90	12	,	,	PUNCT
ejde-320	90	13	a.e	a.e	PROPN
ejde-320	90	14	.	.	PROPN
ejde-320	90	15	t	t	PROPN
ejde-320	90	16	∈	∈	PROPN
ejde-320	91	1	[	[	X
ejde-320	91	2	0	0	NUM
ejde-320	91	3	,	,	PUNCT
ejde-320	91	4	r	r	NOUN
ejde-320	91	5	]	]	X
ejde-320	91	6	,	,	PUNCT
ejde-320	91	7	∀ξ	∀ξ	X
ejde-320	91	8	∈	∈	PROPN
ejde-320	91	9	rn	rn	NOUN
ejde-320	91	10	.	.	PUNCT
ejde-320	92	1	a	a	DET
ejde-320	92	2	subunit	subunit	NOUN
ejde-320	92	3	metric	metric	NOUN
ejde-320	92	4	is	be	AUX
ejde-320	92	5	defined	define	VERB
ejde-320	92	6	by	by	ADP
ejde-320	92	7	d(x	d(x	PROPN
ejde-320	92	8	,	,	PUNCT
ejde-320	92	9	y	y	NOUN
ejde-320	92	10	)	)	PUNCT
ejde-320	93	1	=	=	SYM
ejde-320	93	2	inf{r	inf{r	PROPN
ejde-320	93	3	>	>	X
ejde-320	93	4	0	0	NUM
ejde-320	93	5	:	:	PUNCT
ejde-320	93	6	γ(0	γ(0	PROPN
ejde-320	93	7	)	)	PUNCT
ejde-320	93	8	=	=	SYM
ejde-320	93	9	x	x	NOUN
ejde-320	93	10	,	,	PUNCT
ejde-320	93	11	γ(r	γ(r	PROPN
ejde-320	93	12	)	)	PUNCT
ejde-320	94	1	=	=	SYM
ejde-320	94	2	y	y	PROPN
ejde-320	94	3	,	,	PUNCT
ejde-320	94	4	γ	γ	PROPN
ejde-320	94	5	is	be	AUX
ejde-320	94	6	a	a	DET
ejde-320	94	7	subunit	subunit	NOUN
ejde-320	94	8	in	in	ADP
ejde-320	94	9	ω	ω	NUM
ejde-320	94	10	}	}	PUNCT
ejde-320	94	11	,	,	PUNCT
ejde-320	94	12	and	and	CCONJ
ejde-320	94	13	the	the	DET
ejde-320	94	14	subunit	subunit	NOUN
ejde-320	94	15	ball	ball	NOUN
ejde-320	94	16	centered	center	VERB
ejde-320	94	17	at	at	ADP
ejde-320	94	18	x	x	PUNCT
ejde-320	94	19	with	with	SCONJ
ejde-320	94	20	radius	radius	NOUN
ejde-320	94	21	r	r	NOUN
ejde-320	94	22	is	be	AUX
ejde-320	94	23	b(x	b(x	VERB
ejde-320	94	24	,	,	PUNCT
ejde-320	94	25	r	r	NOUN
ejde-320	94	26	)	)	PUNCT
ejde-320	94	27	=	=	SYM
ejde-320	94	28	{	{	PUNCT
ejde-320	94	29	y	y	PROPN
ejde-320	94	30	∈	∈	PROPN
ejde-320	94	31	ω	ω	NOUN
ejde-320	94	32	:	:	PUNCT
ejde-320	94	33	d(x	d(x	PROPN
ejde-320	94	34	,	,	PUNCT
ejde-320	94	35	y	y	NOUN
ejde-320	94	36	)	)	PUNCT
ejde-320	94	37	<	<	X
ejde-320	94	38	r	r	X
ejde-320	94	39	}	}	PUNCT
ejde-320	94	40	.	.	PUNCT
ejde-320	95	1	franchi	franchi	PROPN
ejde-320	95	2	and	and	CCONJ
ejde-320	95	3	lanconelli	lanconelli	PROPN
ejde-320	95	4	[	[	X
ejde-320	95	5	3	3	X
ejde-320	95	6	]	]	PUNCT
ejde-320	95	7	were	be	AUX
ejde-320	95	8	the	the	DET
ejde-320	95	9	first	first	ADJ
ejde-320	95	10	to	to	PART
ejde-320	95	11	realize	realize	VERB
ejde-320	95	12	that	that	SCONJ
ejde-320	95	13	the	the	DET
ejde-320	95	14	classical	classical	ADJ
ejde-320	95	15	moser	moser	PROPN
ejde-320	95	16	iteration	iteration	PROPN
ejde-320	95	17	scheme	scheme	NOUN
ejde-320	95	18	can	can	AUX
ejde-320	95	19	be	be	AUX
ejde-320	95	20	adapted	adapt	VERB
ejde-320	95	21	to	to	ADP
ejde-320	95	22	certain	certain	ADJ
ejde-320	95	23	degenerate	degenerate	ADJ
ejde-320	95	24	operators	operator	NOUN
ejde-320	95	25	(	(	PUNCT
ejde-320	95	26	with	with	ADP
ejde-320	95	27	one	one	NUM
ejde-320	95	28	fixed	fix	VERB
ejde-320	95	29	constant	constant	ADJ
ejde-320	95	30	eigenvalue	eigenvalue	NOUN
ejde-320	95	31	)	)	PUNCT
ejde-320	95	32	provided	provide	VERB
ejde-320	95	33	the	the	DET
ejde-320	95	34	euclidean	euclidean	PROPN
ejde-320	95	35	rn	rn	PROPN
ejde-320	95	36	is	be	AUX
ejde-320	95	37	replaced	replace	VERB
ejde-320	95	38	by	by	ADP
ejde-320	95	39	the	the	DET
ejde-320	95	40	subunit	subunit	NOUN
ejde-320	95	41	metric	metric	ADJ
ejde-320	95	42	space	space	NOUN
ejde-320	95	43	.	.	PUNCT
ejde-320	96	1	2.2	2.2	NUM
ejde-320	96	2	.	.	PUNCT
ejde-320	97	1	orlicz	orlicz	PROPN
ejde-320	97	2	spaces	space	VERB
ejde-320	97	3	.	.	PUNCT
ejde-320	98	1	as	as	SCONJ
ejde-320	98	2	mentioned	mention	VERB
ejde-320	98	3	in	in	ADP
ejde-320	98	4	the	the	DET
ejde-320	98	5	introduction	introduction	NOUN
ejde-320	98	6	,	,	PUNCT
ejde-320	98	7	we	we	PRON
ejde-320	98	8	will	will	AUX
ejde-320	98	9	work	work	VERB
ejde-320	98	10	with	with	ADP
ejde-320	98	11	orlicz	orlicz	ADJ
ejde-320	98	12	spaces	space	NOUN
ejde-320	98	13	,	,	PUNCT
ejde-320	98	14	which	which	PRON
ejde-320	98	15	can	can	AUX
ejde-320	98	16	be	be	AUX
ejde-320	98	17	seen	see	VERB
ejde-320	98	18	as	as	ADP
ejde-320	98	19	generalizations	generalization	NOUN
ejde-320	98	20	of	of	ADP
ejde-320	98	21	lebesgue	lebesgue	ADJ
ejde-320	98	22	spaces	space	NOUN
ejde-320	98	23	:	:	PUNCT
ejde-320	98	24	power	power	NOUN
ejde-320	98	25	functions	function	NOUN
ejde-320	98	26	used	use	VERB
ejde-320	98	27	do	do	AUX
ejde-320	98	28	define	define	VERB
ejde-320	98	29	lebesgue	lebesgue	NOUN
ejde-320	98	30	spaces	space	NOUN
ejde-320	98	31	are	be	AUX
ejde-320	98	32	replaced	replace	VERB
ejde-320	98	33	by	by	ADP
ejde-320	98	34	more	more	ADJ
ejde-320	98	35	general	general	ADJ
ejde-320	98	36	young	young	ADJ
ejde-320	98	37	functions	function	NOUN
ejde-320	98	38	.	.	PUNCT
ejde-320	99	1	the	the	DET
ejde-320	99	2	material	material	NOUN
ejde-320	99	3	below	below	ADV
ejde-320	99	4	is	be	AUX
ejde-320	99	5	taken	take	VERB
ejde-320	99	6	from	from	ADP
ejde-320	99	7	[	[	X
ejde-320	99	8	8	8	NUM
ejde-320	99	9	]	]	PUNCT
ejde-320	99	10	.	.	PUNCT
ejde-320	100	1	definition	definition	NOUN
ejde-320	100	2	2.3	2.3	NUM
ejde-320	100	3	(	(	PUNCT
ejde-320	100	4	[	[	X
ejde-320	100	5	8	8	NUM
ejde-320	100	6	]	]	NUM
ejde-320	100	7	)	)	PUNCT
ejde-320	100	8	.	.	PUNCT
ejde-320	101	1	a	a	DET
ejde-320	101	2	function	function	NOUN
ejde-320	101	3	θ	θ	NOUN
ejde-320	101	4	:	:	PUNCT
ejde-320	102	1	r→	r→	PROPN
ejde-320	103	1	[	[	X
ejde-320	103	2	0,∞	0,∞	X
ejde-320	103	3	]	]	X
ejde-320	103	4	is	be	AUX
ejde-320	103	5	a	a	DET
ejde-320	103	6	young	young	ADJ
ejde-320	103	7	function	function	NOUN
ejde-320	103	8	if	if	SCONJ
ejde-320	103	9	(	(	PUNCT
ejde-320	103	10	1	1	X
ejde-320	103	11	)	)	PUNCT
ejde-320	103	12	θ	θ	PROPN
ejde-320	103	13	is	be	AUX
ejde-320	103	14	a	a	DET
ejde-320	103	15	convex	convex	NOUN
ejde-320	103	16	,	,	PUNCT
ejde-320	103	17	lower	low	ADJ
ejde-320	103	18	semicontinuous	semicontinuous	ADJ
ejde-320	103	19	,	,	PUNCT
ejde-320	103	20	[	[	X
ejde-320	103	21	0,∞]-valued	0,∞]-value	VERB
ejde-320	103	22	function	function	NOUN
ejde-320	103	23	on	on	ADP
ejde-320	103	24	r.	r.	PROPN
ejde-320	103	25	(	(	PUNCT
ejde-320	103	26	2	2	NUM
ejde-320	103	27	)	)	PUNCT
ejde-320	103	28	θ	θ	NOUN
ejde-320	103	29	is	be	AUX
ejde-320	103	30	even	even	ADV
ejde-320	103	31	and	and	CCONJ
ejde-320	103	32	θ(0	θ(0	PROPN
ejde-320	103	33	)	)	PUNCT
ejde-320	103	34	=	=	SYM
ejde-320	104	1	0	0	X
ejde-320	104	2	.	.	PUNCT
ejde-320	105	1	(	(	PUNCT
ejde-320	105	2	3	3	X
ejde-320	105	3	)	)	PUNCT
ejde-320	105	4	θ	θ	PROPN
ejde-320	105	5	is	be	AUX
ejde-320	105	6	non	non	ADJ
ejde-320	105	7	-	-	ADJ
ejde-320	105	8	trivial	trivial	ADJ
ejde-320	105	9	,	,	PUNCT
ejde-320	105	10	i.e.	i.e.	X
ejde-320	105	11	it	it	PRON
ejde-320	105	12	is	be	AUX
ejde-320	105	13	different	different	ADJ
ejde-320	105	14	from	from	ADP
ejde-320	105	15	the	the	DET
ejde-320	105	16	constant	constant	ADJ
ejde-320	105	17	function	function	NOUN
ejde-320	105	18	θ(s	θ(s	NOUN
ejde-320	105	19	)	)	PUNCT
ejde-320	106	1	=	=	SYM
ejde-320	106	2	0	0	NUM
ejde-320	107	1	for	for	ADP
ejde-320	107	2	s	s	PROPN
ejde-320	107	3	∈	∈	PROPN
ejde-320	107	4	r.	r.	PROPN
ejde-320	107	5	note	note	PROPN
ejde-320	107	6	that	that	SCONJ
ejde-320	107	7	properties	property	NOUN
ejde-320	107	8	(	(	PUNCT
ejde-320	107	9	1	1	NUM
ejde-320	107	10	)	)	PUNCT
ejde-320	107	11	and	and	CCONJ
ejde-320	107	12	(	(	PUNCT
ejde-320	107	13	2	2	X
ejde-320	107	14	)	)	PUNCT
ejde-320	107	15	imply	imply	VERB
ejde-320	107	16	that	that	SCONJ
ejde-320	107	17	any	any	DET
ejde-320	107	18	young	young	ADJ
ejde-320	107	19	function	function	NOUN
ejde-320	107	20	is	be	AUX
ejde-320	107	21	non	non	ADJ
ejde-320	107	22	-	-	ADJ
ejde-320	107	23	decreasing	decrease	VERB
ejde-320	107	24	on	on	ADP
ejde-320	107	25	[	[	X
ejde-320	107	26	0,∞	0,∞	NOUN
ejde-320	107	27	)	)	PUNCT
ejde-320	107	28	.	.	PUNCT
ejde-320	108	1	definition	definition	NOUN
ejde-320	108	2	2.4	2.4	NUM
ejde-320	108	3	(	(	PUNCT
ejde-320	108	4	[	[	X
ejde-320	108	5	8	8	NUM
ejde-320	108	6	]	]	NUM
ejde-320	108	7	)	)	PUNCT
ejde-320	108	8	.	.	PUNCT
ejde-320	109	1	given	give	VERB
ejde-320	109	2	a	a	DET
ejde-320	109	3	young	young	ADJ
ejde-320	109	4	function	function	NOUN
ejde-320	109	5	,	,	PUNCT
ejde-320	109	6	θ	θ	PROPN
ejde-320	109	7	,	,	PUNCT
ejde-320	109	8	the	the	DET
ejde-320	109	9	convex	convex	NOUN
ejde-320	109	10	conjugate	conjugate	NOUN
ejde-320	109	11	of	of	ADP
ejde-320	109	12	θ	θ	PROPN
ejde-320	109	13	,	,	PUNCT
ejde-320	109	14	is	be	AUX
ejde-320	109	15	defined	define	VERB
ejde-320	109	16	as	as	ADP
ejde-320	109	17	θ̃	θ̃	NOUN
ejde-320	109	18	=	=	PUNCT
ejde-320	109	19	sup	sup	NOUN
ejde-320	109	20	s∈r	s∈r	NOUN
ejde-320	109	21	{	{	PUNCT
ejde-320	109	22	st−	st−	X
ejde-320	109	23	θ(s	θ(s	NOUN
ejde-320	109	24	)	)	PUNCT
ejde-320	109	25	}	}	PUNCT
ejde-320	109	26	∈	∈	PROPN
ejde-320	110	1	[	[	X
ejde-320	110	2	0,∞	0,∞	X
ejde-320	110	3	]	]	PUNCT
ejde-320	110	4	for	for	ADP
ejde-320	110	5	t	t	PROPN
ejde-320	110	6	∈	∈	PROPN
ejde-320	110	7	r.	r.	PROPN
ejde-320	110	8	we	we	PRON
ejde-320	110	9	next	next	ADV
ejde-320	110	10	define	define	VERB
ejde-320	110	11	the	the	DET
ejde-320	110	12	luxembourg	luxembourg	PROPN
ejde-320	110	13	norm	norm	NOUN
ejde-320	110	14	,	,	PUNCT
ejde-320	110	15	which	which	PRON
ejde-320	110	16	in	in	ADP
ejde-320	110	17	turn	turn	NOUN
ejde-320	110	18	leads	lead	VERB
ejde-320	110	19	to	to	ADP
ejde-320	110	20	the	the	DET
ejde-320	110	21	definition	definition	NOUN
ejde-320	110	22	of	of	ADP
ejde-320	110	23	an	an	DET
ejde-320	110	24	orlicz	orlicz	ADJ
ejde-320	110	25	space	space	NOUN
ejde-320	110	26	.	.	PUNCT
ejde-320	111	1	ejde-2021/82	ejde-2021/82	ADJ
ejde-320	111	2	orlicz	orlicz	NUM
ejde-320	111	3	-	-	PUNCT
ejde-320	111	4	sobolev	sobolev	NOUN
ejde-320	111	5	inequalities	inequality	NOUN
ejde-320	111	6	and	and	CCONJ
ejde-320	111	7	the	the	DET
ejde-320	111	8	dirichlet	dirichlet	PROPN
ejde-320	111	9	problem	problem	NOUN
ejde-320	111	10	5	5	NUM
ejde-320	111	11	definition	definition	NOUN
ejde-320	111	12	2.5	2.5	NUM
ejde-320	111	13	.	.	PUNCT
ejde-320	112	1	let	let	VERB
ejde-320	112	2	θ	θ	NOUN
ejde-320	112	3	be	be	AUX
ejde-320	112	4	a	a	DET
ejde-320	112	5	young	young	ADJ
ejde-320	112	6	function	function	NOUN
ejde-320	112	7	,	,	PUNCT
ejde-320	112	8	and	and	CCONJ
ejde-320	112	9	ω	ω	NUM
ejde-320	112	10	be	be	AUX
ejde-320	112	11	a	a	DET
ejde-320	112	12	space	space	NOUN
ejde-320	112	13	with	with	ADP
ejde-320	112	14	a	a	DET
ejde-320	112	15	σ	σ	NOUN
ejde-320	112	16	-	-	PUNCT
ejde-320	112	17	field	field	NOUN
ejde-320	112	18	and	and	CCONJ
ejde-320	112	19	a	a	DET
ejde-320	112	20	σ	σ	NOUN
ejde-320	112	21	-	-	ADJ
ejde-320	112	22	finite	finite	ADJ
ejde-320	112	23	positive	positive	ADJ
ejde-320	112	24	measure	measure	NOUN
ejde-320	112	25	µ.	µ.	NOUN
ejde-320	112	26	for	for	ADP
ejde-320	112	27	any	any	DET
ejde-320	112	28	measurable	measurable	ADJ
ejde-320	112	29	function	function	NOUN
ejde-320	112	30	on	on	ADP
ejde-320	112	31	ω	ω	NUM
ejde-320	112	32	we	we	PRON
ejde-320	112	33	define	define	VERB
ejde-320	112	34	the	the	DET
ejde-320	112	35	luxembourg	luxembourg	PROPN
ejde-320	112	36	norm	norm	NOUN
ejde-320	112	37	as	as	ADP
ejde-320	112	38	‖f‖lθ	‖f‖lθ	PROPN
ejde-320	112	39	=	=	SYM
ejde-320	112	40	‖f‖lθ(ω	‖f‖lθ(ω	PROPN
ejde-320	112	41	)	)	PUNCT
ejde-320	112	42	:	:	PUNCT
ejde-320	113	1	=	=	SYM
ejde-320	113	2	inf	inf	PROPN
ejde-320	113	3	{	{	PUNCT
ejde-320	113	4	k	k	X
ejde-320	113	5	>	>	X
ejde-320	113	6	0	0	NUM
ejde-320	113	7	:	:	PUNCT
ejde-320	113	8	∫	∫	PROPN
ejde-320	113	9	ω	ω	NUM
ejde-320	113	10	θ(f	θ(f	PROPN
ejde-320	113	11	/	/	SYM
ejde-320	113	12	k)dµ	k)dµ	PROPN
ejde-320	113	13	≤	≤	NOUN
ejde-320	113	14	1	1	NUM
ejde-320	113	15	}	}	PUNCT
ejde-320	113	16	,	,	PUNCT
ejde-320	113	17	(	(	PUNCT
ejde-320	113	18	2.2	2.2	NUM
ejde-320	113	19	)	)	PUNCT
ejde-320	113	20	where	where	SCONJ
ejde-320	113	21	inf(∅	inf(∅	ADP
ejde-320	113	22	)	)	PUNCT
ejde-320	113	23	=	=	PUNCT
ejde-320	114	1	+	+	NUM
ejde-320	114	2	∞.	∞.	PROPN
ejde-320	114	3	for	for	ADP
ejde-320	114	4	a	a	DET
ejde-320	114	5	young	young	ADJ
ejde-320	114	6	function	function	NOUN
ejde-320	114	7	θ	θ	PROPN
ejde-320	114	8	,	,	PUNCT
ejde-320	114	9	the	the	DET
ejde-320	114	10	associated	associated	ADJ
ejde-320	114	11	orlicz	orlicz	ADJ
ejde-320	114	12	space	space	NOUN
ejde-320	114	13	is	be	AUX
ejde-320	114	14	lθ(ω	lθ(ω	NUM
ejde-320	114	15	)	)	PUNCT
ejde-320	114	16	=	=	PRON
ejde-320	115	1	{	{	PUNCT
ejde-320	115	2	f	f	PROPN
ejde-320	115	3	measurable	measurable	NOUN
ejde-320	115	4	:	:	PUNCT
ejde-320	115	5	‖f‖lθ	‖f‖lθ	PROPN
ejde-320	115	6	<	<	X
ejde-320	115	7	∞	∞	NUM
ejde-320	115	8	}	}	PUNCT
ejde-320	115	9	.	.	PUNCT
ejde-320	116	1	the	the	DET
ejde-320	116	2	following	follow	VERB
ejde-320	116	3	proposition	proposition	NOUN
ejde-320	116	4	follows	follow	VERB
ejde-320	116	5	directly	directly	ADV
ejde-320	116	6	from	from	ADP
ejde-320	116	7	(	(	PUNCT
ejde-320	116	8	2.2	2.2	NUM
ejde-320	116	9	)	)	PUNCT
ejde-320	116	10	.	.	PUNCT
ejde-320	117	1	proposition	proposition	NOUN
ejde-320	117	2	2.6	2.6	NUM
ejde-320	117	3	.	.	PUNCT
ejde-320	118	1	let	let	VERB
ejde-320	118	2	θ1	θ1	PROPN
ejde-320	118	3	and	and	CCONJ
ejde-320	118	4	θ2	θ2	PROPN
ejde-320	118	5	be	be	AUX
ejde-320	118	6	two	two	NUM
ejde-320	118	7	young	young	ADJ
ejde-320	118	8	functions	function	NOUN
ejde-320	118	9	such	such	ADJ
ejde-320	118	10	that	that	DET
ejde-320	118	11	θ1(t	θ1(t	PROPN
ejde-320	118	12	)	)	PUNCT
ejde-320	118	13	≤	≤	NUM
ejde-320	118	14	θ2(t	θ2(t	PROPN
ejde-320	118	15	)	)	PUNCT
ejde-320	118	16	for	for	ADP
ejde-320	118	17	all	all	DET
ejde-320	118	18	t	t	PROPN
ejde-320	118	19	≥	≥	NOUN
ejde-320	118	20	0	0	NUM
ejde-320	118	21	.	.	PUNCT
ejde-320	119	1	then	then	ADV
ejde-320	119	2	lθ2	lθ2	VERB
ejde-320	119	3	⊆	⊆	NUM
ejde-320	119	4	lθ1	lθ1	NOUN
ejde-320	119	5	,	,	PUNCT
ejde-320	119	6	in	in	ADP
ejde-320	119	7	particular	particular	ADJ
ejde-320	119	8	,	,	PUNCT
ejde-320	119	9	for	for	SCONJ
ejde-320	119	10	every	every	DET
ejde-320	119	11	f	f	PROPN
ejde-320	119	12	∈	∈	PROPN
ejde-320	119	13	lθ2	lθ2	ADJ
ejde-320	119	14	it	it	PRON
ejde-320	119	15	holds	hold	VERB
ejde-320	119	16	‖f‖lθ1	‖f‖lθ1	PROPN
ejde-320	119	17	≤	≤	X
ejde-320	119	18	‖f‖lθ2	‖f‖lθ2	VERB
ejde-320	119	19	.	.	PUNCT
ejde-320	120	1	an	an	DET
ejde-320	120	2	equivalent	equivalent	ADJ
ejde-320	120	3	norm	norm	NOUN
ejde-320	120	4	on	on	ADP
ejde-320	120	5	lθ	lθ	NOUN
ejde-320	120	6	given	give	VERB
ejde-320	120	7	below	below	ADV
ejde-320	120	8	is	be	AUX
ejde-320	120	9	based	base	VERB
ejde-320	120	10	on	on	ADP
ejde-320	120	11	duality	duality	NOUN
ejde-320	120	12	and	and	CCONJ
ejde-320	120	13	will	will	AUX
ejde-320	120	14	be	be	AUX
ejde-320	120	15	used	use	VERB
ejde-320	120	16	in	in	ADP
ejde-320	120	17	some	some	PRON
ejde-320	120	18	of	of	ADP
ejde-320	120	19	the	the	DET
ejde-320	120	20	proofs	proof	NOUN
ejde-320	120	21	contained	contain	VERB
ejde-320	120	22	in	in	ADP
ejde-320	120	23	this	this	DET
ejde-320	120	24	paper	paper	NOUN
ejde-320	120	25	.	.	PUNCT
ejde-320	121	1	definition	definition	NOUN
ejde-320	121	2	2.7	2.7	NUM
ejde-320	121	3	.	.	PUNCT
ejde-320	122	1	the	the	DET
ejde-320	122	2	orlicz	orlicz	ADJ
ejde-320	122	3	norm	norm	NOUN
ejde-320	122	4	of	of	ADP
ejde-320	122	5	a	a	DET
ejde-320	122	6	measurable	measurable	ADJ
ejde-320	122	7	function	function	NOUN
ejde-320	122	8	f	f	PROPN
ejde-320	122	9	is	be	AUX
ejde-320	122	10	defined	define	VERB
ejde-320	122	11	as	as	ADP
ejde-320	122	12	|f	|f	PRON
ejde-320	122	13	|lθ	|lθ	NUM
ejde-320	122	14	:	:	PUNCT
ejde-320	122	15	=	=	SYM
ejde-320	122	16	sup	sup	X
ejde-320	122	17	{	{	PUNCT
ejde-320	122	18	∫	∫	PROPN
ejde-320	122	19	ω	ω	PROPN
ejde-320	122	20	fg	fg	PROPN
ejde-320	122	21	dµ	dµ	PROPN
ejde-320	122	22	:	:	PUNCT
ejde-320	122	23	g	g	PROPN
ejde-320	122	24	∈	∈	PROPN
ejde-320	122	25	lθ̃	lθ̃	NOUN
ejde-320	122	26	and	and	CCONJ
ejde-320	122	27	‖g‖lθ̃	‖g‖lθ̃	NOUN
ejde-320	122	28	≤	≤	NOUN
ejde-320	122	29	1	1	NUM
ejde-320	122	30	}	}	PUNCT
ejde-320	122	31	=	=	PUNCT
ejde-320	122	32	sup	sup	NOUN
ejde-320	122	33	{	{	PUNCT
ejde-320	122	34	∫	∫	PROPN
ejde-320	122	35	ω	ω	PROPN
ejde-320	122	36	fg	fg	PROPN
ejde-320	122	37	dµ	dµ	PROPN
ejde-320	122	38	:	:	PUNCT
ejde-320	122	39	g	g	PROPN
ejde-320	122	40	∈	∈	PROPN
ejde-320	122	41	lθ̃	lθ̃	NOUN
ejde-320	122	42	and	and	CCONJ
ejde-320	122	43	∫	∫	PROPN
ejde-320	122	44	ω	ω	NUM
ejde-320	122	45	θ̃(g)dµ	θ̃(g)dµ	PROPN
ejde-320	122	46	≤	≤	ADV
ejde-320	122	47	1	1	NUM
ejde-320	122	48	}	}	PUNCT
ejde-320	122	49	.	.	PUNCT
ejde-320	123	1	the	the	DET
ejde-320	123	2	orclicz	orclicz	ADJ
ejde-320	123	3	norm	norm	NOUN
ejde-320	123	4	and	and	CCONJ
ejde-320	123	5	the	the	DET
ejde-320	123	6	luxembourg	luxembourg	PROPN
ejde-320	123	7	norm	norm	NOUN
ejde-320	123	8	are	be	AUX
ejde-320	123	9	equivalent	equivalent	ADJ
ejde-320	123	10	,	,	PUNCT
ejde-320	123	11	more	more	ADV
ejde-320	123	12	precisely	precisely	ADV
ejde-320	123	13	,	,	PUNCT
ejde-320	123	14	‖f‖lθ	‖f‖lθ	PROPN
ejde-320	123	15	≤	≤	NUM
ejde-320	123	16	|f	|f	ADP
ejde-320	123	17	|lθ	|lθ	X
ejde-320	123	18	≤	≤	NUM
ejde-320	123	19	2‖f‖lθ	2‖f‖lθ	NUM
ejde-320	123	20	.	.	PUNCT
ejde-320	124	1	(	(	PUNCT
ejde-320	124	2	2.3	2.3	NUM
ejde-320	124	3	)	)	PUNCT
ejde-320	124	4	proposition	proposition	NOUN
ejde-320	124	5	2.8	2.8	NUM
ejde-320	124	6	(	(	PUNCT
ejde-320	124	7	hölder	hölder	NOUN
ejde-320	124	8	inequality	inequality	NOUN
ejde-320	124	9	[	[	X
ejde-320	124	10	8	8	NUM
ejde-320	124	11	]	]	NUM
ejde-320	124	12	)	)	PUNCT
ejde-320	124	13	.	.	PUNCT
ejde-320	125	1	given	give	VERB
ejde-320	125	2	a	a	DET
ejde-320	125	3	young	young	ADJ
ejde-320	125	4	function	function	NOUN
ejde-320	125	5	θ	θ	PROPN
ejde-320	125	6	,	,	PUNCT
ejde-320	125	7	for	for	ADP
ejde-320	125	8	any	any	DET
ejde-320	125	9	f	f	PROPN
ejde-320	125	10	∈	∈	PROPN
ejde-320	125	11	lθ(ω	lθ(ω	PROPN
ejde-320	125	12	)	)	PUNCT
ejde-320	125	13	and	and	CCONJ
ejde-320	125	14	g	g	PROPN
ejde-320	125	15	∈	∈	PROPN
ejde-320	125	16	lθ̃(ω	lθ̃(ω	NOUN
ejde-320	125	17	)	)	PUNCT
ejde-320	125	18	it	it	PRON
ejde-320	125	19	holds	hold	VERB
ejde-320	125	20	∫	∫	PROPN
ejde-320	125	21	|fg|dµ	|fg|dµ	PUNCT
ejde-320	125	22	≤	≤	NUM
ejde-320	125	23	2‖f‖θ‖g‖θ̃.	2‖f‖θ‖g‖θ̃.	NUM
ejde-320	125	24	(	(	PUNCT
ejde-320	125	25	2.4	2.4	NUM
ejde-320	125	26	)	)	PUNCT
ejde-320	125	27	in	in	ADP
ejde-320	125	28	particular	particular	ADJ
ejde-320	125	29	,	,	PUNCT
ejde-320	125	30	fg	fg	PROPN
ejde-320	125	31	∈	∈	PROPN
ejde-320	125	32	l1	l1	PROPN
ejde-320	125	33	.	.	PUNCT
ejde-320	126	1	finally	finally	ADV
ejde-320	126	2	,	,	PUNCT
ejde-320	126	3	we	we	PRON
ejde-320	126	4	define	define	VERB
ejde-320	126	5	a	a	DET
ejde-320	126	6	particular	particular	ADJ
ejde-320	126	7	family	family	NOUN
ejde-320	126	8	of	of	ADP
ejde-320	126	9	orlicz	orlicz	PROPN
ejde-320	126	10	functions	function	NOUN
ejde-320	126	11	first	first	ADV
ejde-320	126	12	introduced	introduce	VERB
ejde-320	126	13	in	in	ADP
ejde-320	126	14	[	[	X
ejde-320	126	15	7	7	NUM
ejde-320	126	16	]	]	PUNCT
ejde-320	126	17	and	and	CCONJ
ejde-320	126	18	employed	employ	VERB
ejde-320	126	19	in	in	ADP
ejde-320	126	20	the	the	DET
ejde-320	126	21	adaptation	adaptation	NOUN
ejde-320	126	22	of	of	ADP
ejde-320	126	23	degiorgi	degiorgi	PROPN
ejde-320	126	24	iteration	iteration	NOUN
ejde-320	126	25	in	in	ADP
ejde-320	126	26	the	the	DET
ejde-320	126	27	proof	proof	NOUN
ejde-320	126	28	of	of	ADP
ejde-320	126	29	theorem	theorem	ADJ
ejde-320	126	30	1.1	1.1	NUM
ejde-320	126	31	definition	definition	NOUN
ejde-320	126	32	2.9	2.9	NUM
ejde-320	126	33	.	.	PUNCT
ejde-320	127	1	the	the	DET
ejde-320	127	2	family	family	NOUN
ejde-320	127	3	of	of	ADP
ejde-320	127	4	orlicz	orlicz	ADJ
ejde-320	127	5	bump	bump	NOUN
ejde-320	127	6	functions	function	NOUN
ejde-320	127	7	{	{	PUNCT
ejde-320	127	8	φn}n>1	φn}n>1	PROPN
ejde-320	127	9	is	be	AUX
ejde-320	127	10	given	give	VERB
ejde-320	127	11	by	by	ADP
ejde-320	127	12	φn	φn	PROPN
ejde-320	127	13	(	(	PUNCT
ejde-320	127	14	t	t	NOUN
ejde-320	127	15	)	)	PUNCT
ejde-320	127	16	=	=	PRON
ejde-320	127	17	{	{	PUNCT
ejde-320	127	18	t(ln	t(ln	PROPN
ejde-320	127	19	t)n	t)n	NOUN
ejde-320	127	20	,	,	PUNCT
ejde-320	127	21	if	if	SCONJ
ejde-320	127	22	t	t	PROPN
ejde-320	127	23	≥	≥	X
ejde-320	127	24	e	e	NOUN
ejde-320	127	25	=	=	PUNCT
ejde-320	127	26	en	en	X
ejde-320	127	27	=	=	PUNCT
ejde-320	127	28	e2n	e2n	X
ejde-320	127	29	;	;	PUNCT
ejde-320	127	30	(	(	PUNCT
ejde-320	127	31	lne)n	lne)n	PROPN
ejde-320	127	32	t	t	PROPN
ejde-320	127	33	,	,	PUNCT
ejde-320	127	34	if	if	SCONJ
ejde-320	127	35	0	0	NUM
ejde-320	127	36	≤	≤	NUM
ejde-320	127	37	t	t	NOUN
ejde-320	127	38	≤	≤	NUM
ejde-320	127	39	e	e	NOUN
ejde-320	127	40	=	=	SYM
ejde-320	127	41	en	en	X
ejde-320	127	42	=	=	PUNCT
ejde-320	127	43	e2n	e2n	X
ejde-320	127	44	.	.	PUNCT
ejde-320	128	1	3	3	X
ejde-320	128	2	.	.	X
ejde-320	128	3	sufficiency	sufficiency	NOUN
ejde-320	128	4	this	this	DET
ejde-320	128	5	section	section	NOUN
ejde-320	128	6	is	be	AUX
ejde-320	128	7	devoted	devote	VERB
ejde-320	128	8	to	to	ADP
ejde-320	128	9	the	the	DET
ejde-320	128	10	proof	proof	NOUN
ejde-320	128	11	of	of	ADP
ejde-320	128	12	theorem	theorem	ADJ
ejde-320	128	13	1.1	1.1	NUM
ejde-320	128	14	.	.	PUNCT
ejde-320	129	1	first	first	ADV
ejde-320	129	2	we	we	PRON
ejde-320	129	3	show	show	VERB
ejde-320	129	4	existence	existence	NOUN
ejde-320	129	5	and	and	CCONJ
ejde-320	129	6	uniqueness	uniqueness	NOUN
ejde-320	129	7	of	of	ADP
ejde-320	129	8	weak	weak	ADJ
ejde-320	129	9	solutions	solution	NOUN
ejde-320	129	10	and	and	CCONJ
ejde-320	129	11	then	then	ADV
ejde-320	129	12	establish	establish	VERB
ejde-320	129	13	the	the	DET
ejde-320	129	14	quantitative	quantitative	ADJ
ejde-320	129	15	boundedness	boundedness	NOUN
ejde-320	129	16	estimate	estimate	NOUN
ejde-320	129	17	.	.	PUNCT
ejde-320	130	1	6	6	NUM
ejde-320	130	2	u.	u.	PROPN
ejde-320	130	3	hafeez	hafeez	PROPN
ejde-320	130	4	,	,	PUNCT
ejde-320	130	5	t.	t.	PROPN
ejde-320	130	6	lavier	lavier	PROPN
ejde-320	130	7	,	,	PUNCT
ejde-320	130	8	l.	l.	PROPN
ejde-320	130	9	williams	williams	PROPN
ejde-320	130	10	,	,	PUNCT
ejde-320	130	11	l.	l.	PROPN
ejde-320	130	12	korobenko	korobenko	PROPN
ejde-320	130	13	ejde-2021/82	ejde-2021/82	PROPN
ejde-320	130	14	3.1	3.1	NUM
ejde-320	130	15	.	.	PUNCT
ejde-320	131	1	existence	existence	NOUN
ejde-320	131	2	of	of	ADP
ejde-320	131	3	a	a	DET
ejde-320	131	4	unique	unique	ADJ
ejde-320	131	5	weak	weak	ADJ
ejde-320	131	6	solution	solution	NOUN
ejde-320	131	7	.	.	PUNCT
ejde-320	132	1	the	the	DET
ejde-320	132	2	proof	proof	NOUN
ejde-320	132	3	is	be	AUX
ejde-320	132	4	based	base	VERB
ejde-320	132	5	on	on	ADP
ejde-320	132	6	the	the	DET
ejde-320	132	7	laxmilgram	laxmilgram	NOUN
ejde-320	132	8	theorem	theorem	NOUN
ejde-320	132	9	applied	apply	VERB
ejde-320	132	10	to	to	ADP
ejde-320	132	11	the	the	DET
ejde-320	132	12	bilinear	bilinear	NOUN
ejde-320	132	13	form	form	NOUN
ejde-320	132	14	b[u	b[u	NOUN
ejde-320	132	15	,	,	PUNCT
ejde-320	132	16	v	v	NOUN
ejde-320	132	17	]	]	PUNCT
ejde-320	132	18	defined	define	VERB
ejde-320	132	19	on	on	ADP
ejde-320	132	20	(	(	PUNCT
ejde-320	132	21	w	w	PROPN
ejde-320	132	22	1,2	1,2	NUM
ejde-320	132	23	a	a	PRON
ejde-320	132	24	)	)	PUNCT
ejde-320	132	25	0	0	NUM
ejde-320	132	26	×	×	NOUN
ejde-320	132	27	(	(	PUNCT
ejde-320	132	28	w	w	PROPN
ejde-320	132	29	1,2	1,2	NUM
ejde-320	132	30	a	a	PRON
ejde-320	132	31	)	)	PUNCT
ejde-320	132	32	0	0	NUM
ejde-320	132	33	.	.	PUNCT
ejde-320	133	1	proposition	proposition	NOUN
ejde-320	133	2	3.1	3.1	NUM
ejde-320	133	3	.	.	PUNCT
ejde-320	134	1	let	let	VERB
ejde-320	134	2	ω	ω	PROPN
ejde-320	134	3	⊂	⊂	PROPN
ejde-320	134	4	rn	rn	AUX
ejde-320	134	5	be	be	AUX
ejde-320	134	6	a	a	DET
ejde-320	134	7	bounded	bounded	ADJ
ejde-320	134	8	subset	subset	NOUN
ejde-320	134	9	,	,	PUNCT
ejde-320	134	10	and	and	CCONJ
ejde-320	134	11	a	a	DET
ejde-320	134	12	a	a	DET
ejde-320	134	13	nonnegative	nonnegative	ADJ
ejde-320	134	14	semidefinite	semidefinite	NOUN
ejde-320	134	15	n×	n×	PROPN
ejde-320	134	16	n	n	NOUN
ejde-320	134	17	matrix	matrix	NOUN
ejde-320	134	18	with	with	ADP
ejde-320	134	19	bounded	bounded	ADJ
ejde-320	134	20	measurable	measurable	ADJ
ejde-320	134	21	coefficients	coefficient	NOUN
ejde-320	134	22	.	.	PUNCT
ejde-320	135	1	suppose	suppose	VERB
ejde-320	135	2	that	that	SCONJ
ejde-320	135	3	for	for	ADP
ejde-320	135	4	every	every	DET
ejde-320	135	5	w	w	PROPN
ejde-320	135	6	∈	∈	PROPN
ejde-320	135	7	(	(	PUNCT
ejde-320	135	8	wa	wa	NOUN
ejde-320	135	9	1,2	1,2	NUM
ejde-320	135	10	)	)	PUNCT
ejde-320	135	11	0	0	NUM
ejde-320	135	12	(	(	PUNCT
ejde-320	135	13	ω	ω	NOUN
ejde-320	135	14	)	)	PUNCT
ejde-320	135	15	the	the	DET
ejde-320	135	16	following	follow	VERB
ejde-320	135	17	(	(	PUNCT
ejde-320	135	18	2	2	NUM
ejde-320	135	19	,	,	PUNCT
ejde-320	135	20	2	2	NUM
ejde-320	135	21	)	)	PUNCT
ejde-320	135	22	sobolev	sobolev	NOUN
ejde-320	135	23	inequality	inequality	NOUN
ejde-320	135	24	holds∫	holds∫	VERB
ejde-320	135	25	ω	ω	PROPN
ejde-320	135	26	|w|2dx	|w|2dx	ADJ
ejde-320	135	27	≤	≤	PUNCT
ejde-320	135	28	c(ω	c(ω	PROPN
ejde-320	135	29	)	)	PUNCT
ejde-320	135	30	∫	∫	PROPN
ejde-320	135	31	ω	ω	PROPN
ejde-320	135	32	|∇aw|2dx	|∇aw|2dx	PROPN
ejde-320	135	33	.	.	PUNCT
ejde-320	136	1	(	(	PUNCT
ejde-320	136	2	3.1	3.1	NUM
ejde-320	136	3	)	)	PUNCT
ejde-320	136	4	then	then	ADV
ejde-320	136	5	the	the	DET
ejde-320	136	6	bilinear	bilinear	PROPN
ejde-320	136	7	form	form	NOUN
ejde-320	136	8	b	b	PROPN
ejde-320	136	9	:	:	PUNCT
ejde-320	136	10	(	(	PUNCT
ejde-320	136	11	wa	wa	PROPN
ejde-320	136	12	1,2	1,2	NUM
ejde-320	136	13	)	)	PUNCT
ejde-320	136	14	0	0	NUM
ejde-320	137	1	(	(	PUNCT
ejde-320	137	2	ω)×	ω)×	PROPN
ejde-320	137	3	(	(	PUNCT
ejde-320	137	4	wa	wa	PROPN
ejde-320	137	5	1,2	1,2	NUM
ejde-320	137	6	)	)	PUNCT
ejde-320	137	7	0	0	NUM
ejde-320	138	1	(	(	PUNCT
ejde-320	138	2	ω)→	ω)→	NOUN
ejde-320	138	3	r	r	NOUN
ejde-320	138	4	defined	define	VERB
ejde-320	138	5	by	by	ADP
ejde-320	138	6	b[u	b[u	NOUN
ejde-320	138	7	,	,	PUNCT
ejde-320	138	8	v	v	NOUN
ejde-320	138	9	]	]	PUNCT
ejde-320	138	10	:	:	PUNCT
ejde-320	138	11	=	=	SYM
ejde-320	138	12	∫	∫	PROPN
ejde-320	139	1	ω	ω	X
ejde-320	139	2	∇u	∇u	PROPN
ejde-320	139	3	·	·	PUNCT
ejde-320	139	4	a∇v	a∇v	NOUN
ejde-320	139	5	is	be	AUX
ejde-320	139	6	bounded	bound	VERB
ejde-320	139	7	and	and	CCONJ
ejde-320	139	8	coercive	coercive	ADJ
ejde-320	139	9	,	,	PUNCT
ejde-320	139	10	i.e.	i.e.	X
ejde-320	139	11	(	(	PUNCT
ejde-320	139	12	1	1	X
ejde-320	139	13	)	)	PUNCT
ejde-320	139	14	there	there	PRON
ejde-320	139	15	exists	exist	VERB
ejde-320	139	16	α	α	PROPN
ejde-320	139	17	>	>	X
ejde-320	139	18	0	0	NUM
ejde-320	139	19	such	such	ADJ
ejde-320	139	20	that	that	DET
ejde-320	139	21	|b[u	|b[u	NOUN
ejde-320	139	22	,	,	PUNCT
ejde-320	139	23	v]|	v]|	PROPN
ejde-320	139	24	≤	≤	PROPN
ejde-320	139	25	α‖u‖w	α‖u‖w	NOUN
ejde-320	139	26	1,2	1,2	NUM
ejde-320	139	27	a	a	DET
ejde-320	139	28	‖v‖w	‖v‖w	NOUN
ejde-320	139	29	1,2	1,2	NUM
ejde-320	139	30	a	a	NOUN
ejde-320	139	31	for	for	ADP
ejde-320	139	32	all	all	DET
ejde-320	139	33	u	u	NOUN
ejde-320	139	34	,	,	PUNCT
ejde-320	139	35	v	v	NOUN
ejde-320	139	36	∈	∈	PROPN
ejde-320	139	37	(	(	PUNCT
ejde-320	139	38	wa	wa	NOUN
ejde-320	139	39	1,2	1,2	NUM
ejde-320	139	40	)	)	PUNCT
ejde-320	139	41	0	0	NUM
ejde-320	139	42	(	(	PUNCT
ejde-320	139	43	ω	ω	NOUN
ejde-320	139	44	)	)	PUNCT
ejde-320	139	45	.	.	PUNCT
ejde-320	140	1	(	(	PUNCT
ejde-320	140	2	2	2	X
ejde-320	140	3	)	)	PUNCT
ejde-320	140	4	there	there	PRON
ejde-320	140	5	exists	exist	VERB
ejde-320	140	6	β	β	X
ejde-320	140	7	>	>	X
ejde-320	140	8	0	0	NUM
ejde-320	141	1	such	such	ADJ
ejde-320	141	2	that	that	PRON
ejde-320	141	3	β‖u‖2	β‖u‖2	PROPN
ejde-320	141	4	w	w	ADP
ejde-320	141	5	1,2	1,2	NUM
ejde-320	141	6	a	a	DET
ejde-320	141	7	≤	≤	NUM
ejde-320	141	8	b[u	b[u	NOUN
ejde-320	141	9	,	,	PUNCT
ejde-320	141	10	u	u	NOUN
ejde-320	141	11	]	]	X
ejde-320	141	12	for	for	ADP
ejde-320	141	13	all	all	DET
ejde-320	141	14	u	u	PROPN
ejde-320	141	15	∈	∈	PROPN
ejde-320	141	16	(	(	PUNCT
ejde-320	141	17	wa	wa	NOUN
ejde-320	141	18	1,2	1,2	NUM
ejde-320	141	19	)	)	PUNCT
ejde-320	141	20	0	0	NUM
ejde-320	141	21	(	(	PUNCT
ejde-320	141	22	ω	ω	NOUN
ejde-320	141	23	)	)	PUNCT
ejde-320	141	24	.	.	PUNCT
ejde-320	142	1	proof	proof	NOUN
ejde-320	142	2	.	.	PUNCT
ejde-320	143	1	we	we	PRON
ejde-320	143	2	begin	begin	VERB
ejde-320	143	3	by	by	ADP
ejde-320	143	4	showing	show	VERB
ejde-320	143	5	b	b	NOUN
ejde-320	143	6	is	be	AUX
ejde-320	143	7	bounded	bound	VERB
ejde-320	143	8	.	.	PUNCT
ejde-320	144	1	we	we	PRON
ejde-320	144	2	have	have	AUX
ejde-320	144	3	using	use	VERB
ejde-320	144	4	hölder	hölder	PROPN
ejde-320	144	5	’s	’s	PART
ejde-320	144	6	inequality	inequality	NOUN
ejde-320	144	7	|b[u	|b[u	PROPN
ejde-320	144	8	,	,	PUNCT
ejde-320	144	9	v]|	v]|	PROPN
ejde-320	144	10	=	=	PUNCT
ejde-320	145	1	∣∣	∣∣	NUM
ejde-320	145	2	∫	∫	PROPN
ejde-320	145	3	∇u	∇u	INTJ
ejde-320	145	4	·	·	PUNCT
ejde-320	145	5	a∇v∣∣	a∇v∣∣	X
ejde-320	145	6	≤	≤	NUM
ejde-320	145	7	(	(	PUNCT
ejde-320	145	8	∫	∫	PROPN
ejde-320	145	9	|∇u	|∇u	NOUN
ejde-320	145	10	·	·	PUNCT
ejde-320	145	11	a∇u|)1/2(∫	a∇u|)1/2(∫	PROPN
ejde-320	145	12	|∇v	|∇v	NOUN
ejde-320	145	13	·	·	SYM
ejde-320	145	14	a∇v|	a∇v|	NOUN
ejde-320	145	15	)	)	PUNCT
ejde-320	145	16	1/2	1/2	NUM
ejde-320	145	17	≤	≤	NOUN
ejde-320	145	18	(	(	PUNCT
ejde-320	145	19	∫	∫	PROPN
ejde-320	145	20	u2	u2	PROPN
ejde-320	145	21	+	+	CCONJ
ejde-320	145	22	∫	∫	PROPN
ejde-320	145	23	|∇u	|∇u	NOUN
ejde-320	145	24	·	·	SYM
ejde-320	145	25	a∇u|	a∇u|	NUM
ejde-320	145	26	)	)	PUNCT
ejde-320	145	27	1/2(∫	1/2(∫	NUM
ejde-320	145	28	v2	v2	NOUN
ejde-320	145	29	+	+	NUM
ejde-320	145	30	∫	∫	PROPN
ejde-320	145	31	|∇v	|∇v	NOUN
ejde-320	145	32	·	·	PUNCT
ejde-320	145	33	a∇v|	a∇v|	NOUN
ejde-320	145	34	)	)	PUNCT
ejde-320	145	35	1/2	1/2	NUM
ejde-320	145	36	=	=	SYM
ejde-320	145	37	‖u‖w	‖u‖w	NOUN
ejde-320	145	38	1,2	1,2	NUM
ejde-320	145	39	a	a	DET
ejde-320	145	40	‖v‖w	‖v‖w	NOUN
ejde-320	145	41	1,2	1,2	NUM
ejde-320	145	42	a	a	NOUN
ejde-320	145	43	for	for	ADP
ejde-320	145	44	all	all	DET
ejde-320	145	45	u	u	NOUN
ejde-320	145	46	,	,	PUNCT
ejde-320	145	47	v	v	NOUN
ejde-320	145	48	∈	∈	PROPN
ejde-320	145	49	(	(	PUNCT
ejde-320	145	50	w	w	PROPN
ejde-320	145	51	1,2	1,2	NUM
ejde-320	145	52	a	a	PRON
ejde-320	145	53	)	)	PUNCT
ejde-320	145	54	0	0	NUM
ejde-320	145	55	.	.	PUNCT
ejde-320	146	1	to	to	PART
ejde-320	146	2	show	show	VERB
ejde-320	146	3	the	the	DET
ejde-320	146	4	coercivity	coercivity	NOUN
ejde-320	146	5	of	of	ADP
ejde-320	146	6	the	the	DET
ejde-320	146	7	bilinear	bilinear	NOUN
ejde-320	146	8	form	form	NOUN
ejde-320	146	9	b	b	NOUN
ejde-320	146	10	,	,	PUNCT
ejde-320	146	11	condition	condition	NOUN
ejde-320	146	12	(	(	PUNCT
ejde-320	146	13	2	2	NUM
ejde-320	146	14	)	)	PUNCT
ejde-320	146	15	,	,	PUNCT
ejde-320	146	16	we	we	PRON
ejde-320	146	17	use	use	VERB
ejde-320	146	18	sobolev	sobolev	ADJ
ejde-320	146	19	inequality	inequality	NOUN
ejde-320	146	20	(	(	PUNCT
ejde-320	146	21	3.1	3.1	NUM
ejde-320	146	22	)	)	PUNCT
ejde-320	146	23	to	to	PART
ejde-320	146	24	obtain	obtain	VERB
ejde-320	146	25	b[u	b[u	NOUN
ejde-320	146	26	,	,	PUNCT
ejde-320	146	27	u	u	NOUN
ejde-320	146	28	]	]	X
ejde-320	146	29	=	=	SYM
ejde-320	146	30	1	1	NUM
ejde-320	146	31	2	2	NUM
ejde-320	146	32	b[u	b[u	NOUN
ejde-320	146	33	,	,	PUNCT
ejde-320	146	34	u	u	NOUN
ejde-320	146	35	]	]	X
ejde-320	146	36	+	+	CCONJ
ejde-320	146	37	1	1	NUM
ejde-320	146	38	2	2	NUM
ejde-320	146	39	b[u	b[u	NOUN
ejde-320	146	40	,	,	PUNCT
ejde-320	146	41	u	u	NOUN
ejde-320	146	42	]	]	X
ejde-320	146	43	=	=	SYM
ejde-320	146	44	1	1	NUM
ejde-320	146	45	2	2	NUM
ejde-320	146	46	∫	∫	NOUN
ejde-320	146	47	ω	ω	PROPN
ejde-320	146	48	∇u	∇u	PROPN
ejde-320	146	49	·	·	PUNCT
ejde-320	146	50	a∇u+	a∇u+	PROPN
ejde-320	146	51	1	1	NUM
ejde-320	146	52	2	2	NUM
ejde-320	146	53	b[u	b[u	NOUN
ejde-320	146	54	,	,	PUNCT
ejde-320	146	55	u	u	NOUN
ejde-320	146	56	]	]	X
ejde-320	146	57	=	=	SYM
ejde-320	146	58	1	1	NUM
ejde-320	146	59	2	2	NUM
ejde-320	146	60	∫	∫	NOUN
ejde-320	146	61	ω	ω	NUM
ejde-320	146	62	|∇au|2	|∇au|2	NOUN
ejde-320	146	63	+	+	CCONJ
ejde-320	146	64	1	1	NUM
ejde-320	146	65	2	2	NUM
ejde-320	146	66	b[u	b[u	NOUN
ejde-320	146	67	,	,	PUNCT
ejde-320	146	68	u	u	NOUN
ejde-320	146	69	]	]	X
ejde-320	146	70	≥	≥	NUM
ejde-320	146	71	1	1	NUM
ejde-320	146	72	2c	2c	NUM
ejde-320	146	73	∫	∫	PROPN
ejde-320	146	74	ω	ω	PROPN
ejde-320	146	75	u2	u2	PROPN
ejde-320	147	1	+	+	CCONJ
ejde-320	147	2	1	1	NUM
ejde-320	147	3	2	2	NUM
ejde-320	147	4	b[u	b[u	NOUN
ejde-320	147	5	,	,	PUNCT
ejde-320	147	6	u	u	NOUN
ejde-320	147	7	]	]	X
ejde-320	147	8	=	=	SYM
ejde-320	147	9	1	1	NUM
ejde-320	147	10	2c	2c	NUM
ejde-320	147	11	∫	∫	PROPN
ejde-320	148	1	ω	ω	PROPN
ejde-320	148	2	u2	u2	PROPN
ejde-320	148	3	+	+	CCONJ
ejde-320	148	4	1	1	NUM
ejde-320	148	5	2	2	NUM
ejde-320	148	6	∫	∫	NOUN
ejde-320	148	7	ω	ω	PROPN
ejde-320	148	8	∇u	∇u	PROPN
ejde-320	148	9	·	·	PUNCT
ejde-320	148	10	a∇u	a∇u	PRON
ejde-320	148	11	≥	≥	NOUN
ejde-320	148	12	min	min	PROPN
ejde-320	148	13	{	{	PUNCT
ejde-320	148	14	1	1	NUM
ejde-320	148	15	2c	2c	NUM
ejde-320	148	16	,	,	PUNCT
ejde-320	148	17	1	1	NUM
ejde-320	148	18	2	2	NUM
ejde-320	148	19	}	}	PUNCT
ejde-320	148	20	(	(	PUNCT
ejde-320	148	21	∫	∫	PROPN
ejde-320	148	22	ω	ω	PROPN
ejde-320	148	23	u2	u2	PROPN
ejde-320	148	24	+	+	CCONJ
ejde-320	148	25	∫	∫	PROPN
ejde-320	148	26	ω	ω	PROPN
ejde-320	148	27	∇u	∇u	PROPN
ejde-320	148	28	·	·	PUNCT
ejde-320	148	29	a∇u	a∇u	X
ejde-320	148	30	)	)	PUNCT
ejde-320	149	1	=	=	SYM
ejde-320	150	1	β‖u‖2	β‖u‖2	PROPN
ejde-320	150	2	w	w	ADP
ejde-320	150	3	1,2	1,2	NUM
ejde-320	150	4	a	a	PRON
ejde-320	150	5	,	,	PUNCT
ejde-320	150	6	where	where	SCONJ
ejde-320	150	7	β	β	X
ejde-320	150	8	=	=	SYM
ejde-320	150	9	min	min	X
ejde-320	150	10	{	{	PUNCT
ejde-320	150	11	1	1	NUM
ejde-320	150	12	2c	2c	NUM
ejde-320	150	13	,	,	PUNCT
ejde-320	150	14	1	1	NUM
ejde-320	150	15	2	2	NUM
ejde-320	150	16	}	}	PUNCT
ejde-320	150	17	and	and	CCONJ
ejde-320	150	18	c	c	NOUN
ejde-320	150	19	=	=	SYM
ejde-320	150	20	c(ω	c(ω	PROPN
ejde-320	150	21	)	)	PUNCT
ejde-320	150	22	from	from	ADP
ejde-320	150	23	(	(	PUNCT
ejde-320	150	24	3.1	3.1	NUM
ejde-320	150	25	)	)	PUNCT
ejde-320	150	26	.	.	PUNCT
ejde-320	151	1	thus	thus	ADV
ejde-320	151	2	,	,	PUNCT
ejde-320	151	3	the	the	DET
ejde-320	151	4	bilinear	bilinear	NOUN
ejde-320	151	5	form	form	NOUN
ejde-320	151	6	b	b	PROPN
ejde-320	151	7	is	be	AUX
ejde-320	151	8	bounded	bound	VERB
ejde-320	151	9	and	and	CCONJ
ejde-320	151	10	coercive	coercive	ADJ
ejde-320	151	11	.	.	PUNCT
ejde-320	152	1	�	�	PROPN
ejde-320	152	2	we	we	PRON
ejde-320	152	3	are	be	AUX
ejde-320	152	4	now	now	ADV
ejde-320	152	5	ready	ready	ADJ
ejde-320	152	6	to	to	PART
ejde-320	152	7	show	show	VERB
ejde-320	152	8	the	the	DET
ejde-320	152	9	existence	existence	NOUN
ejde-320	152	10	and	and	CCONJ
ejde-320	152	11	uniqueness	uniqueness	NOUN
ejde-320	152	12	of	of	ADP
ejde-320	152	13	the	the	DET
ejde-320	152	14	weak	weak	ADJ
ejde-320	152	15	solution	solution	NOUN
ejde-320	152	16	claimed	claim	VERB
ejde-320	152	17	in	in	ADP
ejde-320	152	18	theorem	theorem	ADJ
ejde-320	152	19	1.1	1.1	NUM
ejde-320	152	20	.	.	PUNCT
ejde-320	153	1	this	this	DET
ejde-320	153	2	result	result	NOUN
ejde-320	153	3	in	in	ADP
ejde-320	153	4	fact	fact	NOUN
ejde-320	153	5	holds	hold	VERB
ejde-320	153	6	under	under	ADP
ejde-320	153	7	a	a	DET
ejde-320	153	8	weaker	weak	ADJ
ejde-320	153	9	assumption	assumption	NOUN
ejde-320	153	10	on	on	ADP
ejde-320	153	11	the	the	DET
ejde-320	153	12	function	function	NOUN
ejde-320	153	13	f	f	NOUN
ejde-320	153	14	,	,	PUNCT
ejde-320	153	15	we	we	PRON
ejde-320	153	16	only	only	ADV
ejde-320	153	17	need	need	VERB
ejde-320	153	18	to	to	PART
ejde-320	153	19	require	require	VERB
ejde-320	153	20	f	f	PROPN
ejde-320	153	21	∈	∈	PROPN
ejde-320	153	22	l2(b	l2(b	NUM
ejde-320	153	23	)	)	PUNCT
ejde-320	153	24	.	.	PUNCT
ejde-320	154	1	theorem	theorem	ADJ
ejde-320	154	2	3.2	3.2	NUM
ejde-320	154	3	.	.	PUNCT
ejde-320	155	1	let	let	VERB
ejde-320	155	2	l	l	NOUN
ejde-320	155	3	=	=	SYM
ejde-320	155	4	∇·a∇	∇·a∇	PROPN
ejde-320	155	5	with	with	ADP
ejde-320	155	6	bounded	bounded	ADJ
ejde-320	155	7	measurable	measurable	ADJ
ejde-320	155	8	non	non	ADJ
ejde-320	155	9	-	-	ADJ
ejde-320	155	10	negative	negative	ADJ
ejde-320	155	11	semidefinite	semidefinite	NOUN
ejde-320	155	12	matrix	matrix	NOUN
ejde-320	155	13	a	a	PRON
ejde-320	155	14	,	,	PUNCT
ejde-320	155	15	and	and	CCONJ
ejde-320	155	16	d	d	ADP
ejde-320	155	17	a	a	DET
ejde-320	155	18	metric	metric	NOUN
ejde-320	155	19	on	on	ADP
ejde-320	155	20	rn	rn	PROPN
ejde-320	155	21	,	,	PUNCT
ejde-320	155	22	such	such	ADJ
ejde-320	155	23	that	that	SCONJ
ejde-320	155	24	for	for	ADP
ejde-320	155	25	any	any	DET
ejde-320	155	26	metric	metric	ADJ
ejde-320	155	27	ball	ball	NOUN
ejde-320	155	28	b	b	NOUN
ejde-320	155	29	=	=	PUNCT
ejde-320	155	30	b(x	b(x	PROPN
ejde-320	155	31	,	,	PUNCT
ejde-320	155	32	r	r	NOUN
ejde-320	155	33	)	)	PUNCT
ejde-320	155	34	with	with	ADP
ejde-320	155	35	0	0	NUM
ejde-320	155	36	<	<	X
ejde-320	155	37	r	r	NOUN
ejde-320	155	38	<	<	X
ejde-320	155	39	∞	∞	NUM
ejde-320	155	40	it	it	PRON
ejde-320	155	41	holds	hold	VERB
ejde-320	155	42	0	0	NUM
ejde-320	155	43	<	<	X
ejde-320	155	44	|b|	|b|	PROPN
ejde-320	155	45	<	<	X
ejde-320	155	46	∞.	∞.	PROPN
ejde-320	155	47	suppose	suppose	VERB
ejde-320	155	48	also	also	ADV
ejde-320	155	49	that	that	SCONJ
ejde-320	155	50	sobolev	sobolev	NOUN
ejde-320	155	51	inequality	inequality	NOUN
ejde-320	155	52	(	(	PUNCT
ejde-320	155	53	3.1	3.1	NUM
ejde-320	155	54	)	)	PUNCT
ejde-320	155	55	holds	hold	VERB
ejde-320	155	56	ejde-2021/82	ejde-2021/82	ADJ
ejde-320	155	57	orlicz	orlicz	ADJ
ejde-320	155	58	-	-	PUNCT
ejde-320	155	59	sobolev	sobolev	NOUN
ejde-320	155	60	inequalities	inequality	NOUN
ejde-320	155	61	and	and	CCONJ
ejde-320	155	62	the	the	DET
ejde-320	155	63	dirichlet	dirichlet	PROPN
ejde-320	155	64	problem	problem	NOUN
ejde-320	155	65	7	7	NUM
ejde-320	155	66	for	for	ADP
ejde-320	155	67	all	all	DET
ejde-320	155	68	w	w	NOUN
ejde-320	155	69	∈	∈	PROPN
ejde-320	155	70	(	(	PUNCT
ejde-320	155	71	w	w	PROPN
ejde-320	155	72	1,2	1,2	NUM
ejde-320	155	73	a	a	PRON
ejde-320	155	74	)	)	PUNCT
ejde-320	155	75	0	0	PUNCT
ejde-320	156	1	(	(	PUNCT
ejde-320	156	2	b	b	NOUN
ejde-320	156	3	)	)	PUNCT
ejde-320	156	4	and	and	CCONJ
ejde-320	156	5	some	some	DET
ejde-320	156	6	ball	ball	NOUN
ejde-320	156	7	b	b	PROPN
ejde-320	156	8	=	=	SYM
ejde-320	156	9	ω	ω	PROPN
ejde-320	156	10	⊂	⊂	PROPN
ejde-320	156	11	rn	rn	PROPN
ejde-320	156	12	.	.	PROPN
ejde-320	157	1	if	if	SCONJ
ejde-320	157	2	f	f	PROPN
ejde-320	157	3	∈	∈	PROPN
ejde-320	157	4	l2(b	l2(b	NUM
ejde-320	157	5	)	)	PUNCT
ejde-320	157	6	,	,	PUNCT
ejde-320	157	7	then	then	ADV
ejde-320	157	8	there	there	PRON
ejde-320	157	9	exists	exist	VERB
ejde-320	157	10	a	a	DET
ejde-320	157	11	unique	unique	ADJ
ejde-320	157	12	weak	weak	ADJ
ejde-320	157	13	solution	solution	NOUN
ejde-320	157	14	u	u	NOUN
ejde-320	157	15	∈	∈	PROPN
ejde-320	157	16	(	(	PUNCT
ejde-320	157	17	w	w	PROPN
ejde-320	157	18	1,2	1,2	NUM
ejde-320	157	19	a	a	PRON
ejde-320	157	20	)	)	PUNCT
ejde-320	157	21	0	0	PUNCT
ejde-320	158	1	(	(	PUNCT
ejde-320	158	2	b	b	NOUN
ejde-320	158	3	)	)	PUNCT
ejde-320	158	4	to	to	ADP
ejde-320	158	5	the	the	DET
ejde-320	158	6	dirichlet	dirichlet	PROPN
ejde-320	158	7	problem	problem	NOUN
ejde-320	158	8	∇	∇	X
ejde-320	158	9	·	·	PUNCT
ejde-320	158	10	a∇u	a∇u	X
ejde-320	158	11	=	=	SYM
ejde-320	158	12	f	f	PROPN
ejde-320	158	13	in	in	ADP
ejde-320	158	14	b	b	PROPN
ejde-320	158	15	u|∂b	u|∂b	PROPN
ejde-320	158	16	=	=	NOUN
ejde-320	158	17	0	0	PROPN
ejde-320	158	18	.	.	PUNCT
ejde-320	159	1	(	(	PUNCT
ejde-320	159	2	3.2	3.2	NUM
ejde-320	159	3	)	)	PUNCT
ejde-320	159	4	proof	proof	NOUN
ejde-320	159	5	.	.	PUNCT
ejde-320	160	1	consider	consider	VERB
ejde-320	160	2	the	the	DET
ejde-320	160	3	linear	linear	ADJ
ejde-320	160	4	functional	functional	ADJ
ejde-320	160	5	(	(	PUNCT
ejde-320	160	6	f	f	X
ejde-320	160	7	,	,	PUNCT
ejde-320	160	8	·	·	PUNCT
ejde-320	160	9	)	)	PUNCT
ejde-320	160	10	:	:	PUNCT
ejde-320	160	11	(	(	PUNCT
ejde-320	161	1	w	w	PROPN
ejde-320	161	2	1,2	1,2	NUM
ejde-320	161	3	a	a	PRON
ejde-320	161	4	)	)	PUNCT
ejde-320	161	5	0	0	NUM
ejde-320	162	1	(	(	PUNCT
ejde-320	162	2	b)→	b)→	VERB
ejde-320	162	3	r	r	NOUN
ejde-320	162	4	defined	define	VERB
ejde-320	162	5	by	by	ADP
ejde-320	162	6	(	(	PUNCT
ejde-320	162	7	f	f	X
ejde-320	162	8	,	,	PUNCT
ejde-320	162	9	w	w	PROPN
ejde-320	162	10	)	)	PUNCT
ejde-320	162	11	=	=	PUNCT
ejde-320	163	1	−	−	PROPN
ejde-320	163	2	∫	∫	PROPN
ejde-320	163	3	b	b	PROPN
ejde-320	163	4	fw	fw	PROPN
ejde-320	163	5	,	,	PUNCT
ejde-320	163	6	∀w	∀w	ADJ
ejde-320	163	7	∈	∈	PROPN
ejde-320	163	8	(	(	PUNCT
ejde-320	163	9	w	w	PROPN
ejde-320	163	10	1,2	1,2	NUM
ejde-320	163	11	a	a	PRON
ejde-320	163	12	)	)	PUNCT
ejde-320	163	13	0	0	PUNCT
ejde-320	163	14	(	(	PUNCT
ejde-320	163	15	b	b	NOUN
ejde-320	163	16	)	)	PUNCT
ejde-320	163	17	.	.	PUNCT
ejde-320	164	1	since	since	SCONJ
ejde-320	164	2	f	f	PROPN
ejde-320	164	3	∈	∈	PROPN
ejde-320	164	4	l2(b	l2(b	NUM
ejde-320	164	5	)	)	PUNCT
ejde-320	164	6	and	and	CCONJ
ejde-320	164	7	w	w	PROPN
ejde-320	164	8	∈	∈	PROPN
ejde-320	164	9	(	(	PUNCT
ejde-320	164	10	w	w	PROPN
ejde-320	164	11	1,2	1,2	NUM
ejde-320	164	12	a	a	PRON
ejde-320	164	13	)	)	PUNCT
ejde-320	164	14	0	0	PUNCT
ejde-320	164	15	(	(	PUNCT
ejde-320	164	16	b	b	X
ejde-320	164	17	)	)	PUNCT
ejde-320	164	18	⊂	⊂	PROPN
ejde-320	164	19	l2(b	l2(b	NUM
ejde-320	164	20	)	)	PUNCT
ejde-320	164	21	we	we	PRON
ejde-320	164	22	have	have	VERB
ejde-320	164	23	|(f	|(f	PROPN
ejde-320	164	24	,	,	PUNCT
ejde-320	164	25	w)|	w)|	VERB
ejde-320	164	26	≤	≤	PROPN
ejde-320	164	27	c‖f‖l2(b)‖w‖w	c‖f‖l2(b)‖w‖w	VERB
ejde-320	164	28	1,2	1,2	NUM
ejde-320	164	29	a	a	DET
ejde-320	164	30	(	(	PUNCT
ejde-320	164	31	b	b	NOUN
ejde-320	164	32	)	)	PUNCT
ejde-320	164	33	,	,	PUNCT
ejde-320	164	34	which	which	PRON
ejde-320	164	35	shows	show	VERB
ejde-320	164	36	that	that	SCONJ
ejde-320	164	37	this	this	DET
ejde-320	164	38	linear	linear	ADJ
ejde-320	164	39	functional	functional	NOUN
ejde-320	164	40	is	be	AUX
ejde-320	164	41	bounded	bound	VERB
ejde-320	164	42	on	on	ADP
ejde-320	164	43	(	(	PUNCT
ejde-320	164	44	w	w	PROPN
ejde-320	164	45	1,2	1,2	NUM
ejde-320	164	46	a	a	PRON
ejde-320	164	47	)	)	PUNCT
ejde-320	164	48	0	0	PUNCT
ejde-320	165	1	(	(	PUNCT
ejde-320	165	2	b	b	NOUN
ejde-320	165	3	)	)	PUNCT
ejde-320	165	4	.	.	PUNCT
ejde-320	166	1	therefore	therefore	ADV
ejde-320	166	2	,	,	PUNCT
ejde-320	166	3	by	by	ADP
ejde-320	166	4	proposition	proposition	NOUN
ejde-320	166	5	3.1	3.1	NUM
ejde-320	166	6	and	and	CCONJ
ejde-320	166	7	lax	lax	PROPN
ejde-320	166	8	-	-	PUNCT
ejde-320	166	9	milgram	milgram	NOUN
ejde-320	166	10	theorem	theorem	NOUN
ejde-320	166	11	there	there	PRON
ejde-320	166	12	exists	exist	VERB
ejde-320	166	13	a	a	DET
ejde-320	166	14	unique	unique	ADJ
ejde-320	166	15	element	element	NOUN
ejde-320	166	16	,	,	PUNCT
ejde-320	166	17	u	u	NOUN
ejde-320	166	18	∈	∈	PROPN
ejde-320	166	19	(	(	PUNCT
ejde-320	166	20	w	w	PROPN
ejde-320	166	21	1,2	1,2	NUM
ejde-320	166	22	a	a	PRON
ejde-320	166	23	)	)	PUNCT
ejde-320	166	24	0	0	PUNCT
ejde-320	167	1	(	(	PUNCT
ejde-320	167	2	b	b	NOUN
ejde-320	167	3	)	)	PUNCT
ejde-320	167	4	,	,	PUNCT
ejde-320	167	5	such	such	ADJ
ejde-320	167	6	that	that	DET
ejde-320	167	7	b[u	b[u	NOUN
ejde-320	167	8	,	,	PUNCT
ejde-320	167	9	w	w	NOUN
ejde-320	167	10	]	]	X
ejde-320	167	11	=	=	SYM
ejde-320	167	12	(	(	PUNCT
ejde-320	167	13	f	f	X
ejde-320	167	14	,	,	PUNCT
ejde-320	167	15	w	w	NOUN
ejde-320	167	16	)	)	PUNCT
ejde-320	167	17	for	for	ADP
ejde-320	167	18	all	all	DET
ejde-320	167	19	w	w	PROPN
ejde-320	167	20	∈	∈	PROPN
ejde-320	167	21	(	(	PUNCT
ejde-320	167	22	w	w	PROPN
ejde-320	167	23	1,2	1,2	NUM
ejde-320	167	24	a	a	PRON
ejde-320	167	25	)	)	PUNCT
ejde-320	167	26	0	0	PUNCT
ejde-320	168	1	(	(	PUNCT
ejde-320	168	2	b	b	NOUN
ejde-320	168	3	)	)	PUNCT
ejde-320	168	4	.	.	PUNCT
ejde-320	169	1	by	by	ADP
ejde-320	169	2	definition	definition	NOUN
ejde-320	169	3	of	of	ADP
ejde-320	169	4	b[u	b[u	NOUN
ejde-320	169	5	,	,	PUNCT
ejde-320	169	6	w	w	NOUN
ejde-320	169	7	]	]	X
ejde-320	169	8	this	this	PRON
ejde-320	169	9	means	mean	VERB
ejde-320	169	10	∫	∫	PROPN
ejde-320	169	11	∇u	∇u	PROPN
ejde-320	169	12	·	·	SYM
ejde-320	169	13	a∇w	a∇w	PROPN
ejde-320	169	14	=	=	SYM
ejde-320	170	1	−	−	PROPN
ejde-320	171	1	∫	∫	PROPN
ejde-320	171	2	fw	fw	INTJ
ejde-320	171	3	for	for	ADP
ejde-320	171	4	all	all	DET
ejde-320	171	5	w	w	PROPN
ejde-320	171	6	∈	∈	PROPN
ejde-320	171	7	(	(	PUNCT
ejde-320	171	8	w	w	PROPN
ejde-320	171	9	1,2	1,2	NUM
ejde-320	171	10	a	a	PRON
ejde-320	171	11	)	)	PUNCT
ejde-320	171	12	0	0	PUNCT
ejde-320	172	1	(	(	PUNCT
ejde-320	172	2	b	b	NOUN
ejde-320	172	3	)	)	PUNCT
ejde-320	172	4	,	,	PUNCT
ejde-320	172	5	and	and	CCONJ
ejde-320	172	6	we	we	PRON
ejde-320	172	7	conclude	conclude	VERB
ejde-320	172	8	that	that	SCONJ
ejde-320	172	9	u	u	PROPN
ejde-320	172	10	is	be	AUX
ejde-320	172	11	the	the	DET
ejde-320	172	12	unique	unique	ADJ
ejde-320	172	13	weak	weak	ADJ
ejde-320	172	14	solution	solution	NOUN
ejde-320	172	15	to	to	ADP
ejde-320	172	16	(	(	PUNCT
ejde-320	172	17	3.2	3.2	NUM
ejde-320	172	18	)	)	PUNCT
ejde-320	172	19	.	.	PUNCT
ejde-320	173	1	�	�	PROPN
ejde-320	173	2	corollary	corollary	ADJ
ejde-320	173	3	3.3	3.3	NUM
ejde-320	173	4	.	.	PUNCT
ejde-320	174	1	under	under	ADP
ejde-320	174	2	the	the	DET
ejde-320	174	3	assumptions	assumption	NOUN
ejde-320	174	4	of	of	ADP
ejde-320	174	5	theorem	theorem	ADJ
ejde-320	174	6	1.1	1.1	NUM
ejde-320	174	7	there	there	ADV
ejde-320	174	8	exists	exist	VERB
ejde-320	174	9	a	a	DET
ejde-320	174	10	unique	unique	ADJ
ejde-320	174	11	weak	weak	ADJ
ejde-320	174	12	solution	solution	NOUN
ejde-320	174	13	to	to	ADP
ejde-320	174	14	(	(	PUNCT
ejde-320	174	15	3.2	3.2	NUM
ejde-320	174	16	)	)	PUNCT
ejde-320	174	17	.	.	PUNCT
ejde-320	175	1	proof	proof	NOUN
ejde-320	175	2	.	.	PUNCT
ejde-320	176	1	suppose	suppose	VERB
ejde-320	176	2	orlicz	orlicz	PROPN
ejde-320	176	3	-	-	PUNCT
ejde-320	176	4	sobolev	sobolev	NOUN
ejde-320	176	5	inequality	inequality	NOUN
ejde-320	176	6	(	(	PUNCT
ejde-320	176	7	1.3	1.3	NUM
ejde-320	176	8	)	)	PUNCT
ejde-320	176	9	holds	hold	VERB
ejde-320	176	10	with	with	ADP
ejde-320	176	11	ϕ(t	ϕ(t	NUM
ejde-320	176	12	)	)	PUNCT
ejde-320	176	13	≥	≥	NOUN
ejde-320	176	14	t2	t2	NOUN
ejde-320	176	15	for	for	ADP
ejde-320	176	16	all	all	DET
ejde-320	176	17	t	t	PROPN
ejde-320	176	18	≥	≥	NOUN
ejde-320	176	19	0	0	NUM
ejde-320	176	20	.	.	PUNCT
ejde-320	177	1	moreover	moreover	ADV
ejde-320	177	2	,	,	PUNCT
ejde-320	177	3	the	the	DET
ejde-320	177	4	orlicz	orlicz	ADJ
ejde-320	177	5	space	space	NOUN
ejde-320	177	6	defined	define	VERB
ejde-320	177	7	by	by	ADP
ejde-320	177	8	ψ(t	ψ(t	PROPN
ejde-320	177	9	)	)	PUNCT
ejde-320	177	10	=	=	SYM
ejde-320	177	11	t2	t2	PROPN
ejde-320	177	12	coincides	coincide	VERB
ejde-320	177	13	with	with	ADP
ejde-320	177	14	l2	l2	NOUN
ejde-320	177	15	.	.	PUNCT
ejde-320	178	1	therefore	therefore	ADV
ejde-320	178	2	,	,	PUNCT
ejde-320	178	3	by	by	ADP
ejde-320	178	4	proposition	proposition	NOUN
ejde-320	178	5	2.6	2.6	NUM
ejde-320	178	6	the	the	DET
ejde-320	178	7	(	(	PUNCT
ejde-320	178	8	2	2	NUM
ejde-320	178	9	,	,	PUNCT
ejde-320	178	10	2	2	NUM
ejde-320	178	11	)	)	PUNCT
ejde-320	178	12	sobolev	sobolev	NOUN
ejde-320	178	13	inequality	inequality	NOUN
ejde-320	178	14	(	(	PUNCT
ejde-320	178	15	3.1	3.1	NUM
ejde-320	178	16	)	)	PUNCT
ejde-320	178	17	holds	hold	VERB
ejde-320	178	18	and	and	CCONJ
ejde-320	178	19	since	since	SCONJ
ejde-320	178	20	f	f	PROPN
ejde-320	178	21	∈	∈	PROPN
ejde-320	178	22	l∞(b	l∞(b	X
ejde-320	178	23	)	)	PUNCT
ejde-320	178	24	⊂	⊂	PROPN
ejde-320	178	25	l2(b	l2(b	NUM
ejde-320	178	26	)	)	PUNCT
ejde-320	178	27	,	,	PUNCT
ejde-320	178	28	theorem	theorem	VERB
ejde-320	178	29	3.2	3.2	NUM
ejde-320	178	30	applies	applie	NOUN
ejde-320	178	31	.	.	PUNCT
ejde-320	179	1	�	�	PROPN
ejde-320	179	2	3.2	3.2	NUM
ejde-320	179	3	.	.	PUNCT
ejde-320	180	1	global	global	ADJ
ejde-320	180	2	boundedness	boundedness	NOUN
ejde-320	180	3	of	of	ADP
ejde-320	180	4	weak	weak	ADJ
ejde-320	180	5	solutions	solution	NOUN
ejde-320	180	6	.	.	PUNCT
ejde-320	181	1	we	we	PRON
ejde-320	181	2	now	now	ADV
ejde-320	181	3	arrive	arrive	VERB
ejde-320	181	4	at	at	ADP
ejde-320	181	5	the	the	DET
ejde-320	181	6	proof	proof	NOUN
ejde-320	181	7	of	of	ADP
ejde-320	181	8	the	the	DET
ejde-320	181	9	global	global	ADJ
ejde-320	181	10	boundedness	boundedness	NOUN
ejde-320	181	11	estimate	estimate	NOUN
ejde-320	181	12	for	for	ADP
ejde-320	181	13	weak	weak	ADJ
ejde-320	181	14	solutions	solution	NOUN
ejde-320	181	15	.	.	PUNCT
ejde-320	182	1	the	the	DET
ejde-320	182	2	proof	proof	NOUN
ejde-320	182	3	closely	closely	ADV
ejde-320	182	4	follows	follow	VERB
ejde-320	182	5	the	the	DET
ejde-320	182	6	argument	argument	NOUN
ejde-320	182	7	of	of	ADP
ejde-320	182	8	[	[	X
ejde-320	182	9	7	7	NUM
ejde-320	182	10	,	,	PUNCT
ejde-320	182	11	chapter	chapter	NOUN
ejde-320	182	12	4	4	NUM
ejde-320	182	13	]	]	PUNCT
ejde-320	182	14	.	.	PUNCT
ejde-320	183	1	however	however	ADV
ejde-320	183	2	,	,	PUNCT
ejde-320	183	3	the	the	DET
ejde-320	183	4	orlicz	orlicz	PROPN
ejde-320	183	5	-	-	PUNCT
ejde-320	183	6	sobolev	sobolev	NOUN
ejde-320	183	7	inequality	inequality	NOUN
ejde-320	183	8	we	we	PRON
ejde-320	183	9	assume	assume	VERB
ejde-320	183	10	is	be	AUX
ejde-320	183	11	weaker	weak	ADJ
ejde-320	183	12	than	than	ADP
ejde-320	183	13	the	the	DET
ejde-320	183	14	one	one	NUM
ejde-320	183	15	in	in	ADP
ejde-320	183	16	[	[	X
ejde-320	183	17	7	7	NUM
ejde-320	183	18	]	]	PUNCT
ejde-320	183	19	,	,	PUNCT
ejde-320	183	20	while	while	SCONJ
ejde-320	183	21	our	our	PRON
ejde-320	183	22	assumption	assumption	NOUN
ejde-320	183	23	on	on	ADP
ejde-320	183	24	the	the	DET
ejde-320	183	25	right	right	ADJ
ejde-320	183	26	hand	hand	NOUN
ejde-320	183	27	side	side	NOUN
ejde-320	183	28	f	f	PROPN
ejde-320	183	29	is	be	AUX
ejde-320	183	30	stronger	strong	ADJ
ejde-320	183	31	.	.	PUNCT
ejde-320	184	1	we	we	PRON
ejde-320	184	2	therefore	therefore	ADV
ejde-320	184	3	provide	provide	VERB
ejde-320	184	4	the	the	DET
ejde-320	184	5	details	detail	NOUN
ejde-320	184	6	of	of	ADP
ejde-320	184	7	the	the	DET
ejde-320	184	8	arguments	argument	NOUN
ejde-320	184	9	that	that	PRON
ejde-320	184	10	are	be	AUX
ejde-320	184	11	necessary	necessary	ADJ
ejde-320	184	12	to	to	PART
ejde-320	184	13	verify	verify	VERB
ejde-320	184	14	in	in	ADP
ejde-320	184	15	this	this	DET
ejde-320	184	16	new	new	ADJ
ejde-320	184	17	setting	setting	NOUN
ejde-320	184	18	.	.	PUNCT
ejde-320	185	1	we	we	PRON
ejde-320	185	2	start	start	VERB
ejde-320	185	3	with	with	ADP
ejde-320	185	4	a	a	DET
ejde-320	185	5	caccioppli	caccioppli	NOUN
ejde-320	185	6	inequality	inequality	NOUN
ejde-320	185	7	,	,	PUNCT
ejde-320	185	8	which	which	PRON
ejde-320	185	9	is	be	AUX
ejde-320	185	10	an	an	DET
ejde-320	185	11	analogue	analogue	NOUN
ejde-320	185	12	of	of	ADP
ejde-320	185	13	[	[	X
ejde-320	185	14	7	7	NUM
ejde-320	185	15	,	,	PUNCT
ejde-320	185	16	proposition	proposition	NOUN
ejde-320	185	17	24	24	NUM
ejde-320	185	18	and	and	CCONJ
ejde-320	185	19	corollary	corollary	ADJ
ejde-320	185	20	25	25	NUM
ejde-320	185	21	]	]	PUNCT
ejde-320	185	22	.	.	PUNCT
ejde-320	186	1	proposition	proposition	NOUN
ejde-320	186	2	3.4	3.4	NUM
ejde-320	186	3	.	.	PUNCT
ejde-320	187	1	let	let	VERB
ejde-320	187	2	u	u	PRON
ejde-320	187	3	be	be	AUX
ejde-320	187	4	a	a	DET
ejde-320	187	5	weak	weak	ADJ
ejde-320	187	6	solution	solution	NOUN
ejde-320	187	7	to	to	ADP
ejde-320	187	8	(	(	PUNCT
ejde-320	187	9	1.1	1.1	NUM
ejde-320	187	10	)	)	PUNCT
ejde-320	187	11	on	on	ADP
ejde-320	187	12	ω	ω	PROPN
ejde-320	187	13	=	=	SYM
ejde-320	187	14	b	b	PROPN
ejde-320	187	15	and	and	CCONJ
ejde-320	187	16	define	define	VERB
ejde-320	187	17	u+	u+	NOUN
ejde-320	187	18	=	=	VERB
ejde-320	187	19	max{u	max{u	NOUN
ejde-320	187	20	,	,	PUNCT
ejde-320	187	21	0	0	NUM
ejde-320	187	22	}	}	PUNCT
ejde-320	187	23	,	,	PUNCT
ejde-320	187	24	then	then	ADV
ejde-320	187	25	the	the	DET
ejde-320	187	26	following	follow	VERB
ejde-320	187	27	caccioppoli	caccioppoli	NOUN
ejde-320	187	28	inequality	inequality	NOUN
ejde-320	187	29	holds	hold	VERB
ejde-320	187	30	on	on	ADP
ejde-320	187	31	the	the	DET
ejde-320	187	32	ball	ball	NOUN
ejde-320	187	33	b∫	b∫	PROPN
ejde-320	187	34	{	{	PUNCT
ejde-320	187	35	x∈b	x∈b	NOUN
ejde-320	187	36	:	:	PUNCT
ejde-320	187	37	u(x)>0	u(x)>0	PROPN
ejde-320	187	38	}	}	PUNCT
ejde-320	187	39	|∇au+|2	|∇au+|2	X
ejde-320	187	40	dµ	dµ	ADP
ejde-320	187	41	≤	≤	NUM
ejde-320	187	42	∫	∫	PROPN
ejde-320	187	43	{	{	PUNCT
ejde-320	187	44	x∈b	x∈b	NOUN
ejde-320	187	45	:	:	PUNCT
ejde-320	187	46	u(x)>0	u(x)>0	PROPN
ejde-320	187	47	}	}	PUNCT
ejde-320	187	48	u+‖f‖l∞	u+‖f‖l∞	VERB
ejde-320	187	49	dµ	dµ	ADP
ejde-320	187	50	,	,	PUNCT
ejde-320	187	51	where	where	SCONJ
ejde-320	187	52	dµ	dµ	ADP
ejde-320	188	1	=	=	SYM
ejde-320	188	2	dx|b|	dx|b|	PROPN
ejde-320	188	3	.	.	PUNCT
ejde-320	188	4	proof	proof	NOUN
ejde-320	188	5	.	.	PUNCT
ejde-320	189	1	let	let	VERB
ejde-320	189	2	v	v	NOUN
ejde-320	189	3	=	=	VERB
ejde-320	189	4	u+	u+	NOUN
ejde-320	189	5	then	then	ADV
ejde-320	189	6	we	we	PRON
ejde-320	189	7	have	have	VERB
ejde-320	189	8	v	v	NUM
ejde-320	189	9	∈	∈	NOUN
ejde-320	189	10	(	(	PUNCT
ejde-320	189	11	w	w	PROPN
ejde-320	189	12	1,2	1,2	NUM
ejde-320	189	13	a	a	PRON
ejde-320	189	14	)	)	PUNCT
ejde-320	189	15	0	0	PUNCT
ejde-320	190	1	(	(	PUNCT
ejde-320	190	2	b	b	NOUN
ejde-320	190	3	)	)	PUNCT
ejde-320	190	4	and	and	CCONJ
ejde-320	190	5	therefore,∫	therefore,∫	ADJ
ejde-320	190	6	b	b	X
ejde-320	191	1	∇u	∇u	PROPN
ejde-320	191	2	·	·	SYM
ejde-320	191	3	a∇v	a∇v	NOUN
ejde-320	191	4	dµ	dµ	PROPN
ejde-320	191	5	=	=	PUNCT
ejde-320	192	1	−	−	PROPN
ejde-320	192	2	∫	∫	PROPN
ejde-320	192	3	b	b	PROPN
ejde-320	192	4	fv	fv	PROPN
ejde-320	192	5	dµ,∫	dµ,∫	PROPN
ejde-320	192	6	{	{	PUNCT
ejde-320	192	7	x∈b	x∈b	NOUN
ejde-320	192	8	:	:	PUNCT
ejde-320	192	9	u(x)>0	u(x)>0	PROPN
ejde-320	192	10	}	}	PUNCT
ejde-320	192	11	∇u	∇u	NOUN
ejde-320	192	12	·	·	PUNCT
ejde-320	192	13	a∇u+	a∇u+	PROPN
ejde-320	192	14	dµ	dµ	PROPN
ejde-320	192	15	=	=	PUNCT
ejde-320	192	16	−	−	PROPN
ejde-320	192	17	∫	∫	PROPN
ejde-320	192	18	{	{	PUNCT
ejde-320	192	19	x∈b	x∈b	NOUN
ejde-320	192	20	:	:	PUNCT
ejde-320	192	21	u(x)>0	u(x)>0	PROPN
ejde-320	192	22	}	}	PUNCT
ejde-320	192	23	fu+	fu+	NOUN
ejde-320	192	24	dµ	dµ	PROPN
ejde-320	192	25	,	,	PUNCT
ejde-320	192	26	8	8	PROPN
ejde-320	192	27	u.	u.	PROPN
ejde-320	192	28	hafeez	hafeez	PROPN
ejde-320	192	29	,	,	PUNCT
ejde-320	192	30	t.	t.	PROPN
ejde-320	192	31	lavier	lavier	PROPN
ejde-320	192	32	,	,	PUNCT
ejde-320	192	33	l.	l.	PROPN
ejde-320	192	34	williams	williams	PROPN
ejde-320	192	35	,	,	PUNCT
ejde-320	192	36	l.	l.	PROPN
ejde-320	192	37	korobenko	korobenko	VERB
ejde-320	192	38	ejde-2021/82∫	ejde-2021/82∫	PROPN
ejde-320	192	39	{	{	PUNCT
ejde-320	192	40	x∈b	x∈b	NOUN
ejde-320	192	41	:	:	PUNCT
ejde-320	192	42	u(x)>0	u(x)>0	PROPN
ejde-320	192	43	}	}	PUNCT
ejde-320	192	44	|∇au+|2	|∇au+|2	X
ejde-320	192	45	dµ	dµ	ADP
ejde-320	192	46	≤	≤	NUM
ejde-320	192	47	∫	∫	PROPN
ejde-320	192	48	{	{	PUNCT
ejde-320	192	49	x∈b	x∈b	NOUN
ejde-320	192	50	:	:	PUNCT
ejde-320	192	51	u(x)>0	u(x)>0	PROPN
ejde-320	192	52	}	}	PUNCT
ejde-320	192	53	u+‖f‖l∞	u+‖f‖l∞	PROPN
ejde-320	192	54	dµ.	dµ.	NOUN
ejde-320	192	55	�	�	PROPN
ejde-320	192	56	corollary	corollary	NOUN
ejde-320	192	57	3.5	3.5	NUM
ejde-320	192	58	.	.	PUNCT
ejde-320	193	1	let	let	VERB
ejde-320	193	2	u	u	PRON
ejde-320	193	3	be	be	AUX
ejde-320	193	4	a	a	DET
ejde-320	193	5	weak	weak	ADJ
ejde-320	193	6	solution	solution	NOUN
ejde-320	193	7	to	to	ADP
ejde-320	193	8	(	(	PUNCT
ejde-320	193	9	1.1	1.1	NUM
ejde-320	193	10	)	)	PUNCT
ejde-320	193	11	in	in	ADP
ejde-320	193	12	b	b	NUM
ejde-320	193	13	,	,	PUNCT
ejde-320	193	14	and	and	CCONJ
ejde-320	193	15	suppose	suppose	VERB
ejde-320	193	16	that	that	SCONJ
ejde-320	193	17	for	for	ADP
ejde-320	193	18	some	some	DET
ejde-320	193	19	p	p	NOUN
ejde-320	193	20	>	>	X
ejde-320	193	21	0	0	PROPN
ejde-320	193	22	and	and	CCONJ
ejde-320	193	23	a	a	DET
ejde-320	193	24	non	non	ADJ
ejde-320	193	25	-	-	ADJ
ejde-320	193	26	negative	negative	ADJ
ejde-320	193	27	function	function	NOUN
ejde-320	193	28	v	v	ADP
ejde-320	193	29	∈w	∈w	VERB
ejde-320	193	30	1,2	1,2	NUM
ejde-320	193	31	a	a	DET
ejde-320	193	32	(	(	PUNCT
ejde-320	193	33	b	b	NOUN
ejde-320	193	34	)	)	PUNCT
ejde-320	193	35	it	it	PRON
ejde-320	193	36	holds	hold	VERB
ejde-320	193	37	‖f‖l∞	‖f‖l∞	NOUN
ejde-320	193	38	≤	≤	ADJ
ejde-320	193	39	pv(x	pv(x	NUM
ejde-320	193	40	)	)	PUNCT
ejde-320	193	41	,	,	PUNCT
ejde-320	193	42	a.e	a.e	PROPN
ejde-320	193	43	.	.	PROPN
ejde-320	193	44	x	x	SYM
ejde-320	193	45	∈	∈	PROPN
ejde-320	193	46	{	{	PUNCT
ejde-320	193	47	u	u	NOUN
ejde-320	193	48	>	>	X
ejde-320	193	49	0	0	NUM
ejde-320	193	50	}	}	PUNCT
ejde-320	193	51	∩b	∩b	NOUN
ejde-320	193	52	.	.	PUNCT
ejde-320	194	1	then	then	ADV
ejde-320	194	2	‖∇au+‖2l2	‖∇au+‖2l2	PROPN
ejde-320	194	3	≤	≤	PROPN
ejde-320	195	1	p	p	PRON
ejde-320	195	2	∫	∫	PROPN
ejde-320	195	3	(	(	PUNCT
ejde-320	195	4	u+v	u+v	PROPN
ejde-320	195	5	)	)	PUNCT
ejde-320	195	6	dµ.	dµ.	NOUN
ejde-320	195	7	(	(	PUNCT
ejde-320	195	8	3.3	3.3	NUM
ejde-320	195	9	)	)	PUNCT
ejde-320	195	10	proof	proof	NOUN
ejde-320	195	11	.	.	PUNCT
ejde-320	196	1	by	by	ADP
ejde-320	196	2	proposition	proposition	NOUN
ejde-320	196	3	3.4	3.4	NUM
ejde-320	196	4	we	we	PRON
ejde-320	196	5	have∫	have∫	AUX
ejde-320	196	6	{	{	PUNCT
ejde-320	196	7	u>0	u>0	PROPN
ejde-320	196	8	}	}	PUNCT
ejde-320	196	9	|∇au+|2	|∇au+|2	X
ejde-320	196	10	dµ	dµ	ADP
ejde-320	196	11	≤	≤	NUM
ejde-320	196	12	∫	∫	PROPN
ejde-320	196	13	{	{	PUNCT
ejde-320	196	14	u>0	u>0	PROPN
ejde-320	196	15	}	}	PUNCT
ejde-320	196	16	u+‖f‖l∞	u+‖f‖l∞	NOUN
ejde-320	196	17	dµ	dµ	ADP
ejde-320	196	18	,	,	PUNCT
ejde-320	196	19	and	and	CCONJ
ejde-320	196	20	using	use	VERB
ejde-320	196	21	the	the	DET
ejde-320	196	22	assumption	assumption	NOUN
ejde-320	196	23	gives∫	gives∫	NOUN
ejde-320	196	24	{	{	PUNCT
ejde-320	196	25	u>0	u>0	PROPN
ejde-320	196	26	}	}	PUNCT
ejde-320	196	27	|∇au+|2	|∇au+|2	X
ejde-320	196	28	dµ	dµ	ADP
ejde-320	196	29	≤	≤	NUM
ejde-320	196	30	∫	∫	PROPN
ejde-320	196	31	{	{	PUNCT
ejde-320	196	32	u>0	u>0	PROPN
ejde-320	196	33	}	}	PUNCT
ejde-320	196	34	u+pv	u+pv	PROPN
ejde-320	196	35	dµ	dµ	PROPN
ejde-320	196	36	,	,	PUNCT
ejde-320	196	37	‖∇au+‖2l2	‖∇au+‖2l2	PROPN
ejde-320	196	38	≤	≤	PROPN
ejde-320	196	39	p	p	PRON
ejde-320	196	40	∫	∫	PROPN
ejde-320	196	41	(	(	PUNCT
ejde-320	196	42	u+v	u+v	PROPN
ejde-320	196	43	)	)	PUNCT
ejde-320	196	44	dµ.	dµ.	PROPN
ejde-320	196	45	�	�	PROPN
ejde-320	196	46	lemma	lemma	PROPN
ejde-320	196	47	3.6	3.6	NUM
ejde-320	196	48	.	.	PUNCT
ejde-320	197	1	let	let	VERB
ejde-320	197	2	ϕ	ϕ	NOUN
ejde-320	197	3	be	be	AUX
ejde-320	197	4	a	a	DET
ejde-320	197	5	young	young	ADJ
ejde-320	197	6	function	function	NOUN
ejde-320	197	7	and	and	CCONJ
ejde-320	197	8	let	let	VERB
ejde-320	197	9	φ	φ	PROPN
ejde-320	197	10	be	be	AUX
ejde-320	197	11	defined	define	VERB
ejde-320	197	12	by	by	ADP
ejde-320	197	13	φ(t	φ(t	PROPN
ejde-320	197	14	)	)	PUNCT
ejde-320	197	15	=	=	SYM
ejde-320	197	16	ϕ(t2	ϕ(t2	NOUN
ejde-320	197	17	)	)	PUNCT
ejde-320	197	18	.	.	PUNCT
ejde-320	198	1	then	then	ADV
ejde-320	198	2	for	for	ADP
ejde-320	198	3	all	all	PRON
ejde-320	198	4	u	u	NOUN
ejde-320	198	5	∈	∈	NOUN
ejde-320	198	6	lφ(b	lφ(b	NUM
ejde-320	198	7	)	)	PUNCT
ejde-320	198	8	,	,	PUNCT
ejde-320	198	9	‖u2‖lϕ	‖u2‖lϕ	X
ejde-320	198	10	≤	≤	X
ejde-320	198	11	‖u‖2lφ	‖u‖2lφ	PUNCT
ejde-320	198	12	≤	≤	PROPN
ejde-320	198	13	4‖u2‖lϕ	4‖u2‖lϕ	NUM
ejde-320	198	14	.	.	PUNCT
ejde-320	199	1	proof	proof	NOUN
ejde-320	199	2	.	.	PUNCT
ejde-320	200	1	for	for	ADP
ejde-320	200	2	the	the	DET
ejde-320	200	3	first	first	ADJ
ejde-320	200	4	inequality	inequality	NOUN
ejde-320	200	5	using	use	VERB
ejde-320	200	6	definition	definition	NOUN
ejde-320	200	7	2.5	2.5	NUM
ejde-320	200	8	we	we	PRON
ejde-320	200	9	need	need	VERB
ejde-320	200	10	to	to	PART
ejde-320	200	11	show	show	VERB
ejde-320	200	12	that∫	that∫	PROPN
ejde-320	200	13	b	b	PROPN
ejde-320	200	14	ϕ	ϕ	X
ejde-320	200	15	(	(	PUNCT
ejde-320	200	16	u2	u2	PROPN
ejde-320	200	17	‖u‖2	‖u‖2	PROPN
ejde-320	200	18	lφ	lφ	PROPN
ejde-320	200	19	)	)	PUNCT
ejde-320	200	20	dµ	dµ	VERB
ejde-320	200	21	≤	≤	NUM
ejde-320	200	22	1	1	NUM
ejde-320	200	23	.	.	PUNCT
ejde-320	200	24	using	use	VERB
ejde-320	200	25	φ(t	φ(t	PROPN
ejde-320	200	26	)	)	PUNCT
ejde-320	201	1	=	=	SYM
ejde-320	201	2	ϕ(t2	ϕ(t2	X
ejde-320	201	3	)	)	PUNCT
ejde-320	201	4	we	we	PRON
ejde-320	201	5	have∫	have∫	VERB
ejde-320	201	6	b	b	X
ejde-320	201	7	ϕ	ϕ	X
ejde-320	201	8	(	(	PUNCT
ejde-320	201	9	u2	u2	PROPN
ejde-320	201	10	‖u‖2	‖u‖2	PROPN
ejde-320	201	11	lφ	lφ	PROPN
ejde-320	201	12	)	)	PUNCT
ejde-320	201	13	dµ	dµ	PROPN
ejde-320	202	1	=	=	SYM
ejde-320	202	2	∫	∫	PROPN
ejde-320	202	3	b	b	PROPN
ejde-320	203	1	φ	φ	PROPN
ejde-320	203	2	(	(	PUNCT
ejde-320	203	3	u	u	PROPN
ejde-320	203	4	‖u‖lφ	‖u‖lφ	PROPN
ejde-320	203	5	)	)	PUNCT
ejde-320	203	6	dµ	dµ	VERB
ejde-320	203	7	≤	≤	NUM
ejde-320	203	8	1	1	NUM
ejde-320	203	9	,	,	PUNCT
ejde-320	203	10	which	which	PRON
ejde-320	203	11	implies	imply	VERB
ejde-320	203	12	‖u2‖lϕ	‖u2‖lϕ	X
ejde-320	203	13	≤	≤	NOUN
ejde-320	203	14	‖u‖2lφ	‖u‖2lφ	PUNCT
ejde-320	203	15	.	.	PUNCT
ejde-320	204	1	to	to	PART
ejde-320	204	2	show	show	VERB
ejde-320	204	3	the	the	DET
ejde-320	204	4	second	second	ADJ
ejde-320	204	5	inequality	inequality	NOUN
ejde-320	204	6	we	we	PRON
ejde-320	204	7	need	need	VERB
ejde-320	204	8	to	to	PART
ejde-320	204	9	show	show	VERB
ejde-320	204	10	that	that	SCONJ
ejde-320	204	11	∫	∫	PROPN
ejde-320	204	12	ϕ	ϕ	PROPN
ejde-320	204	13	(	(	PUNCT
ejde-320	204	14	4u2	4u2	NUM
ejde-320	204	15	‖u‖2	‖u‖2	ADJ
ejde-320	204	16	lφ	lφ	PROPN
ejde-320	204	17	)	)	PUNCT
ejde-320	204	18	dµ	dµ	ADP
ejde-320	204	19	≥	≥	NOUN
ejde-320	204	20	1	1	NUM
ejde-320	204	21	.	.	PUNCT
ejde-320	205	1	once	once	ADV
ejde-320	205	2	again	again	ADV
ejde-320	205	3	using	use	VERB
ejde-320	205	4	φ(t	φ(t	NOUN
ejde-320	205	5	)	)	PUNCT
ejde-320	205	6	=	=	SYM
ejde-320	205	7	ϕ(t2	ϕ(t2	X
ejde-320	205	8	)	)	PUNCT
ejde-320	205	9	we	we	PRON
ejde-320	205	10	have∫	have∫	VERB
ejde-320	205	11	ϕ	ϕ	X
ejde-320	205	12	(	(	PUNCT
ejde-320	205	13	4u2	4u2	NUM
ejde-320	205	14	‖u‖2	‖u‖2	ADJ
ejde-320	205	15	lφ	lφ	NOUN
ejde-320	205	16	)	)	PUNCT
ejde-320	206	1	=	=	SYM
ejde-320	206	2	∫	∫	PROPN
ejde-320	206	3	φ	φ	PROPN
ejde-320	206	4	(	(	PUNCT
ejde-320	206	5	2u	2u	PROPN
ejde-320	206	6	‖u‖lφ	‖u‖lφ	PROPN
ejde-320	206	7	)	)	PUNCT
ejde-320	206	8	.	.	PUNCT
ejde-320	207	1	by	by	ADP
ejde-320	207	2	definition	definition	NOUN
ejde-320	207	3	(	(	PUNCT
ejde-320	207	4	2.2	2.2	NUM
ejde-320	207	5	)	)	PUNCT
ejde-320	207	6	,	,	PUNCT
ejde-320	207	7	‖u‖lφ	‖u‖lφ	PROPN
ejde-320	207	8	is	be	AUX
ejde-320	207	9	the	the	DET
ejde-320	207	10	smallest	small	ADJ
ejde-320	207	11	number	number	NOUN
ejde-320	207	12	such	such	ADJ
ejde-320	207	13	that	that	DET
ejde-320	207	14	∫	∫	PROPN
ejde-320	207	15	φ	φ	PROPN
ejde-320	207	16	(	(	PUNCT
ejde-320	207	17	u	u	NOUN
ejde-320	207	18	‖u‖	‖u‖	PROPN
ejde-320	207	19	lφ	lφ	ADV
ejde-320	207	20	)	)	PUNCT
ejde-320	207	21	≤	≤	NUM
ejde-320	207	22	1	1	NUM
ejde-320	207	23	,	,	PUNCT
ejde-320	207	24	and	and	CCONJ
ejde-320	207	25	therefore	therefore	ADV
ejde-320	207	26	∫	∫	PROPN
ejde-320	207	27	ϕ	ϕ	PROPN
ejde-320	207	28	(	(	PUNCT
ejde-320	207	29	4u2	4u2	NUM
ejde-320	207	30	‖w‖2	‖w‖2	VERB
ejde-320	207	31	lφ	lφ	ADV
ejde-320	207	32	)	)	PUNCT
ejde-320	208	1	=	=	SYM
ejde-320	208	2	∫	∫	PROPN
ejde-320	209	1	ϕ	ϕ	PROPN
ejde-320	209	2	(	(	PUNCT
ejde-320	209	3	u2	u2	PROPN
ejde-320	209	4	‖u‖2	‖u‖2	PROPN
ejde-320	209	5	lφ	lφ	PROPN
ejde-320	209	6	/4	/4	NOUN
ejde-320	209	7	)	)	PUNCT
ejde-320	210	1	=	=	SYM
ejde-320	211	1	∫	∫	PROPN
ejde-320	211	2	φ	φ	PROPN
ejde-320	211	3	(	(	PUNCT
ejde-320	211	4	u	u	PROPN
ejde-320	211	5	‖u‖lφ/2	‖u‖lφ/2	PROPN
ejde-320	211	6	)	)	PUNCT
ejde-320	211	7	≥	≥	NOUN
ejde-320	211	8	1	1	NUM
ejde-320	211	9	,	,	PUNCT
ejde-320	211	10	which	which	PRON
ejde-320	211	11	concludes	conclude	VERB
ejde-320	211	12	the	the	DET
ejde-320	211	13	proof	proof	NOUN
ejde-320	211	14	.	.	PUNCT
ejde-320	212	1	�	�	PROPN
ejde-320	212	2	ejde-2021/82	ejde-2021/82	VERB
ejde-320	212	3	orlicz	orlicz	ADJ
ejde-320	212	4	-	-	PUNCT
ejde-320	212	5	sobolev	sobolev	NOUN
ejde-320	212	6	inequalities	inequality	NOUN
ejde-320	212	7	and	and	CCONJ
ejde-320	212	8	the	the	DET
ejde-320	212	9	dirichlet	dirichlet	PROPN
ejde-320	212	10	problem	problem	NOUN
ejde-320	212	11	9	9	NUM
ejde-320	212	12	we	we	PRON
ejde-320	212	13	are	be	AUX
ejde-320	212	14	now	now	ADV
ejde-320	212	15	ready	ready	ADJ
ejde-320	212	16	to	to	PART
ejde-320	212	17	prove	prove	VERB
ejde-320	212	18	the	the	DET
ejde-320	212	19	l∞	l∞	NOUN
ejde-320	212	20	estimate	estimate	NOUN
ejde-320	212	21	in	in	ADP
ejde-320	212	22	theorem	theorem	ADJ
ejde-320	212	23	1.1	1.1	NUM
ejde-320	212	24	,	,	PUNCT
ejde-320	212	25	and	and	CCONJ
ejde-320	212	26	the	the	DET
ejde-320	212	27	argument	argument	NOUN
ejde-320	212	28	follows	follow	VERB
ejde-320	212	29	closely	closely	ADV
ejde-320	212	30	the	the	DET
ejde-320	212	31	proof	proof	NOUN
ejde-320	212	32	of	of	ADP
ejde-320	212	33	[	[	X
ejde-320	212	34	7	7	NUM
ejde-320	212	35	,	,	PUNCT
ejde-320	212	36	proposition	proposition	NOUN
ejde-320	212	37	27	27	NUM
ejde-320	212	38	]	]	PUNCT
ejde-320	212	39	.	.	PUNCT
ejde-320	213	1	theorem	theorem	VERB
ejde-320	213	2	3.7	3.7	NUM
ejde-320	213	3	.	.	PUNCT
ejde-320	214	1	let	let	VERB
ejde-320	214	2	l	l	NOUN
ejde-320	214	3	=	=	SYM
ejde-320	214	4	∇·a∇	∇·a∇	PROPN
ejde-320	214	5	with	with	ADP
ejde-320	214	6	bounded	bounded	ADJ
ejde-320	214	7	measurable	measurable	ADJ
ejde-320	214	8	non	non	ADJ
ejde-320	214	9	-	-	ADJ
ejde-320	214	10	negative	negative	ADJ
ejde-320	214	11	semidefinite	semidefinite	NOUN
ejde-320	214	12	matrix	matrix	NOUN
ejde-320	214	13	a	a	PRON
ejde-320	214	14	,	,	PUNCT
ejde-320	214	15	and	and	CCONJ
ejde-320	214	16	d	d	ADP
ejde-320	214	17	a	a	DET
ejde-320	214	18	metric	metric	NOUN
ejde-320	214	19	on	on	ADP
ejde-320	214	20	rn	rn	PROPN
ejde-320	214	21	,	,	PUNCT
ejde-320	214	22	such	such	ADJ
ejde-320	214	23	that	that	SCONJ
ejde-320	214	24	for	for	ADP
ejde-320	214	25	any	any	DET
ejde-320	214	26	metric	metric	ADJ
ejde-320	214	27	ball	ball	NOUN
ejde-320	214	28	b	b	NOUN
ejde-320	214	29	=	=	PUNCT
ejde-320	214	30	b(x	b(x	PROPN
ejde-320	214	31	,	,	PUNCT
ejde-320	214	32	r	r	NOUN
ejde-320	214	33	)	)	PUNCT
ejde-320	214	34	with	with	ADP
ejde-320	214	35	0	0	NUM
ejde-320	214	36	<	<	X
ejde-320	214	37	r	r	NOUN
ejde-320	214	38	<	<	X
ejde-320	214	39	∞	∞	NUM
ejde-320	214	40	it	it	PRON
ejde-320	214	41	holds	hold	VERB
ejde-320	214	42	0	0	NUM
ejde-320	214	43	<	<	X
ejde-320	214	44	|b|	|b|	PROPN
ejde-320	214	45	<	<	X
ejde-320	214	46	∞.	∞.	PROPN
ejde-320	214	47	suppose	suppose	VERB
ejde-320	214	48	also	also	ADV
ejde-320	214	49	that	that	SCONJ
ejde-320	214	50	the	the	DET
ejde-320	214	51	following	follow	VERB
ejde-320	214	52	orlicz	orlicz	ADJ
ejde-320	214	53	-	-	PUNCT
ejde-320	214	54	sobolev	sobolev	NOUN
ejde-320	214	55	inequality	inequality	NOUN
ejde-320	214	56	holds	hold	VERB
ejde-320	214	57	for	for	ADP
ejde-320	214	58	all	all	PRON
ejde-320	214	59	v	v	NOUN
ejde-320	214	60	∈	∈	NOUN
ejde-320	214	61	(	(	PUNCT
ejde-320	214	62	w	w	PROPN
ejde-320	214	63	1,2	1,2	NUM
ejde-320	214	64	a	a	PRON
ejde-320	214	65	)	)	PUNCT
ejde-320	214	66	0	0	PUNCT
ejde-320	215	1	(	(	PUNCT
ejde-320	215	2	b	b	NOUN
ejde-320	215	3	)	)	PUNCT
ejde-320	215	4	and	and	CCONJ
ejde-320	215	5	the	the	DET
ejde-320	215	6	metric	metric	ADJ
ejde-320	215	7	ball	ball	PROPN
ejde-320	215	8	b	b	PROPN
ejde-320	215	9	⊂	⊂	PROPN
ejde-320	215	10	rn	rn	PROPN
ejde-320	215	11	,	,	PUNCT
ejde-320	215	12	‖v‖lφ(b	‖v‖lφ(b	ADJ
ejde-320	215	13	)	)	PUNCT
ejde-320	215	14	≤	≤	NOUN
ejde-320	216	1	c(b)‖∇av‖l2(b	c(b)‖∇av‖l2(b	NUM
ejde-320	216	2	)	)	PUNCT
ejde-320	216	3	,	,	PUNCT
ejde-320	216	4	(	(	PUNCT
ejde-320	216	5	3.4	3.4	NUM
ejde-320	216	6	)	)	PUNCT
ejde-320	216	7	where	where	SCONJ
ejde-320	216	8	φ	φ	PROPN
ejde-320	216	9	is	be	AUX
ejde-320	216	10	defined	define	VERB
ejde-320	216	11	by	by	ADP
ejde-320	216	12	φ(t	φ(t	PROPN
ejde-320	216	13	)	)	PUNCT
ejde-320	216	14	=	=	SYM
ejde-320	216	15	φ(t2	φ(t2	NOUN
ejde-320	216	16	)	)	PUNCT
ejde-320	216	17	with	with	ADP
ejde-320	216	18	φ	φ	PROPN
ejde-320	216	19	=	=	SYM
ejde-320	216	20	φn	φn	PROPN
ejde-320	216	21	from	from	ADP
ejde-320	216	22	definition	definition	NOUN
ejde-320	216	23	2.9	2.9	NUM
ejde-320	216	24	,	,	PUNCT
ejde-320	216	25	for	for	ADP
ejde-320	216	26	some	some	PRON
ejde-320	216	27	n	n	NOUN
ejde-320	216	28	>	>	X
ejde-320	216	29	1	1	NUM
ejde-320	216	30	.	.	PUNCT
ejde-320	217	1	then	then	ADV
ejde-320	217	2	the	the	DET
ejde-320	217	3	unique	unique	ADJ
ejde-320	217	4	weak	weak	ADJ
ejde-320	217	5	solution	solution	NOUN
ejde-320	217	6	u	u	NOUN
ejde-320	217	7	to	to	ADP
ejde-320	217	8	(	(	PUNCT
ejde-320	217	9	3.2	3.2	NUM
ejde-320	217	10	)	)	PUNCT
ejde-320	217	11	satisfies	satisfie	NOUN
ejde-320	217	12	sup	sup	PROPN
ejde-320	217	13	b	b	PROPN
ejde-320	217	14	|u|	|u|	ADJ
ejde-320	217	15	≤	≤	NOUN
ejde-320	217	16	c‖f‖l∞(b	c‖f‖l∞(b	NOUN
ejde-320	217	17	)	)	PUNCT
ejde-320	217	18	.	.	PUNCT
ejde-320	218	1	proof	proof	NOUN
ejde-320	218	2	.	.	PUNCT
ejde-320	219	1	we	we	PRON
ejde-320	219	2	first	first	ADV
ejde-320	219	3	define	define	VERB
ejde-320	219	4	the	the	DET
ejde-320	219	5	family	family	NOUN
ejde-320	219	6	of	of	ADP
ejde-320	219	7	truncations	truncation	NOUN
ejde-320	219	8	uk	uk	PROPN
ejde-320	219	9	=	=	SYM
ejde-320	219	10	(	(	PUNCT
ejde-320	219	11	u−	u−	PROPN
ejde-320	219	12	ck)+	ck)+	NOUN
ejde-320	219	13	,	,	PUNCT
ejde-320	219	14	where	where	SCONJ
ejde-320	219	15	ck	ck	ADV
ejde-320	219	16	=	=	X
ejde-320	219	17	τ‖f‖l∞	τ‖f‖l∞	X
ejde-320	219	18	(	(	PUNCT
ejde-320	219	19	1−	1−	NUM
ejde-320	219	20	c(k	c(k	NOUN
ejde-320	219	21	+	+	CCONJ
ejde-320	219	22	1)−ε/2	1)−ε/2	NUM
ejde-320	219	23	)	)	PUNCT
ejde-320	219	24	,	,	PUNCT
ejde-320	219	25	τ	τ	X
ejde-320	219	26	≥	≥	NUM
ejde-320	219	27	1	1	NUM
ejde-320	219	28	,	,	PUNCT
ejde-320	219	29	and	and	CCONJ
ejde-320	219	30	denote	denote	VERB
ejde-320	219	31	uk	uk	PROPN
ejde-320	219	32	≡	≡	PROPN
ejde-320	219	33	∫	∫	PROPN
ejde-320	219	34	b	b	PROPN
ejde-320	219	35	|uk|2	|uk|2	PROPN
ejde-320	219	36	dµ	dµ	PROPN
ejde-320	219	37	,	,	PUNCT
ejde-320	219	38	where	where	SCONJ
ejde-320	219	39	dµ	dµ	PROPN
ejde-320	219	40	=	=	PUNCT
ejde-320	219	41	dx/|b|	dx/|b|	PROPN
ejde-320	219	42	.	.	PUNCT
ejde-320	220	1	since	since	SCONJ
ejde-320	220	2	uk	uk	PROPN
ejde-320	220	3	∈	∈	PROPN
ejde-320	220	4	(	(	PUNCT
ejde-320	220	5	w	w	PROPN
ejde-320	220	6	1,2	1,2	NUM
ejde-320	220	7	a	a	PRON
ejde-320	220	8	)	)	PUNCT
ejde-320	220	9	0	0	PUNCT
ejde-320	220	10	(	(	PUNCT
ejde-320	220	11	b	b	NOUN
ejde-320	220	12	)	)	PUNCT
ejde-320	220	13	for	for	ADP
ejde-320	220	14	all	all	DET
ejde-320	220	15	k	k	PROPN
ejde-320	220	16	,	,	PUNCT
ejde-320	220	17	using	use	VERB
ejde-320	220	18	hölder	hölder	NOUN
ejde-320	220	19	’s	’s	PART
ejde-320	220	20	inequality	inequality	NOUN
ejde-320	220	21	for	for	ADP
ejde-320	220	22	orlicz	orlicz	ADJ
ejde-320	220	23	spaces	space	NOUN
ejde-320	220	24	(	(	PUNCT
ejde-320	220	25	2.4	2.4	NUM
ejde-320	220	26	)	)	PUNCT
ejde-320	220	27	we	we	PRON
ejde-320	220	28	can	can	AUX
ejde-320	220	29	write∫	write∫	VERB
ejde-320	220	30	u2	u2	PROPN
ejde-320	220	31	k+1	k+1	PROPN
ejde-320	220	32	dµ	dµ	PROPN
ejde-320	220	33	≤	≤	NUM
ejde-320	220	34	c‖u2	c‖u2	PROPN
ejde-320	220	35	k+1‖lφ	k+1‖lφ	PROPN
ejde-320	220	36	·	·	PUNCT
ejde-320	220	37	‖1‖	‖1‖	PROPN
ejde-320	220	38	lφ̃	lφ̃	PUNCT
ejde-320	220	39	{	{	PUNCT
ejde-320	220	40	uk+1>0	uk+1>0	NOUN
ejde-320	220	41	}	}	PUNCT
ejde-320	220	42	,	,	PUNCT
ejde-320	220	43	(	(	PUNCT
ejde-320	220	44	3.5	3.5	NUM
ejde-320	220	45	)	)	PUNCT
ejde-320	220	46	where	where	SCONJ
ejde-320	220	47	the	the	DET
ejde-320	220	48	norms	norm	NOUN
ejde-320	220	49	are	be	AUX
ejde-320	220	50	taken	take	VERB
ejde-320	220	51	with	with	ADP
ejde-320	220	52	respect	respect	NOUN
ejde-320	220	53	to	to	ADP
ejde-320	220	54	the	the	DET
ejde-320	220	55	measure	measure	NOUN
ejde-320	220	56	µ.	µ.	VERB
ejde-320	220	57	our	our	PRON
ejde-320	220	58	first	first	ADJ
ejde-320	220	59	goal	goal	NOUN
ejde-320	220	60	is	be	AUX
ejde-320	220	61	to	to	PART
ejde-320	220	62	bound	bind	VERB
ejde-320	220	63	the	the	DET
ejde-320	220	64	first	first	ADJ
ejde-320	220	65	factor	factor	NOUN
ejde-320	220	66	on	on	ADP
ejde-320	220	67	the	the	DET
ejde-320	220	68	right	right	NOUN
ejde-320	220	69	.	.	PUNCT
ejde-320	221	1	note	note	VERB
ejde-320	221	2	that	that	SCONJ
ejde-320	221	3	if	if	SCONJ
ejde-320	221	4	uk+1	uk+1	ADP
ejde-320	221	5	>	>	X
ejde-320	221	6	0	0	NUM
ejde-320	221	7	we	we	PRON
ejde-320	221	8	have	have	VERB
ejde-320	221	9	u	u	NOUN
ejde-320	221	10	>	>	X
ejde-320	221	11	ck+1	ck+1	PROPN
ejde-320	221	12	=	=	SYM
ejde-320	221	13	τ‖f‖l∞	τ‖f‖l∞	X
ejde-320	221	14	(	(	PUNCT
ejde-320	221	15	1−	1−	NUM
ejde-320	221	16	c(k	c(k	NOUN
ejde-320	221	17	+	+	CCONJ
ejde-320	221	18	2)−ε/2	2)−ε/2	NUM
ejde-320	221	19	)	)	PUNCT
ejde-320	221	20	,	,	PUNCT
ejde-320	221	21	which	which	PRON
ejde-320	221	22	implies	imply	VERB
ejde-320	221	23	that	that	SCONJ
ejde-320	221	24	uk	uk	PROPN
ejde-320	221	25	=	=	SYM
ejde-320	221	26	(	(	PUNCT
ejde-320	221	27	u−	u−	PROPN
ejde-320	221	28	ck)+	ck)+	NOUN
ejde-320	221	29	>	>	X
ejde-320	221	30	cτ‖f‖l∞	cτ‖f‖l∞	X
ejde-320	221	31	[	[	PUNCT
ejde-320	221	32	(	(	PUNCT
ejde-320	221	33	k	k	X
ejde-320	221	34	+	+	PROPN
ejde-320	221	35	1)−ε/2	1)−ε/2	NUM
ejde-320	221	36	−	−	PROPN
ejde-320	221	37	(	(	PUNCT
ejde-320	221	38	k	k	X
ejde-320	221	39	+	+	X
ejde-320	221	40	2)−ε/2	2)−ε/2	NUM
ejde-320	221	41	]	]	PUNCT
ejde-320	221	42	=	=	PUNCT
ejde-320	221	43	cτ‖f‖l∞(k	cτ‖f‖l∞(k	PUNCT
ejde-320	222	1	+	+	NUM
ejde-320	222	2	1)−ε/2	1)−ε/2	NUM
ejde-320	222	3	[	[	PUNCT
ejde-320	222	4	1−	1−	NUM
ejde-320	222	5	(	(	PUNCT
ejde-320	222	6	k	k	NOUN
ejde-320	222	7	+	+	PROPN
ejde-320	222	8	1	1	NUM
ejde-320	222	9	k	k	NOUN
ejde-320	222	10	+	+	ADJ
ejde-320	222	11	2	2	NUM
ejde-320	222	12	)	)	PUNCT
ejde-320	222	13	ε/2	ε/2	PROPN
ejde-320	222	14	]	]	PUNCT
ejde-320	222	15	≥	≥	X
ejde-320	222	16	cτ‖f‖l∞(k	cτ‖f‖l∞(k	CCONJ
ejde-320	223	1	+	+	NUM
ejde-320	223	2	1)−ε/2	1)−ε/2	NUM
ejde-320	223	3	(	(	PUNCT
ejde-320	223	4	1−	1−	NUM
ejde-320	223	5	k	k	NOUN
ejde-320	224	1	+	+	CCONJ
ejde-320	224	2	1	1	NUM
ejde-320	224	3	k	k	NOUN
ejde-320	224	4	+	+	CCONJ
ejde-320	224	5	2	2	X
ejde-320	224	6	)	)	PUNCT
ejde-320	224	7	ε	ε	PROPN
ejde-320	224	8	2	2	NUM
ejde-320	224	9	(	(	PUNCT
ejde-320	224	10	k	k	PROPN
ejde-320	224	11	+	+	PROPN
ejde-320	224	12	1	1	NUM
ejde-320	224	13	k	k	NOUN
ejde-320	224	14	+	+	CCONJ
ejde-320	224	15	2	2	X
ejde-320	224	16	)	)	PUNCT
ejde-320	224	17	ε	ε	PROPN
ejde-320	224	18	2−1	2−1	NUM
ejde-320	224	19	.	.	PUNCT
ejde-320	225	1	note	note	VERB
ejde-320	225	2	that	that	SCONJ
ejde-320	225	3	k+1	k+1	X
ejde-320	225	4	k+2	k+2	PROPN
ejde-320	225	5	<	<	X
ejde-320	225	6	1	1	NUM
ejde-320	225	7	which	which	PRON
ejde-320	225	8	allows	allow	VERB
ejde-320	225	9	us	we	PRON
ejde-320	225	10	to	to	PART
ejde-320	225	11	conclude	conclude	VERB
ejde-320	225	12	that	that	SCONJ
ejde-320	225	13	uk	uk	PROPN
ejde-320	225	14	≥	≥	NOUN
ejde-320	225	15	ε	ε	PROPN
ejde-320	225	16	2	2	NUM
ejde-320	225	17	cτ‖f‖l∞(k	cτ‖f‖l∞(k	CCONJ
ejde-320	225	18	+	+	NUM
ejde-320	225	19	2)−1−	2)−1−	PROPN
ejde-320	225	20	ε2	ε2	NOUN
ejde-320	225	21	on	on	ADP
ejde-320	225	22	the	the	DET
ejde-320	225	23	set	set	NOUN
ejde-320	225	24	where	where	SCONJ
ejde-320	225	25	uk+1	uk+1	ADP
ejde-320	225	26	>	>	X
ejde-320	225	27	0	0	NUM
ejde-320	225	28	;	;	PUNCT
ejde-320	225	29	thus	thus	ADV
ejde-320	225	30	‖f‖l∞	‖f‖l∞	X
ejde-320	225	31	≤	≤	ADJ
ejde-320	225	32	2	2	NUM
ejde-320	225	33	cτε	cτε	NOUN
ejde-320	225	34	(	(	PUNCT
ejde-320	225	35	k	k	NOUN
ejde-320	226	1	+	+	CCONJ
ejde-320	226	2	2)1	2)1	NUM
ejde-320	226	3	+	+	CCONJ
ejde-320	226	4	ε	ε	PROPN
ejde-320	226	5	2uk	2uk	NOUN
ejde-320	226	6	≤	≤	ADV
ejde-320	226	7	2	2	NUM
ejde-320	226	8	cε	cε	NOUN
ejde-320	226	9	(	(	PUNCT
ejde-320	226	10	k	k	NOUN
ejde-320	226	11	+	+	CCONJ
ejde-320	226	12	2)1	2)1	NUM
ejde-320	226	13	+	+	SYM
ejde-320	226	14	ε	ε	PROPN
ejde-320	226	15	2uk	2uk	NOUN
ejde-320	226	16	,	,	PUNCT
ejde-320	226	17	(	(	PUNCT
ejde-320	226	18	3.6	3.6	NUM
ejde-320	226	19	)	)	PUNCT
ejde-320	226	20	since	since	SCONJ
ejde-320	226	21	τ	τ	PROPN
ejde-320	226	22	≥	≥	NUM
ejde-320	226	23	1	1	NUM
ejde-320	226	24	.	.	PUNCT
ejde-320	227	1	next	next	ADV
ejde-320	227	2	,	,	PUNCT
ejde-320	227	3	since	since	SCONJ
ejde-320	227	4	u	u	NOUN
ejde-320	227	5	is	be	AUX
ejde-320	227	6	a	a	DET
ejde-320	227	7	weak	weak	ADJ
ejde-320	227	8	solution	solution	NOUN
ejde-320	227	9	it	it	PRON
ejde-320	227	10	follows	follow	VERB
ejde-320	227	11	that	that	SCONJ
ejde-320	227	12	u−ck+1	u−ck+1	NOUN
ejde-320	227	13	is	be	AUX
ejde-320	227	14	also	also	ADV
ejde-320	227	15	a	a	DET
ejde-320	227	16	weak	weak	ADJ
ejde-320	227	17	solution	solution	NOUN
ejde-320	227	18	so	so	SCONJ
ejde-320	227	19	(	(	PUNCT
ejde-320	227	20	3.6	3.6	NUM
ejde-320	227	21	)	)	PUNCT
ejde-320	227	22	implies	imply	VERB
ejde-320	227	23	we	we	PRON
ejde-320	227	24	can	can	AUX
ejde-320	227	25	use	use	VERB
ejde-320	227	26	(	(	PUNCT
ejde-320	227	27	3.3	3.3	NUM
ejde-320	227	28	)	)	PUNCT
ejde-320	227	29	with	with	ADP
ejde-320	227	30	v	v	NOUN
ejde-320	227	31	=	=	SYM
ejde-320	227	32	uk	uk	PROPN
ejde-320	227	33	and	and	CCONJ
ejde-320	227	34	p	p	NOUN
ejde-320	227	35	=	=	PROPN
ejde-320	227	36	2	2	NUM
ejde-320	227	37	cε	cε	NOUN
ejde-320	227	38	(	(	PUNCT
ejde-320	227	39	k+	k+	NOUN
ejde-320	228	1	2)1	2)1	NUM
ejde-320	228	2	+	+	CCONJ
ejde-320	228	3	ε	ε	PROPN
ejde-320	228	4	2	2	NUM
ejde-320	228	5	,	,	PUNCT
ejde-320	228	6	which	which	PRON
ejde-320	228	7	gives	give	VERB
ejde-320	228	8	∫	∫	PROPN
ejde-320	228	9	|∇auk+1|2	|∇auk+1|2	PROPN
ejde-320	228	10	dµ	dµ	PROPN
ejde-320	228	11	=	=	SYM
ejde-320	228	12	∫	∫	PROPN
ejde-320	228	13	∣∣∇a(u−	∣∣∇a(u−	X
ejde-320	228	14	ck+1	ck+1	X
ejde-320	228	15	)	)	PUNCT
ejde-320	229	1	+	+	CCONJ
ejde-320	229	2	∣∣2	∣∣2	PROPN
ejde-320	229	3	dµ	dµ	ADJ
ejde-320	229	4	≤	≤	NUM
ejde-320	229	5	c	c	NOUN
ejde-320	229	6	2	2	NUM
ejde-320	229	7	cε	cε	X
ejde-320	229	8	(	(	PUNCT
ejde-320	229	9	k	k	NOUN
ejde-320	229	10	+	+	CCONJ
ejde-320	229	11	2)1	2)1	NUM
ejde-320	229	12	+	+	SYM
ejde-320	229	13	ε	ε	PROPN
ejde-320	229	14	2	2	NUM
ejde-320	229	15	∫	∫	PROPN
ejde-320	229	16	(	(	PUNCT
ejde-320	229	17	uk+1uk	uk+1uk	PROPN
ejde-320	229	18	)	)	PUNCT
ejde-320	229	19	dµ	dµ	VERB
ejde-320	229	20	10	10	NUM
ejde-320	229	21	u.	u.	PROPN
ejde-320	229	22	hafeez	hafeez	PROPN
ejde-320	229	23	,	,	PUNCT
ejde-320	229	24	t.	t.	PROPN
ejde-320	229	25	lavier	lavier	PROPN
ejde-320	229	26	,	,	PUNCT
ejde-320	229	27	l.	l.	PROPN
ejde-320	229	28	williams	williams	PROPN
ejde-320	229	29	,	,	PUNCT
ejde-320	229	30	l.	l.	PROPN
ejde-320	229	31	korobenko	korobenko	PROPN
ejde-320	229	32	ejde-2021/82	ejde-2021/82	VERB
ejde-320	229	33	≤	≤	NUM
ejde-320	229	34	c	c	NOUN
ejde-320	229	35	2	2	NUM
ejde-320	229	36	cε	cε	X
ejde-320	229	37	(	(	PUNCT
ejde-320	229	38	k	k	NOUN
ejde-320	229	39	+	+	CCONJ
ejde-320	229	40	2)1	2)1	NUM
ejde-320	229	41	+	+	SYM
ejde-320	229	42	ε	ε	PROPN
ejde-320	229	43	2	2	NUM
ejde-320	229	44	∫	∫	PROPN
ejde-320	229	45	u2	u2	PROPN
ejde-320	229	46	k	k	PROPN
ejde-320	229	47	dµ.	dµ.	PROPN
ejde-320	229	48	applying	apply	VERB
ejde-320	229	49	(	(	PUNCT
ejde-320	229	50	3.4	3.4	NUM
ejde-320	229	51	)	)	PUNCT
ejde-320	229	52	and	and	CCONJ
ejde-320	229	53	using	use	VERB
ejde-320	229	54	lemma	lemma	PROPN
ejde-320	229	55	3.6	3.6	NUM
ejde-320	229	56	with	with	ADP
ejde-320	229	57	ϕ	ϕ	NOUN
ejde-320	229	58	=	=	SYM
ejde-320	229	59	φ	φ	PROPN
ejde-320	229	60	we	we	PRON
ejde-320	229	61	have	have	AUX
ejde-320	229	62	‖u2	‖u2	VERB
ejde-320	229	63	k+1‖lφ	k+1‖lφ	PROPN
ejde-320	229	64	≤	≤	NOUN
ejde-320	229	65	‖uk+1‖2lφ	‖uk+1‖2lφ	NOUN
ejde-320	229	66	≤	≤	ADJ
ejde-320	229	67	c‖∇auk+1‖2	c‖∇auk+1‖2	NOUN
ejde-320	229	68	,	,	PUNCT
ejde-320	229	69	which	which	PRON
ejde-320	229	70	combining	combine	VERB
ejde-320	229	71	with	with	ADP
ejde-320	229	72	the	the	DET
ejde-320	229	73	above	above	ADJ
ejde-320	229	74	inequality	inequality	NOUN
ejde-320	229	75	gives	give	VERB
ejde-320	229	76	‖u2	‖u2	PROPN
ejde-320	229	77	k+1‖lφ	k+1‖lφ	VERB
ejde-320	229	78	≤	≤	ADJ
ejde-320	229	79	c(k	c(k	NOUN
ejde-320	229	80	+	+	CCONJ
ejde-320	229	81	2	2	NUM
ejde-320	229	82	)	)	PUNCT
ejde-320	229	83	2+ε	2+ε	NUM
ejde-320	229	84	2	2	NUM
ejde-320	229	85	∫	∫	NOUN
ejde-320	229	86	u2	u2	PROPN
ejde-320	229	87	k	k	PROPN
ejde-320	229	88	dµ.	dµ.	PROPN
ejde-320	229	89	(	(	PUNCT
ejde-320	229	90	3.7	3.7	NUM
ejde-320	229	91	)	)	PUNCT
ejde-320	229	92	now	now	ADV
ejde-320	229	93	we	we	PRON
ejde-320	229	94	want	want	VERB
ejde-320	229	95	to	to	PART
ejde-320	229	96	bound	bound	VERB
ejde-320	229	97	the	the	DET
ejde-320	229	98	second	second	ADJ
ejde-320	229	99	factor	factor	NOUN
ejde-320	229	100	on	on	ADP
ejde-320	229	101	the	the	DET
ejde-320	229	102	right	right	ADJ
ejde-320	229	103	hand	hand	NOUN
ejde-320	229	104	side	side	NOUN
ejde-320	229	105	of	of	ADP
ejde-320	229	106	(	(	PUNCT
ejde-320	229	107	3.5	3.5	NUM
ejde-320	229	108	)	)	PUNCT
ejde-320	229	109	,	,	PUNCT
ejde-320	229	110	‖1‖lφ̃	‖1‖lφ̃	PROPN
ejde-320	229	111	.	.	PUNCT
ejde-320	230	1	consider	consider	VERB
ejde-320	230	2	the	the	DET
ejde-320	230	3	function	function	NOUN
ejde-320	230	4	γ(t	γ(t	NOUN
ejde-320	230	5	)	)	PUNCT
ejde-320	230	6	:	:	PUNCT
ejde-320	231	1	=	=	NOUN
ejde-320	231	2	1	1	NUM
ejde-320	231	3	φ̃−1	φ̃−1	NOUN
ejde-320	231	4	(	(	PUNCT
ejde-320	231	5	1	1	NUM
ejde-320	231	6	t	t	NOUN
ejde-320	231	7	)	)	PUNCT
ejde-320	231	8	,	,	PUNCT
ejde-320	231	9	and	and	CCONJ
ejde-320	231	10	note	note	VERB
ejde-320	231	11	that	that	SCONJ
ejde-320	231	12	∫	∫	PROPN
ejde-320	231	13	{	{	PUNCT
ejde-320	231	14	uk+1>0	uk+1>0	NOUN
ejde-320	231	15	}	}	PUNCT
ejde-320	231	16	φ̃	φ̃	PROPN
ejde-320	231	17	(	(	PUNCT
ejde-320	231	18	1	1	NUM
ejde-320	231	19	a	a	PRON
ejde-320	231	20	)	)	PUNCT
ejde-320	231	21	dµ	dµ	PROPN
ejde-320	231	22	=	=	PUNCT
ejde-320	231	23	φ̃	φ̃	PROPN
ejde-320	231	24	(	(	PUNCT
ejde-320	231	25	1	1	NUM
ejde-320	231	26	a	a	X
ejde-320	231	27	)	)	PUNCT
ejde-320	231	28	µ({uk+1	µ({uk+1	SYM
ejde-320	231	29	>	>	NOUN
ejde-320	231	30	0}big	0}big	X
ejde-320	231	31	)	)	PUNCT
ejde-320	231	32	for	for	ADP
ejde-320	231	33	all	all	DET
ejde-320	231	34	a	a	DET
ejde-320	231	35	>	>	X
ejde-320	231	36	0	0	X
ejde-320	231	37	.	.	PUNCT
ejde-320	231	38	now	now	ADV
ejde-320	231	39	let	let	VERB
ejde-320	231	40	a	a	DET
ejde-320	231	41	=	=	SYM
ejde-320	231	42	γ	γ	X
ejde-320	231	43	(	(	PUNCT
ejde-320	231	44	µ({uk+1	µ({uk+1	INTJ
ejde-320	231	45	>	>	PUNCT
ejde-320	231	46	0	0	NUM
ejde-320	231	47	}	}	PUNCT
ejde-320	231	48	)	)	PUNCT
ejde-320	231	49	)	)	PUNCT
ejde-320	232	1	=	=	SYM
ejde-320	232	2	1	1	NUM
ejde-320	232	3	φ̃−1	φ̃−1	X
ejde-320	232	4	(	(	PUNCT
ejde-320	232	5	1	1	NUM
ejde-320	232	6	µ({uk+1>0	µ({uk+1>0	NUM
ejde-320	232	7	}	}	PUNCT
ejde-320	232	8	)	)	PUNCT
ejde-320	232	9	)	)	PUNCT
ejde-320	232	10	,	,	PUNCT
ejde-320	232	11	so	so	SCONJ
ejde-320	232	12	that	that	SCONJ
ejde-320	232	13	∫	∫	PROPN
ejde-320	232	14	{	{	PUNCT
ejde-320	232	15	uk+1>0	uk+1>0	NOUN
ejde-320	232	16	}	}	PUNCT
ejde-320	232	17	φ̃	φ̃	PROPN
ejde-320	232	18	(	(	PUNCT
ejde-320	232	19	1	1	NUM
ejde-320	232	20	a	a	PRON
ejde-320	232	21	)	)	PUNCT
ejde-320	232	22	dµ	dµ	PROPN
ejde-320	232	23	=	=	SYM
ejde-320	232	24	1	1	NUM
ejde-320	232	25	,	,	PUNCT
ejde-320	232	26	and	and	CCONJ
ejde-320	232	27	therefore	therefore	ADV
ejde-320	232	28	‖1‖	‖1‖	PROPN
ejde-320	232	29	lφ̃	lφ̃	PUNCT
ejde-320	233	1	(	(	PUNCT
ejde-320	233	2	{	{	PUNCT
ejde-320	233	3	uk+1>0	uk+1>0	NOUN
ejde-320	233	4	}	}	PUNCT
ejde-320	233	5	)	)	PUNCT
ejde-320	233	6	≤	≤	NOUN
ejde-320	233	7	a	a	DET
ejde-320	233	8	=	=	SYM
ejde-320	233	9	γ	γ	X
ejde-320	233	10	(	(	PUNCT
ejde-320	233	11	µ({uk+1	µ({uk+1	INTJ
ejde-320	233	12	>	>	PUNCT
ejde-320	233	13	0	0	NUM
ejde-320	233	14	}	}	PUNCT
ejde-320	233	15	)	)	PUNCT
ejde-320	233	16	)	)	PUNCT
ejde-320	233	17	.	.	PUNCT
ejde-320	234	1	(	(	PUNCT
ejde-320	234	2	3.8	3.8	NUM
ejde-320	234	3	)	)	PUNCT
ejde-320	234	4	now	now	ADV
ejde-320	234	5	recall	recall	VERB
ejde-320	234	6	that	that	SCONJ
ejde-320	234	7	we	we	PRON
ejde-320	234	8	showed	show	VERB
ejde-320	234	9	{	{	PUNCT
ejde-320	234	10	uk+1	uk+1	X
ejde-320	234	11	>	>	X
ejde-320	234	12	0	0	NUM
ejde-320	234	13	}	}	PUNCT
ejde-320	234	14	⊂	⊂	PROPN
ejde-320	234	15	{	{	PUNCT
ejde-320	234	16	uk	uk	PROPN
ejde-320	234	17	>	>	X
ejde-320	234	18	ε	ε	PROPN
ejde-320	234	19	2	2	NUM
ejde-320	234	20	cτ‖f‖l∞(k	cτ‖f‖l∞(k	CCONJ
ejde-320	234	21	+	+	NUM
ejde-320	234	22	2)−1−	2)−1−	PROPN
ejde-320	234	23	ε2	ε2	NOUN
ejde-320	234	24	}	}	PUNCT
ejde-320	234	25	,	,	PUNCT
ejde-320	234	26	where	where	SCONJ
ejde-320	234	27	τ	τ	PROPN
ejde-320	234	28	≥	≥	NUM
ejde-320	234	29	1	1	NUM
ejde-320	234	30	which	which	PRON
ejde-320	234	31	follows	follow	VERB
ejde-320	234	32	from	from	ADP
ejde-320	234	33	the	the	DET
ejde-320	234	34	observation	observation	NOUN
ejde-320	234	35	that	that	SCONJ
ejde-320	234	36	uk+1	uk+1	X
ejde-320	234	37	>	>	SYM
ejde-320	234	38	0	0	NUM
ejde-320	234	39	implies	imply	VERB
ejde-320	234	40	uk	uk	PROPN
ejde-320	234	41	>	>	X
ejde-320	234	42	τ‖f‖l∞	τ‖f‖l∞	X
ejde-320	234	43	(	(	PUNCT
ejde-320	234	44	1−	1−	NUM
ejde-320	234	45	c(k	c(k	NOUN
ejde-320	234	46	+	+	CCONJ
ejde-320	234	47	2)−ε/2	2)−ε/2	NUM
ejde-320	234	48	)	)	PUNCT
ejde-320	234	49	.	.	PUNCT
ejde-320	235	1	using	use	VERB
ejde-320	235	2	chebyshev	chebyshev	PROPN
ejde-320	235	3	’s	’s	PART
ejde-320	235	4	inequality	inequality	NOUN
ejde-320	235	5	thus	thus	ADV
ejde-320	235	6	gives	give	VERB
ejde-320	235	7	µ	µ	PROPN
ejde-320	235	8	(	(	PUNCT
ejde-320	235	9	{	{	PUNCT
ejde-320	235	10	uk+1	uk+1	X
ejde-320	235	11	>	>	X
ejde-320	235	12	0	0	NUM
ejde-320	235	13	}	}	PUNCT
ejde-320	235	14	)	)	PUNCT
ejde-320	235	15	≤	≤	NUM
ejde-320	235	16	µ	µ	X
ejde-320	235	17	(	(	PUNCT
ejde-320	235	18	{	{	PUNCT
ejde-320	235	19	uk	uk	PROPN
ejde-320	235	20	>	>	X
ejde-320	235	21	ε	ε	PROPN
ejde-320	235	22	2	2	NUM
ejde-320	235	23	cτ‖f‖l∞(k	cτ‖f‖l∞(k	CCONJ
ejde-320	235	24	+	+	NUM
ejde-320	235	25	2)−1−	2)−1−	PROPN
ejde-320	235	26	ε2	ε2	NOUN
ejde-320	235	27	}	}	PUNCT
ejde-320	235	28	)	)	PUNCT
ejde-320	235	29	≤	≤	ADV
ejde-320	235	30	4	4	NUM
ejde-320	235	31	c2τ2‖f‖2l∞ε2	c2τ2‖f‖2l∞ε2	NOUN
ejde-320	235	32	(	(	PUNCT
ejde-320	235	33	k	k	PROPN
ejde-320	235	34	+	+	CCONJ
ejde-320	235	35	2)2+ε	2)2+ε	NUM
ejde-320	235	36	∫	∫	PROPN
ejde-320	235	37	u2	u2	PROPN
ejde-320	235	38	k	k	PROPN
ejde-320	235	39	dµ.	dµ.	PROPN
ejde-320	235	40	(	(	PUNCT
ejde-320	235	41	3.9	3.9	NUM
ejde-320	235	42	)	)	PUNCT
ejde-320	235	43	combining	combine	VERB
ejde-320	235	44	(	(	PUNCT
ejde-320	235	45	3.8	3.8	NUM
ejde-320	235	46	)	)	PUNCT
ejde-320	235	47	and	and	CCONJ
ejde-320	235	48	(	(	PUNCT
ejde-320	235	49	3.9	3.9	NUM
ejde-320	235	50	)	)	PUNCT
ejde-320	235	51	we	we	PRON
ejde-320	235	52	obtain	obtain	VERB
ejde-320	235	53	‖1‖	‖1‖	PROPN
ejde-320	235	54	lφ̃	lφ̃	X
ejde-320	235	55	{	{	PUNCT
ejde-320	235	56	uk+1>0	uk+1>0	NOUN
ejde-320	235	57	}	}	PUNCT
ejde-320	235	58	≤	≤	NUM
ejde-320	235	59	γ	γ	X
ejde-320	235	60	(	(	PUNCT
ejde-320	235	61	c(k	c(k	PROPN
ejde-320	235	62	+	+	CCONJ
ejde-320	235	63	2)2+ε	2)2+ε	NUM
ejde-320	235	64	∫	∫	PROPN
ejde-320	235	65	u2	u2	PROPN
ejde-320	235	66	k	k	PROPN
ejde-320	235	67	)	)	PUNCT
ejde-320	235	68	.	.	PUNCT
ejde-320	236	1	(	(	PUNCT
ejde-320	236	2	3.10	3.10	NUM
ejde-320	236	3	)	)	PUNCT
ejde-320	236	4	finally	finally	ADV
ejde-320	236	5	substituting	substitute	VERB
ejde-320	236	6	(	(	PUNCT
ejde-320	236	7	3.7	3.7	NUM
ejde-320	236	8	)	)	PUNCT
ejde-320	236	9	and	and	CCONJ
ejde-320	236	10	(	(	PUNCT
ejde-320	236	11	3.10	3.10	NUM
ejde-320	236	12	)	)	PUNCT
ejde-320	236	13	into	into	ADP
ejde-320	236	14	(	(	PUNCT
ejde-320	236	15	3.5	3.5	NUM
ejde-320	236	16	)	)	PUNCT
ejde-320	236	17	we	we	PRON
ejde-320	236	18	conclude	conclude	VERB
ejde-320	236	19	that∫	that∫	PROPN
ejde-320	236	20	u2	u2	PROPN
ejde-320	236	21	k+1	k+1	AUX
ejde-320	236	22	dµ	dµ	PROPN
ejde-320	236	23	≤	≤	NUM
ejde-320	236	24	c(k	c(k	NOUN
ejde-320	236	25	+	+	CCONJ
ejde-320	236	26	2	2	NUM
ejde-320	236	27	)	)	PUNCT
ejde-320	236	28	2+ε	2+ε	NUM
ejde-320	236	29	2	2	NUM
ejde-320	236	30	∫	∫	NOUN
ejde-320	236	31	u2	u2	PROPN
ejde-320	236	32	k	k	PROPN
ejde-320	236	33	·	·	PUNCT
ejde-320	236	34	γ	γ	X
ejde-320	236	35	(	(	PUNCT
ejde-320	236	36	c(k	c(k	PROPN
ejde-320	236	37	+	+	CCONJ
ejde-320	236	38	2)2+ε	2)2+ε	NUM
ejde-320	236	39	∫	∫	PROPN
ejde-320	236	40	u2	u2	PROPN
ejde-320	236	41	k	k	PROPN
ejde-320	236	42	)	)	PUNCT
ejde-320	236	43	,	,	PUNCT
ejde-320	236	44	uk+1	uk+1	ADJ
ejde-320	236	45	≤	≤	NUM
ejde-320	236	46	c(k	c(k	NOUN
ejde-320	236	47	+	+	CCONJ
ejde-320	236	48	2	2	NUM
ejde-320	236	49	)	)	PUNCT
ejde-320	236	50	2+ε	2+ε	NUM
ejde-320	236	51	2	2	NUM
ejde-320	236	52	ukγ	ukγ	NOUN
ejde-320	236	53	(	(	PUNCT
ejde-320	236	54	c(k	c(k	NOUN
ejde-320	236	55	+	+	CCONJ
ejde-320	236	56	2)2+εuk	2)2+εuk	NUM
ejde-320	236	57	)	)	PUNCT
ejde-320	236	58	.	.	PUNCT
ejde-320	237	1	ejde-2021/82	ejde-2021/82	ADJ
ejde-320	237	2	orlicz	orlicz	NUM
ejde-320	237	3	-	-	PUNCT
ejde-320	237	4	sobolev	sobolev	NOUN
ejde-320	237	5	inequalities	inequality	NOUN
ejde-320	237	6	and	and	CCONJ
ejde-320	237	7	the	the	DET
ejde-320	237	8	dirichlet	dirichlet	PROPN
ejde-320	237	9	problem	problem	NOUN
ejde-320	237	10	11	11	NUM
ejde-320	237	11	this	this	DET
ejde-320	237	12	estimate	estimate	NOUN
ejde-320	237	13	is	be	AUX
ejde-320	237	14	the	the	DET
ejde-320	237	15	same	same	ADJ
ejde-320	237	16	as	as	ADP
ejde-320	237	17	the	the	DET
ejde-320	237	18	one	one	NOUN
ejde-320	237	19	obtained	obtain	VERB
ejde-320	237	20	in	in	ADP
ejde-320	237	21	the	the	DET
ejde-320	237	22	proof	proof	NOUN
ejde-320	237	23	of	of	ADP
ejde-320	237	24	[	[	X
ejde-320	237	25	7	7	NUM
ejde-320	237	26	,	,	PUNCT
ejde-320	237	27	theorem	theorem	VERB
ejde-320	237	28	30	30	NUM
ejde-320	237	29	]	]	PUNCT
ejde-320	237	30	,	,	PUNCT
ejde-320	237	31	so	so	ADV
ejde-320	237	32	the	the	DET
ejde-320	237	33	rest	rest	NOUN
ejde-320	237	34	of	of	ADP
ejde-320	237	35	the	the	DET
ejde-320	237	36	proof	proof	NOUN
ejde-320	237	37	can	can	AUX
ejde-320	237	38	be	be	AUX
ejde-320	237	39	repeated	repeat	VERB
ejde-320	237	40	verbatim	verbatim	ADJ
ejde-320	237	41	to	to	PART
ejde-320	237	42	conclude	conclude	VERB
ejde-320	237	43	that	that	DET
ejde-320	237	44	sup	sup	PROPN
ejde-320	237	45	b	b	PROPN
ejde-320	237	46	|u|	|u|	ADJ
ejde-320	237	47	≤	≤	NOUN
ejde-320	237	48	c‖f‖l∞(b	c‖f‖l∞(b	NOUN
ejde-320	237	49	)	)	PUNCT
ejde-320	237	50	.	.	PUNCT
ejde-320	238	1	�	�	PROPN
ejde-320	238	2	4	4	NUM
ejde-320	238	3	.	.	PUNCT
ejde-320	238	4	almost	almost	ADV
ejde-320	238	5	necessity	necessity	NOUN
ejde-320	238	6	in	in	ADP
ejde-320	238	7	this	this	DET
ejde-320	238	8	section	section	NOUN
ejde-320	238	9	we	we	PRON
ejde-320	238	10	demonstrate	demonstrate	VERB
ejde-320	238	11	the	the	DET
ejde-320	238	12	almost	almost	ADV
ejde-320	238	13	necessity	necessity	NOUN
ejde-320	238	14	of	of	ADP
ejde-320	238	15	an	an	DET
ejde-320	238	16	orlicz	orlicz	ADJ
ejde-320	238	17	-	-	PUNCT
ejde-320	238	18	sobolev	sobolev	NOUN
ejde-320	238	19	inequality	inequality	NOUN
ejde-320	238	20	for	for	ADP
ejde-320	238	21	the	the	DET
ejde-320	238	22	existence	existence	NOUN
ejde-320	238	23	,	,	PUNCT
ejde-320	238	24	uniqueness	uniqueness	NOUN
ejde-320	238	25	,	,	PUNCT
ejde-320	238	26	and	and	CCONJ
ejde-320	238	27	boundedness	boundedness	NOUN
ejde-320	238	28	of	of	ADP
ejde-320	238	29	solutions	solution	NOUN
ejde-320	238	30	to	to	ADP
ejde-320	238	31	(	(	PUNCT
ejde-320	238	32	1.1	1.1	NUM
ejde-320	238	33	)	)	PUNCT
ejde-320	238	34	,	,	PUNCT
ejde-320	238	35	namely	namely	ADV
ejde-320	238	36	we	we	PRON
ejde-320	238	37	prove	prove	VERB
ejde-320	238	38	theorem	theorem	VERB
ejde-320	238	39	1.2	1.2	NUM
ejde-320	238	40	.	.	PUNCT
ejde-320	239	1	we	we	PRON
ejde-320	239	2	start	start	VERB
ejde-320	239	3	with	with	ADP
ejde-320	239	4	two	two	NUM
ejde-320	239	5	simple	simple	ADJ
ejde-320	239	6	technical	technical	ADJ
ejde-320	239	7	lemmas	lemmas	NOUN
ejde-320	239	8	.	.	PUNCT
ejde-320	240	1	lemma	lemma	PROPN
ejde-320	240	2	4.1	4.1	NUM
ejde-320	240	3	.	.	PUNCT
ejde-320	241	1	let	let	VERB
ejde-320	241	2	ϕ	ϕ	NOUN
ejde-320	241	3	:	:	PUNCT
ejde-320	241	4	r	r	NOUN
ejde-320	241	5	→	→	X
ejde-320	241	6	[	[	X
ejde-320	241	7	0,∞	0,∞	X
ejde-320	241	8	]	]	PUNCT
ejde-320	241	9	be	be	VERB
ejde-320	241	10	a	a	DET
ejde-320	241	11	young	young	ADJ
ejde-320	241	12	function	function	NOUN
ejde-320	241	13	and	and	CCONJ
ejde-320	241	14	let	let	VERB
ejde-320	241	15	ϕ̃	ϕ̃	PROPN
ejde-320	241	16	be	be	AUX
ejde-320	241	17	the	the	DET
ejde-320	241	18	convex	convex	NOUN
ejde-320	241	19	conjugate	conjugate	NOUN
ejde-320	241	20	,	,	PUNCT
ejde-320	241	21	or	or	CCONJ
ejde-320	241	22	dual	dual	ADJ
ejde-320	241	23	,	,	PUNCT
ejde-320	241	24	of	of	ADP
ejde-320	241	25	ϕ	ϕ	NOUN
ejde-320	241	26	as	as	SCONJ
ejde-320	241	27	defined	define	VERB
ejde-320	241	28	in	in	ADP
ejde-320	241	29	definition	definition	NOUN
ejde-320	241	30	2.4	2.4	NUM
ejde-320	241	31	.	.	PUNCT
ejde-320	242	1	let	let	VERB
ejde-320	242	2	b	b	PROPN
ejde-320	242	3	⊂	⊂	PROPN
ejde-320	242	4	rn	rn	AUX
ejde-320	242	5	be	be	AUX
ejde-320	242	6	any	any	DET
ejde-320	242	7	ball	ball	NOUN
ejde-320	242	8	,	,	PUNCT
ejde-320	242	9	and	and	CCONJ
ejde-320	242	10	define	define	VERB
ejde-320	242	11	x	x	X
ejde-320	242	12	=	=	PRON
ejde-320	242	13	{	{	PUNCT
ejde-320	242	14	f	f	PROPN
ejde-320	242	15	∈	∈	PROPN
ejde-320	242	16	lϕ̃	lϕ̃	NUM
ejde-320	242	17	:	:	PUNCT
ejde-320	242	18	∫	∫	PROPN
ejde-320	242	19	b	b	PROPN
ejde-320	242	20	ϕ̃(|f	ϕ̃(|f	PROPN
ejde-320	242	21	|)dµ	|)dµ	PROPN
ejde-320	242	22	≤	≤	ADV
ejde-320	242	23	1	1	NUM
ejde-320	242	24	}	}	PUNCT
ejde-320	242	25	,	,	PUNCT
ejde-320	242	26	y	y	PROPN
ejde-320	242	27	=	=	PRON
ejde-320	242	28	{	{	PUNCT
ejde-320	242	29	f	f	PROPN
ejde-320	242	30	∈	∈	PROPN
ejde-320	242	31	lϕ̃	lϕ̃	PRON
ejde-320	242	32	:	:	PUNCT
ejde-320	242	33	f	f	PROPN
ejde-320	242	34	≥	≥	NOUN
ejde-320	242	35	0	0	NUM
ejde-320	242	36	and	and	CCONJ
ejde-320	242	37	∫	∫	PROPN
ejde-320	242	38	b	b	PROPN
ejde-320	242	39	ϕ̃(f)dµ	ϕ̃(f)dµ	PROPN
ejde-320	242	40	≤	≤	NUM
ejde-320	242	41	1	1	NUM
ejde-320	242	42	}	}	PUNCT
ejde-320	242	43	.	.	PUNCT
ejde-320	243	1	then	then	ADV
ejde-320	243	2	sup	sup	NOUN
ejde-320	243	3	x	x	SYM
ejde-320	243	4	∫	∫	PROPN
ejde-320	243	5	b	b	PROPN
ejde-320	243	6	w2f	w2f	PROPN
ejde-320	243	7	dµ	dµ	ADJ
ejde-320	243	8	=	=	PUNCT
ejde-320	243	9	sup	sup	NOUN
ejde-320	243	10	y	y	PROPN
ejde-320	243	11	∫	∫	PROPN
ejde-320	243	12	b	b	PROPN
ejde-320	243	13	w2f	w2f	PROPN
ejde-320	243	14	dµ	dµ	PROPN
ejde-320	243	15	(	(	PUNCT
ejde-320	243	16	4.1	4.1	NUM
ejde-320	243	17	)	)	PUNCT
ejde-320	243	18	for	for	ADP
ejde-320	243	19	any	any	DET
ejde-320	243	20	w	w	PROPN
ejde-320	243	21	∈	∈	PROPN
ejde-320	243	22	lip0(b	lip0(b	NOUN
ejde-320	243	23	)	)	PUNCT
ejde-320	243	24	.	.	PUNCT
ejde-320	244	1	proof	proof	NOUN
ejde-320	244	2	.	.	PUNCT
ejde-320	245	1	first	first	ADV
ejde-320	245	2	note	note	VERB
ejde-320	245	3	that	that	SCONJ
ejde-320	245	4	y	y	PROPN
ejde-320	245	5	⊂	⊂	PROPN
ejde-320	245	6	x.	x.	PROPN
ejde-320	246	1	thus	thus	ADV
ejde-320	246	2	,	,	PUNCT
ejde-320	246	3	it	it	PRON
ejde-320	246	4	suffices	suffice	VERB
ejde-320	246	5	to	to	PART
ejde-320	246	6	show	show	VERB
ejde-320	246	7	that∫	that∫	PROPN
ejde-320	246	8	b	b	PROPN
ejde-320	246	9	w2	w2	NOUN
ejde-320	246	10	g	g	PROPN
ejde-320	246	11	≤	≤	NUM
ejde-320	246	12	sup	sup	NOUN
ejde-320	246	13	y	y	PROPN
ejde-320	246	14	∫	∫	PROPN
ejde-320	246	15	b	b	PROPN
ejde-320	246	16	w2f	w2f	VERB
ejde-320	246	17	for	for	ADP
ejde-320	246	18	all	all	PRON
ejde-320	246	19	g	g	NOUN
ejde-320	246	20	∈	∈	PRON
ejde-320	246	21	x\y	x\y	PROPN
ejde-320	246	22	.	.	PUNCT
ejde-320	247	1	let	let	VERB
ejde-320	247	2	g	g	PROPN
ejde-320	247	3	∈	∈	PROPN
ejde-320	247	4	x\y	x\y	PROPN
ejde-320	247	5	,	,	PUNCT
ejde-320	247	6	and	and	CCONJ
ejde-320	247	7	write	write	VERB
ejde-320	247	8	g+	g+	NOUN
ejde-320	247	9	and	and	CCONJ
ejde-320	247	10	g−	g−	PROPN
ejde-320	247	11	for	for	ADP
ejde-320	247	12	the	the	DET
ejde-320	247	13	positive	positive	ADJ
ejde-320	247	14	and	and	CCONJ
ejde-320	247	15	negative	negative	ADJ
ejde-320	247	16	parts	part	NOUN
ejde-320	247	17	of	of	ADP
ejde-320	247	18	g	g	NOUN
ejde-320	247	19	respectively	respectively	ADV
ejde-320	247	20	.	.	PUNCT
ejde-320	248	1	note	note	VERB
ejde-320	248	2	that	that	SCONJ
ejde-320	248	3	since	since	SCONJ
ejde-320	248	4	ϕ̃	ϕ̃	PROPN
ejde-320	248	5	is	be	AUX
ejde-320	248	6	even	even	ADV
ejde-320	248	7	and	and	CCONJ
ejde-320	248	8	non	non	ADJ
ejde-320	248	9	-	-	ADJ
ejde-320	248	10	decreasing	decrease	VERB
ejde-320	248	11	on	on	ADP
ejde-320	248	12	[	[	X
ejde-320	248	13	0,∞	0,∞	NOUN
ejde-320	248	14	)	)	PUNCT
ejde-320	248	15	and	and	CCONJ
ejde-320	248	16	non	non	ADJ
ejde-320	248	17	-	-	ADJ
ejde-320	248	18	negative	negative	ADJ
ejde-320	248	19	on	on	ADP
ejde-320	248	20	r	r	NOUN
ejde-320	248	21	,	,	PUNCT
ejde-320	248	22	we	we	PRON
ejde-320	248	23	have	have	VERB
ejde-320	248	24	∫	∫	PROPN
ejde-320	248	25	b	b	PROPN
ejde-320	248	26	ϕ̃(g+	ϕ̃(g+	PROPN
ejde-320	248	27	)	)	PUNCT
ejde-320	248	28	≤	≤	NUM
ejde-320	249	1	∫	∫	PROPN
ejde-320	250	1	b	b	PROPN
ejde-320	250	2	ϕ̃(g+	ϕ̃(g+	PROPN
ejde-320	250	3	+	+	CCONJ
ejde-320	250	4	g−	g−	PROPN
ejde-320	250	5	)	)	PUNCT
ejde-320	250	6	=	=	SYM
ejde-320	250	7	∫	∫	PROPN
ejde-320	250	8	b	b	NOUN
ejde-320	250	9	ϕ̃(|g|	ϕ̃(|g|	ADJ
ejde-320	250	10	)	)	PUNCT
ejde-320	250	11	≤	≤	NUM
ejde-320	250	12	1	1	NUM
ejde-320	250	13	for	for	ADP
ejde-320	250	14	all	all	DET
ejde-320	250	15	g	g	PROPN
ejde-320	250	16	∈	∈	PROPN
ejde-320	250	17	x.	x.	NOUN
ejde-320	250	18	in	in	ADP
ejde-320	250	19	particular	particular	ADJ
ejde-320	250	20	,	,	PUNCT
ejde-320	250	21	we	we	PRON
ejde-320	250	22	conclude	conclude	VERB
ejde-320	250	23	that	that	SCONJ
ejde-320	250	24	g+	g+	PUNCT
ejde-320	250	25	∈	∈	PROPN
ejde-320	250	26	y	y	PROPN
ejde-320	250	27	.	.	PUNCT
ejde-320	250	28	therefore,∫	therefore,∫	PROPN
ejde-320	251	1	b	b	X
ejde-320	251	2	w2	w2	NOUN
ejde-320	251	3	g	g	PROPN
ejde-320	251	4	=	=	SYM
ejde-320	251	5	∫	∫	PROPN
ejde-320	251	6	b	b	PROPN
ejde-320	252	1	w2g+	w2g+	ADV
ejde-320	252	2	−	−	PROPN
ejde-320	252	3	∫	∫	PROPN
ejde-320	252	4	b	b	PROPN
ejde-320	252	5	w2g−	w2g−	PROPN
ejde-320	252	6	≤	≤	NUM
ejde-320	252	7	sup	sup	NUM
ejde-320	252	8	f∈y	f∈y	NUM
ejde-320	252	9	∫	∫	PROPN
ejde-320	252	10	b	b	PROPN
ejde-320	252	11	w2f	w2f	VERB
ejde-320	252	12	−	−	NUM
ejde-320	252	13	∫	∫	PROPN
ejde-320	252	14	w2g−	w2g−	PROPN
ejde-320	252	15	≤	≤	NUM
ejde-320	252	16	sup	sup	NUM
ejde-320	252	17	f∈y	f∈y	NUM
ejde-320	252	18	∫	∫	PROPN
ejde-320	252	19	b	b	PROPN
ejde-320	252	20	w2f	w2f	PROPN
ejde-320	252	21	.	.	PUNCT
ejde-320	253	1	the	the	DET
ejde-320	253	2	last	last	ADJ
ejde-320	253	3	inequality	inequality	NOUN
ejde-320	253	4	follows	follow	VERB
ejde-320	253	5	from	from	ADP
ejde-320	253	6	the	the	DET
ejde-320	253	7	fact	fact	NOUN
ejde-320	253	8	that	that	SCONJ
ejde-320	253	9	since	since	SCONJ
ejde-320	253	10	w2	w2	NOUN
ejde-320	253	11	and	and	CCONJ
ejde-320	253	12	g−	g−	PROPN
ejde-320	253	13	are	be	AUX
ejde-320	253	14	both	both	PRON
ejde-320	253	15	non	non	ADJ
ejde-320	253	16	-	-	ADJ
ejde-320	253	17	negative	negative	ADJ
ejde-320	253	18	,	,	PUNCT
ejde-320	253	19	it	it	PRON
ejde-320	253	20	must	must	AUX
ejde-320	253	21	be	be	AUX
ejde-320	253	22	the	the	DET
ejde-320	253	23	case	case	NOUN
ejde-320	253	24	that	that	SCONJ
ejde-320	253	25	∫	∫	PROPN
ejde-320	253	26	b	b	X
ejde-320	253	27	w2g−	w2g−	PROPN
ejde-320	253	28	is	be	AUX
ejde-320	253	29	also	also	ADV
ejde-320	253	30	non	non	ADJ
ejde-320	253	31	-	-	ADJ
ejde-320	253	32	negative	negative	ADJ
ejde-320	253	33	.	.	PUNCT
ejde-320	254	1	�	�	PROPN
ejde-320	254	2	lemma	lemma	PROPN
ejde-320	254	3	4.2	4.2	NUM
ejde-320	254	4	.	.	PUNCT
ejde-320	255	1	let	let	VERB
ejde-320	255	2	ϕ	ϕ	NOUN
ejde-320	255	3	:	:	PUNCT
ejde-320	255	4	r	r	NOUN
ejde-320	255	5	→	→	X
ejde-320	255	6	[	[	X
ejde-320	255	7	0,∞	0,∞	X
ejde-320	255	8	]	]	PUNCT
ejde-320	255	9	be	be	VERB
ejde-320	255	10	a	a	DET
ejde-320	255	11	young	young	ADJ
ejde-320	255	12	function	function	NOUN
ejde-320	255	13	and	and	CCONJ
ejde-320	255	14	ϕ̃	ϕ̃	PROPN
ejde-320	255	15	be	be	AUX
ejde-320	255	16	its	its	PRON
ejde-320	255	17	dual	dual	ADJ
ejde-320	255	18	.	.	PUNCT
ejde-320	256	1	furthermore	furthermore	ADV
ejde-320	256	2	,	,	PUNCT
ejde-320	256	3	define	define	VERB
ejde-320	256	4	the	the	DET
ejde-320	256	5	sets	set	NOUN
ejde-320	256	6	x	x	PUNCT
ejde-320	256	7	and	and	CCONJ
ejde-320	256	8	y	y	PROPN
ejde-320	256	9	as	as	ADP
ejde-320	256	10	in	in	ADP
ejde-320	256	11	lemma	lemma	PROPN
ejde-320	256	12	4.1	4.1	NUM
ejde-320	256	13	.	.	PUNCT
ejde-320	257	1	then	then	ADV
ejde-320	257	2	for	for	ADP
ejde-320	257	3	all	all	DET
ejde-320	257	4	w	w	PROPN
ejde-320	257	5	∈	∈	PROPN
ejde-320	257	6	lip0(b	lip0(b	NOUN
ejde-320	257	7	)	)	PUNCT
ejde-320	257	8	,	,	PUNCT
ejde-320	257	9	‖u‖lϕ	‖u‖lϕ	X
ejde-320	257	10	≤	≤	PROPN
ejde-320	257	11	sup	sup	PROPN
ejde-320	257	12	f∈y	f∈y	NUM
ejde-320	257	13	∫	∫	PROPN
ejde-320	257	14	b	b	PROPN
ejde-320	257	15	uf	uf	PROPN
ejde-320	257	16	dµ.	dµ.	PROPN
ejde-320	257	17	proof	proof	NOUN
ejde-320	257	18	.	.	PUNCT
ejde-320	258	1	as	as	SCONJ
ejde-320	258	2	before	before	ADV
ejde-320	258	3	let	let	VERB
ejde-320	258	4	dµ	dµ	VERB
ejde-320	258	5	=	=	PUNCT
ejde-320	258	6	dx/|b|	dx/|b|	PROPN
ejde-320	258	7	,	,	PUNCT
ejde-320	258	8	and	and	CCONJ
ejde-320	258	9	recall	recall	VERB
ejde-320	258	10	from	from	ADP
ejde-320	258	11	(	(	PUNCT
ejde-320	258	12	2.3	2.3	NUM
ejde-320	258	13	)	)	PUNCT
ejde-320	259	1	that	that	SCONJ
ejde-320	259	2	‖u‖lϕ	‖u‖lϕ	X
ejde-320	259	3	≤	≤	PROPN
ejde-320	259	4	|u|lϕ	|u|lϕ	PROPN
ejde-320	259	5	=	=	PRON
ejde-320	259	6	sup	sup	INTJ
ejde-320	259	7	{	{	PUNCT
ejde-320	259	8	∫	∫	PROPN
ejde-320	259	9	ug	ug	ADP
ejde-320	259	10	dµ	dµ	PROPN
ejde-320	259	11	:	:	PUNCT
ejde-320	259	12	∫	∫	PROPN
ejde-320	259	13	ϕ̃(g	ϕ̃(g	PROPN
ejde-320	259	14	)	)	PUNCT
ejde-320	259	15	≤	≤	NUM
ejde-320	259	16	1	1	NUM
ejde-320	259	17	}	}	PUNCT
ejde-320	259	18	=	=	PUNCT
ejde-320	259	19	sup	sup	NOUN
ejde-320	259	20	{	{	PUNCT
ejde-320	259	21	∫	∫	PROPN
ejde-320	259	22	ug	ug	ADP
ejde-320	259	23	dµ	dµ	PROPN
ejde-320	259	24	:	:	PUNCT
ejde-320	259	25	∫	∫	PROPN
ejde-320	259	26	ϕ̃(|g|	ϕ̃(|g|	NOUN
ejde-320	259	27	)	)	PUNCT
ejde-320	259	28	≤	≤	NUM
ejde-320	259	29	1	1	NUM
ejde-320	259	30	}	}	PUNCT
ejde-320	259	31	,	,	PUNCT
ejde-320	259	32	where	where	SCONJ
ejde-320	259	33	the	the	DET
ejde-320	259	34	last	last	ADJ
ejde-320	259	35	equality	equality	NOUN
ejde-320	259	36	holds	hold	VERB
ejde-320	259	37	because	because	SCONJ
ejde-320	259	38	ϕ̃	ϕ̃	PROPN
ejde-320	259	39	is	be	AUX
ejde-320	259	40	even	even	ADV
ejde-320	259	41	by	by	ADP
ejde-320	259	42	definition	definition	NOUN
ejde-320	259	43	of	of	ADP
ejde-320	259	44	a	a	DET
ejde-320	259	45	young	young	ADJ
ejde-320	259	46	function	function	NOUN
ejde-320	259	47	.	.	PUNCT
ejde-320	260	1	finally	finally	ADV
ejde-320	260	2	,	,	PUNCT
ejde-320	260	3	using	use	VERB
ejde-320	260	4	lemma	lemma	PROPN
ejde-320	260	5	4.1	4.1	NUM
ejde-320	260	6	we	we	PRON
ejde-320	260	7	have	have	VERB
ejde-320	260	8	sup	sup	NOUN
ejde-320	260	9	{	{	PUNCT
ejde-320	260	10	∫	∫	PROPN
ejde-320	260	11	ug	ug	ADP
ejde-320	260	12	dµ	dµ	PROPN
ejde-320	260	13	:	:	PUNCT
ejde-320	260	14	∫	∫	PROPN
ejde-320	260	15	ϕ̃(|g|	ϕ̃(|g|	NOUN
ejde-320	260	16	)	)	PUNCT
ejde-320	260	17	≤	≤	NUM
ejde-320	260	18	1	1	NUM
ejde-320	260	19	}	}	PUNCT
ejde-320	260	20	=	=	SYM
ejde-320	260	21	sup	sup	NUM
ejde-320	260	22	g∈x	g∈x	NOUN
ejde-320	260	23	∫	∫	PROPN
ejde-320	260	24	b	b	PROPN
ejde-320	260	25	ug	ug	ADP
ejde-320	260	26	dµ	dµ	PROPN
ejde-320	260	27	≤	≤	NUM
ejde-320	260	28	sup	sup	NOUN
ejde-320	260	29	g∈y	g∈y	NOUN
ejde-320	260	30	∫	∫	PROPN
ejde-320	260	31	b	b	PROPN
ejde-320	260	32	ug	ug	ADP
ejde-320	260	33	dµ	dµ	PROPN
ejde-320	260	34	,	,	PUNCT
ejde-320	260	35	which	which	PRON
ejde-320	260	36	concludes	conclude	VERB
ejde-320	260	37	the	the	DET
ejde-320	260	38	result	result	NOUN
ejde-320	260	39	.	.	PUNCT
ejde-320	261	1	�	�	PROPN
ejde-320	261	2	12	12	NUM
ejde-320	261	3	u.	u.	PROPN
ejde-320	261	4	hafeez	hafeez	PROPN
ejde-320	261	5	,	,	PUNCT
ejde-320	261	6	t.	t.	PROPN
ejde-320	261	7	lavier	lavier	PROPN
ejde-320	261	8	,	,	PUNCT
ejde-320	261	9	l.	l.	PROPN
ejde-320	261	10	williams	williams	PROPN
ejde-320	261	11	,	,	PUNCT
ejde-320	261	12	l.	l.	PROPN
ejde-320	261	13	korobenko	korobenko	PROPN
ejde-320	261	14	ejde-2021/82	ejde-2021/82	VERB
ejde-320	261	15	we	we	PRON
ejde-320	261	16	are	be	AUX
ejde-320	261	17	now	now	ADV
ejde-320	261	18	ready	ready	ADJ
ejde-320	261	19	to	to	PART
ejde-320	261	20	prove	prove	VERB
ejde-320	261	21	theorem	theorem	ADJ
ejde-320	261	22	1.2	1.2	NUM
ejde-320	261	23	,	,	PUNCT
ejde-320	261	24	which	which	PRON
ejde-320	261	25	we	we	PRON
ejde-320	261	26	state	state	VERB
ejde-320	261	27	again	again	ADV
ejde-320	261	28	here	here	ADV
ejde-320	261	29	for	for	ADP
ejde-320	261	30	convenience	convenience	NOUN
ejde-320	261	31	.	.	PUNCT
ejde-320	262	1	theorem	theorem	VERB
ejde-320	262	2	4.3	4.3	NUM
ejde-320	262	3	.	.	PUNCT
ejde-320	263	1	let	let	VERB
ejde-320	263	2	ϕ	ϕ	NOUN
ejde-320	263	3	be	be	AUX
ejde-320	263	4	a	a	DET
ejde-320	263	5	young	young	ADJ
ejde-320	263	6	function	function	NOUN
ejde-320	263	7	that	that	PRON
ejde-320	263	8	satisfies	satisfy	VERB
ejde-320	263	9	ϕ(t	ϕ(t	NUM
ejde-320	263	10	)	)	PUNCT
ejde-320	263	11	>	>	X
ejde-320	264	1	t	t	PROPN
ejde-320	264	2	for	for	ADP
ejde-320	264	3	all	all	DET
ejde-320	264	4	t	t	PROPN
ejde-320	264	5	>	>	X
ejde-320	264	6	0	0	NUM
ejde-320	264	7	,	,	PUNCT
ejde-320	264	8	and	and	CCONJ
ejde-320	264	9	let	let	VERB
ejde-320	264	10	φ(t	φ(t	NUM
ejde-320	264	11	)	)	PUNCT
ejde-320	264	12	=	=	SYM
ejde-320	264	13	ϕ(t2	ϕ(t2	NOUN
ejde-320	264	14	)	)	PUNCT
ejde-320	264	15	.	.	PUNCT
ejde-320	265	1	additionally	additionally	ADV
ejde-320	265	2	,	,	PUNCT
ejde-320	265	3	let	let	VERB
ejde-320	265	4	f	f	PROPN
ejde-320	265	5	∈	∈	PROPN
ejde-320	265	6	lϕ̃(b	lϕ̃(b	PROPN
ejde-320	265	7	)	)	PUNCT
ejde-320	265	8	and	and	CCONJ
ejde-320	265	9	assume	assume	VERB
ejde-320	265	10	that	that	SCONJ
ejde-320	265	11	all	all	DET
ejde-320	265	12	weak	weak	ADJ
ejde-320	265	13	solutions	solution	NOUN
ejde-320	265	14	u	u	NOUN
ejde-320	265	15	∈	∈	PROPN
ejde-320	265	16	(	(	PUNCT
ejde-320	265	17	w	w	PROPN
ejde-320	265	18	1,2	1,2	NUM
ejde-320	265	19	a	a	PRON
ejde-320	265	20	)	)	PUNCT
ejde-320	265	21	0	0	PUNCT
ejde-320	265	22	(	(	PUNCT
ejde-320	265	23	b	b	NOUN
ejde-320	265	24	)	)	PUNCT
ejde-320	265	25	to	to	ADP
ejde-320	265	26	(	(	PUNCT
ejde-320	265	27	3.2	3.2	NUM
ejde-320	265	28	)	)	PUNCT
ejde-320	265	29	satisfy	satisfy	VERB
ejde-320	265	30	the	the	DET
ejde-320	265	31	global	global	ADJ
ejde-320	265	32	boundedness	boundedness	PROPN
ejde-320	265	33	estimate	estimate	NOUN
ejde-320	265	34	supb	supb	VERB
ejde-320	265	35	|u|	|u|	ADV
ejde-320	265	36	≤	≤	NUM
ejde-320	265	37	c‖f‖lϕ̃(b	c‖f‖lϕ̃(b	NOUN
ejde-320	265	38	)	)	PUNCT
ejde-320	265	39	.	.	PUNCT
ejde-320	266	1	then	then	ADV
ejde-320	266	2	the	the	DET
ejde-320	266	3	following	follow	VERB
ejde-320	266	4	orlicz	orlicz	ADJ
ejde-320	266	5	-	-	PUNCT
ejde-320	266	6	sobolev	sobolev	NOUN
ejde-320	266	7	inequality	inequality	NOUN
ejde-320	266	8	holds	hold	VERB
ejde-320	266	9	:	:	PUNCT
ejde-320	266	10	‖v‖lφ(b	‖v‖lφ(b	VERB
ejde-320	266	11	,	,	PUNCT
ejde-320	266	12	dµ	dµ	ADJ
ejde-320	266	13	)	)	PUNCT
ejde-320	266	14	≤	≤	NOUN
ejde-320	266	15	c‖∇av‖l2(b	c‖∇av‖l2(b	PROPN
ejde-320	266	16	,	,	PUNCT
ejde-320	266	17	dµ	dµ	PROPN
ejde-320	266	18	)	)	PUNCT
ejde-320	266	19	for	for	ADP
ejde-320	266	20	all	all	PRON
ejde-320	266	21	v	v	ADP
ejde-320	266	22	∈	∈	NOUN
ejde-320	266	23	(	(	PUNCT
ejde-320	266	24	w	w	PROPN
ejde-320	266	25	1,2	1,2	NUM
ejde-320	266	26	a	a	PRON
ejde-320	266	27	)	)	PUNCT
ejde-320	266	28	0	0	PUNCT
ejde-320	267	1	(	(	PUNCT
ejde-320	267	2	b	b	NOUN
ejde-320	267	3	)	)	PUNCT
ejde-320	267	4	.	.	PUNCT
ejde-320	268	1	note	note	VERB
ejde-320	268	2	that	that	SCONJ
ejde-320	268	3	the	the	DET
ejde-320	268	4	global	global	ADJ
ejde-320	268	5	boundedness	boundedness	PROPN
ejde-320	268	6	condition	condition	NOUN
ejde-320	268	7	is	be	AUX
ejde-320	268	8	different	different	ADJ
ejde-320	268	9	from	from	ADP
ejde-320	268	10	that	that	PRON
ejde-320	268	11	in	in	ADP
ejde-320	268	12	the	the	DET
ejde-320	268	13	sufficiency	sufficiency	NOUN
ejde-320	268	14	result	result	NOUN
ejde-320	268	15	.	.	PUNCT
ejde-320	269	1	we	we	PRON
ejde-320	269	2	previously	previously	ADV
ejde-320	269	3	demonstrated	demonstrate	VERB
ejde-320	269	4	that	that	SCONJ
ejde-320	269	5	a	a	DET
ejde-320	269	6	(	(	PUNCT
ejde-320	269	7	φ	φ	PROPN
ejde-320	269	8	,	,	PUNCT
ejde-320	269	9	2	2	NUM
ejde-320	269	10	)	)	PUNCT
ejde-320	269	11	orlicz	orlicz	ADJ
ejde-320	269	12	-	-	PUNCT
ejde-320	269	13	sobolev	sobolev	NOUN
ejde-320	269	14	inequality	inequality	NOUN
ejde-320	269	15	with	with	ADP
ejde-320	269	16	sufficiently	sufficiently	ADV
ejde-320	269	17	large	large	ADJ
ejde-320	269	18	φ	φ	PROPN
ejde-320	269	19	gives	give	VERB
ejde-320	269	20	the	the	DET
ejde-320	269	21	estimate	estimate	NOUN
ejde-320	269	22	,	,	PUNCT
ejde-320	269	23	supb	supb	VERB
ejde-320	269	24	|u|	|u|	ADJ
ejde-320	269	25	≤	≤	NUM
ejde-320	269	26	c‖f‖l∞(b	c‖f‖l∞(b	NOUN
ejde-320	269	27	)	)	PUNCT
ejde-320	269	28	for	for	ADP
ejde-320	269	29	all	all	DET
ejde-320	269	30	weak	weak	ADJ
ejde-320	269	31	solutions	solution	NOUN
ejde-320	269	32	u	u	NOUN
ejde-320	269	33	to	to	ADP
ejde-320	269	34	(	(	PUNCT
ejde-320	269	35	3.2	3.2	NUM
ejde-320	269	36	)	)	PUNCT
ejde-320	269	37	with	with	ADP
ejde-320	269	38	f	f	PROPN
ejde-320	269	39	∈	∈	PROPN
ejde-320	269	40	l∞(b	l∞(b	PROPN
ejde-320	269	41	)	)	PUNCT
ejde-320	269	42	.	.	PUNCT
ejde-320	270	1	however	however	ADV
ejde-320	270	2	,	,	PUNCT
ejde-320	270	3	in	in	ADP
ejde-320	270	4	order	order	NOUN
ejde-320	270	5	to	to	PART
ejde-320	270	6	prove	prove	VERB
ejde-320	270	7	necessity	necessity	NOUN
ejde-320	270	8	of	of	ADP
ejde-320	270	9	a	a	DET
ejde-320	270	10	(	(	PUNCT
ejde-320	270	11	φ	φ	PROPN
ejde-320	270	12	,	,	PUNCT
ejde-320	270	13	2	2	NUM
ejde-320	270	14	)	)	PUNCT
ejde-320	270	15	orlicz	orlicz	ADJ
ejde-320	270	16	-	-	PUNCT
ejde-320	270	17	sobolev	sobolev	NOUN
ejde-320	270	18	inequality	inequality	NOUN
ejde-320	270	19	we	we	PRON
ejde-320	270	20	require	require	VERB
ejde-320	270	21	a	a	DET
ejde-320	270	22	stronger	strong	ADJ
ejde-320	270	23	condition	condition	NOUN
ejde-320	270	24	;	;	PUNCT
ejde-320	270	25	namely	namely	ADV
ejde-320	270	26	,	,	PUNCT
ejde-320	270	27	that	that	SCONJ
ejde-320	270	28	all	all	DET
ejde-320	270	29	weak	weak	ADJ
ejde-320	270	30	solutions	solution	NOUN
ejde-320	270	31	to	to	ADP
ejde-320	270	32	(	(	PUNCT
ejde-320	270	33	1.1	1.1	NUM
ejde-320	270	34	)	)	PUNCT
ejde-320	270	35	with	with	ADP
ejde-320	270	36	the	the	DET
ejde-320	270	37	right	right	ADJ
ejde-320	270	38	hand	hand	NOUN
ejde-320	270	39	side	side	NOUN
ejde-320	270	40	in	in	ADP
ejde-320	270	41	a	a	DET
ejde-320	270	42	larger	large	ADJ
ejde-320	270	43	class	class	NOUN
ejde-320	270	44	,	,	PUNCT
ejde-320	270	45	i.e.	i.e.	X
ejde-320	270	46	f	f	X
ejde-320	270	47	∈	∈	PROPN
ejde-320	270	48	lϕ̃(b	lϕ̃(b	PROPN
ejde-320	270	49	)	)	PUNCT
ejde-320	270	50	,	,	PUNCT
ejde-320	270	51	are	be	AUX
ejde-320	270	52	bounded	bound	VERB
ejde-320	270	53	.	.	PUNCT
ejde-320	271	1	hence	hence	ADV
ejde-320	271	2	the	the	DET
ejde-320	271	3	term	term	NOUN
ejde-320	271	4	“	"	PUNCT
ejde-320	271	5	almost	almost	ADV
ejde-320	271	6	necessity	necessity	NOUN
ejde-320	271	7	”	"	PUNCT
ejde-320	271	8	.	.	PUNCT
ejde-320	272	1	proof	proof	NOUN
ejde-320	272	2	.	.	PUNCT
ejde-320	273	1	the	the	DET
ejde-320	273	2	proof	proof	NOUN
ejde-320	273	3	is	be	AUX
ejde-320	273	4	similar	similar	ADJ
ejde-320	273	5	to	to	ADP
ejde-320	273	6	the	the	DET
ejde-320	273	7	proof	proof	NOUN
ejde-320	273	8	of	of	ADP
ejde-320	273	9	[	[	X
ejde-320	273	10	11	11	NUM
ejde-320	273	11	,	,	PUNCT
ejde-320	273	12	lemma	lemma	PROPN
ejde-320	273	13	102	102	NUM
ejde-320	273	14	]	]	PUNCT
ejde-320	273	15	,	,	PUNCT
ejde-320	273	16	and	and	CCONJ
ejde-320	273	17	the	the	DET
ejde-320	273	18	proof	proof	NOUN
ejde-320	273	19	in	in	ADP
ejde-320	273	20	[	[	X
ejde-320	273	21	6	6	NUM
ejde-320	273	22	,	,	PUNCT
ejde-320	273	23	sections	section	NOUN
ejde-320	273	24	1	1	NUM
ejde-320	273	25	and	and	CCONJ
ejde-320	273	26	2	2	NUM
ejde-320	273	27	of	of	ADP
ejde-320	273	28	chapter	chapter	NOUN
ejde-320	273	29	9	9	NUM
ejde-320	273	30	]	]	PUNCT
ejde-320	273	31	.	.	PUNCT
ejde-320	274	1	let	let	VERB
ejde-320	274	2	u	u	PRON
ejde-320	274	3	∈	∈	PROPN
ejde-320	274	4	(	(	PUNCT
ejde-320	274	5	w	w	PROPN
ejde-320	274	6	1,2	1,2	NUM
ejde-320	274	7	a	a	PRON
ejde-320	274	8	)	)	PUNCT
ejde-320	274	9	0	0	PUNCT
ejde-320	274	10	be	be	AUX
ejde-320	274	11	a	a	DET
ejde-320	274	12	weak	weak	ADJ
ejde-320	274	13	solution	solution	NOUN
ejde-320	274	14	to	to	ADP
ejde-320	274	15	(	(	PUNCT
ejde-320	274	16	3.2	3.2	NUM
ejde-320	274	17	)	)	PUNCT
ejde-320	274	18	,	,	PUNCT
ejde-320	274	19	i.e.	i.e.	X
ejde-320	274	20	,∫	,∫	X
ejde-320	274	21	b	b	X
ejde-320	274	22	∇ψ	∇ψ	PROPN
ejde-320	274	23	·	·	PUNCT
ejde-320	274	24	a∇u	a∇u	PROPN
ejde-320	274	25	dµ	dµ	PROPN
ejde-320	274	26	=	=	PUNCT
ejde-320	275	1	−	−	PROPN
ejde-320	275	2	∫	∫	PROPN
ejde-320	275	3	b	b	X
ejde-320	275	4	ψf	ψf	X
ejde-320	275	5	dµ	dµ	PROPN
ejde-320	275	6	for	for	ADP
ejde-320	275	7	all	all	DET
ejde-320	275	8	ψ	ψ	PRON
ejde-320	275	9	∈	∈	PROPN
ejde-320	275	10	lip0(b	lip0(b	NOUN
ejde-320	275	11	)	)	PUNCT
ejde-320	275	12	,	,	PUNCT
ejde-320	275	13	and	and	CCONJ
ejde-320	275	14	assume	assume	VERB
ejde-320	275	15	f	f	PROPN
ejde-320	275	16	≥	≥	PROPN
ejde-320	275	17	0	0	NUM
ejde-320	275	18	.	.	PUNCT
ejde-320	276	1	for	for	ADP
ejde-320	276	2	any	any	DET
ejde-320	276	3	w	w	PROPN
ejde-320	276	4	∈	∈	PROPN
ejde-320	276	5	lip0(b	lip0(b	PROPN
ejde-320	276	6	)	)	PUNCT
ejde-320	276	7	we	we	PRON
ejde-320	276	8	therefore	therefore	ADV
ejde-320	276	9	have	have	VERB
ejde-320	276	10	−	−	PROPN
ejde-320	276	11	∫	∫	PROPN
ejde-320	276	12	b	b	PROPN
ejde-320	276	13	∇w2	∇w2	X
ejde-320	276	14	·	·	SYM
ejde-320	276	15	a∇u	a∇u	X
ejde-320	276	16	=	=	SYM
ejde-320	276	17	∫	∫	PROPN
ejde-320	276	18	b	b	PROPN
ejde-320	276	19	w2f	w2f	PROPN
ejde-320	276	20	,	,	PUNCT
ejde-320	276	21	since	since	SCONJ
ejde-320	276	22	w2	w2	PROPN
ejde-320	276	23	∈	∈	PROPN
ejde-320	276	24	lip0(b	lip0(b	PROPN
ejde-320	276	25	)	)	PUNCT
ejde-320	276	26	.	.	PUNCT
ejde-320	277	1	by	by	ADP
ejde-320	277	2	applying	apply	VERB
ejde-320	277	3	the	the	DET
ejde-320	277	4	chain	chain	NOUN
ejde-320	277	5	rule	rule	NOUN
ejde-320	277	6	to	to	ADP
ejde-320	277	7	∇w2	∇w2	NOUN
ejde-320	277	8	and	and	CCONJ
ejde-320	277	9	using	use	VERB
ejde-320	277	10	the	the	DET
ejde-320	277	11	inner	inner	ADJ
ejde-320	277	12	product	product	NOUN
ejde-320	277	13	from	from	ADP
ejde-320	277	14	definition	definition	NOUN
ejde-320	277	15	2.1	2.1	NUM
ejde-320	277	16	,	,	PUNCT
ejde-320	277	17	we	we	PRON
ejde-320	277	18	see	see	VERB
ejde-320	277	19	that∫	that∫	PROPN
ejde-320	277	20	b	b	PROPN
ejde-320	277	21	w2f	w2f	PROPN
ejde-320	277	22	=	=	PUNCT
ejde-320	277	23	−2	−2	NOUN
ejde-320	277	24	∫	∫	PROPN
ejde-320	277	25	b	b	PROPN
ejde-320	277	26	w〈∇w,∇u	w〈∇w,∇u	PROPN
ejde-320	277	27	〉	〉	NOUN
ejde-320	277	28	≤	≤	NUM
ejde-320	277	29	2	2	NUM
ejde-320	277	30	(	(	PUNCT
ejde-320	277	31	∫	∫	PROPN
ejde-320	277	32	b	b	PROPN
ejde-320	277	33	w2[∇u]2a	w2[∇u]2a	PROPN
ejde-320	277	34	)	)	PUNCT
ejde-320	278	1	1/2(∫	1/2(∫	NUM
ejde-320	278	2	b	b	NOUN
ejde-320	279	1	[	[	X
ejde-320	279	2	∇w]2a	∇w]2a	NOUN
ejde-320	279	3	)	)	PUNCT
ejde-320	279	4	1/2	1/2	NUM
ejde-320	279	5	,	,	PUNCT
ejde-320	279	6	(	(	PUNCT
ejde-320	279	7	4.2	4.2	NUM
ejde-320	279	8	)	)	PUNCT
ejde-320	279	9	where	where	SCONJ
ejde-320	279	10	the	the	DET
ejde-320	279	11	inequality	inequality	NOUN
ejde-320	279	12	follows	follow	VERB
ejde-320	279	13	from	from	ADP
ejde-320	279	14	an	an	DET
ejde-320	279	15	application	application	NOUN
ejde-320	279	16	of	of	ADP
ejde-320	279	17	the	the	DET
ejde-320	279	18	cauchy	cauchy	PROPN
ejde-320	279	19	-	-	PUNCT
ejde-320	279	20	schwartz	schwartz	PROPN
ejde-320	279	21	inequality	inequality	NOUN
ejde-320	279	22	followed	follow	VERB
ejde-320	279	23	by	by	ADP
ejde-320	279	24	the	the	DET
ejde-320	279	25	hölder	hölder	NOUN
ejde-320	279	26	’s	’s	PART
ejde-320	279	27	inequality	inequality	NOUN
ejde-320	279	28	.	.	PUNCT
ejde-320	280	1	now	now	ADV
ejde-320	280	2	analyzing	analyze	VERB
ejde-320	280	3	the	the	DET
ejde-320	280	4	first	first	ADJ
ejde-320	280	5	term	term	NOUN
ejde-320	280	6	in	in	ADP
ejde-320	280	7	the	the	DET
ejde-320	280	8	above	above	ADJ
ejde-320	280	9	inequality	inequality	NOUN
ejde-320	280	10	we	we	PRON
ejde-320	280	11	observe	observe	VERB
ejde-320	280	12	that∫	that∫	PROPN
ejde-320	280	13	b	b	PROPN
ejde-320	280	14	w2[∇u]2a	w2[∇u]2a	PROPN
ejde-320	280	15	=	=	SYM
ejde-320	280	16	∫	∫	PROPN
ejde-320	280	17	b	b	PROPN
ejde-320	280	18	w2∇u	w2∇u	PROPN
ejde-320	280	19	·	·	SYM
ejde-320	280	20	a∇u	a∇u	X
ejde-320	280	21	.	.	PUNCT
ejde-320	281	1	(	(	PUNCT
ejde-320	281	2	4.3	4.3	NUM
ejde-320	281	3	)	)	PUNCT
ejde-320	281	4	furthermore	furthermore	ADV
ejde-320	281	5	,	,	PUNCT
ejde-320	281	6	since	since	SCONJ
ejde-320	281	7	lip0(b	lip0(b	NOUN
ejde-320	281	8	)	)	PUNCT
ejde-320	281	9	is	be	AUX
ejde-320	281	10	dense	dense	ADJ
ejde-320	281	11	in	in	ADP
ejde-320	281	12	(	(	PUNCT
ejde-320	281	13	w	w	PROPN
ejde-320	281	14	1,2	1,2	NUM
ejde-320	281	15	a	a	PRON
ejde-320	281	16	)	)	PUNCT
ejde-320	281	17	0	0	PUNCT
ejde-320	282	1	we	we	PRON
ejde-320	282	2	can	can	AUX
ejde-320	282	3	take	take	VERB
ejde-320	282	4	w2u	w2u	ADV
ejde-320	282	5	as	as	ADP
ejde-320	282	6	a	a	DET
ejde-320	282	7	test	test	NOUN
ejde-320	282	8	function	function	NOUN
ejde-320	282	9	in	in	ADP
ejde-320	282	10	the	the	DET
ejde-320	282	11	definition	definition	NOUN
ejde-320	282	12	of	of	ADP
ejde-320	282	13	a	a	DET
ejde-320	282	14	weak	weak	ADJ
ejde-320	282	15	solution	solution	NOUN
ejde-320	282	16	to	to	PART
ejde-320	282	17	obtain	obtain	VERB
ejde-320	282	18	−	−	PROPN
ejde-320	282	19	∫	∫	PROPN
ejde-320	282	20	w2uf	w2uf	X
ejde-320	283	1	=	=	SYM
ejde-320	283	2	∫	∫	PROPN
ejde-320	283	3	∇(w2u	∇(w2u	PROPN
ejde-320	283	4	)	)	PUNCT
ejde-320	283	5	·	·	PUNCT
ejde-320	283	6	a∇u	a∇u	X
ejde-320	283	7	=	=	SYM
ejde-320	283	8	∫	∫	PROPN
ejde-320	283	9	(	(	PUNCT
ejde-320	283	10	2w∇wu+	2w∇wu+	NUM
ejde-320	283	11	w2∇u	w2∇u	PROPN
ejde-320	283	12	)	)	PUNCT
ejde-320	283	13	·	·	PUNCT
ejde-320	283	14	a∇u	a∇u	X
ejde-320	283	15	=	=	SYM
ejde-320	283	16	2	2	NUM
ejde-320	283	17	∫	∫	NOUN
ejde-320	283	18	wu∇w	wu∇w	PROPN
ejde-320	283	19	·	·	PUNCT
ejde-320	283	20	a∇u+	a∇u+	PROPN
ejde-320	283	21	∫	∫	PROPN
ejde-320	283	22	b	b	PROPN
ejde-320	283	23	w2∇u	w2∇u	PROPN
ejde-320	283	24	·	·	SYM
ejde-320	283	25	a∇u	a∇u	X
ejde-320	283	26	.	.	PUNCT
ejde-320	284	1	therefore	therefore	ADV
ejde-320	284	2	,	,	PUNCT
ejde-320	284	3	∫	∫	PROPN
ejde-320	284	4	w2∇u	w2∇u	PROPN
ejde-320	284	5	·	·	PUNCT
ejde-320	284	6	a∇u	a∇u	X
ejde-320	284	7	=	=	SYM
ejde-320	284	8	−2	−2	PROPN
ejde-320	284	9	∫	∫	PROPN
ejde-320	284	10	wu∇w	wu∇w	PROPN
ejde-320	284	11	·	·	PUNCT
ejde-320	284	12	a∇u−	a∇u−	PROPN
ejde-320	284	13	∫	∫	PROPN
ejde-320	284	14	w2uf	w2uf	X
ejde-320	284	15	.	.	PUNCT
ejde-320	285	1	hence	hence	ADV
ejde-320	285	2	,	,	PUNCT
ejde-320	285	3	(	(	PUNCT
ejde-320	285	4	4.3	4.3	NUM
ejde-320	285	5	)	)	PUNCT
ejde-320	285	6	becomes∫	becomes∫	X
ejde-320	285	7	w2[∇u]2a	w2[∇u]2a	PROPN
ejde-320	285	8	=	=	SYM
ejde-320	285	9	∫	∫	PROPN
ejde-320	285	10	w2∇u	w2∇u	PROPN
ejde-320	285	11	·	·	PUNCT
ejde-320	285	12	a∇u	a∇u	X
ejde-320	285	13	=	=	SYM
ejde-320	285	14	−2	−2	PROPN
ejde-320	285	15	∫	∫	PROPN
ejde-320	286	1	wu∇w	wu∇w	PROPN
ejde-320	286	2	·	·	PUNCT
ejde-320	286	3	a∇u−	a∇u−	PROPN
ejde-320	286	4	∫	∫	PROPN
ejde-320	286	5	w2uf	w2uf	X
ejde-320	287	1	ejde-2021/82	ejde-2021/82	PRON
ejde-320	287	2	orlicz	orlicz	ADJ
ejde-320	287	3	-	-	PUNCT
ejde-320	287	4	sobolev	sobolev	NOUN
ejde-320	287	5	inequalities	inequality	NOUN
ejde-320	287	6	and	and	CCONJ
ejde-320	287	7	the	the	DET
ejde-320	287	8	dirichlet	dirichlet	PROPN
ejde-320	287	9	problem	problem	NOUN
ejde-320	287	10	13	13	NUM
ejde-320	287	11	=	=	SYM
ejde-320	287	12	−2	−2	PROPN
ejde-320	287	13	∫	∫	X
ejde-320	287	14	〈	〈	PROPN
ejde-320	287	15	u∇w	u∇w	PROPN
ejde-320	287	16	,	,	PUNCT
ejde-320	287	17	w∇u	w∇u	PROPN
ejde-320	287	18	〉	〉	PROPN
ejde-320	287	19	−	−	NOUN
ejde-320	287	20	∫	∫	NOUN
ejde-320	287	21	uw2f	uw2f	ADJ
ejde-320	287	22	≤	≤	ADJ
ejde-320	287	23	1	1	NUM
ejde-320	287	24	2	2	NUM
ejde-320	287	25	∫	∫	NOUN
ejde-320	287	26	w2[∇u]2a	w2[∇u]2a	PROPN
ejde-320	288	1	+	+	CCONJ
ejde-320	288	2	8	8	NUM
ejde-320	288	3	∫	∫	NOUN
ejde-320	288	4	u2[∇w]2a	u2[∇w]2a	PROPN
ejde-320	288	5	+	+	CCONJ
ejde-320	288	6	∫	∫	PROPN
ejde-320	288	7	|u|w2|f	|u|w2|f	NOUN
ejde-320	288	8	|	|	NOUN
ejde-320	288	9	,	,	PUNCT
ejde-320	288	10	where	where	SCONJ
ejde-320	288	11	the	the	DET
ejde-320	288	12	final	final	ADJ
ejde-320	288	13	estimate	estimate	NOUN
ejde-320	288	14	follows	follow	VERB
ejde-320	288	15	from	from	ADP
ejde-320	288	16	the	the	DET
ejde-320	288	17	cauchy	cauchy	PROPN
ejde-320	288	18	-	-	PUNCT
ejde-320	288	19	schwartz	schwartz	PROPN
ejde-320	288	20	inequality	inequality	NOUN
ejde-320	288	21	followed	follow	VERB
ejde-320	288	22	by	by	ADP
ejde-320	288	23	young	young	PROPN
ejde-320	288	24	’s	’s	PART
ejde-320	288	25	inequality	inequality	NOUN
ejde-320	288	26	.	.	PUNCT
ejde-320	289	1	absorbing	absorb	VERB
ejde-320	289	2	the	the	DET
ejde-320	289	3	first	first	ADJ
ejde-320	289	4	term	term	NOUN
ejde-320	289	5	on	on	ADP
ejde-320	289	6	the	the	DET
ejde-320	289	7	right	right	NOUN
ejde-320	289	8	to	to	ADP
ejde-320	289	9	the	the	DET
ejde-320	289	10	left	left	ADJ
ejde-320	289	11	-	-	PUNCT
ejde-320	289	12	hand	hand	NOUN
ejde-320	289	13	side	side	NOUN
ejde-320	289	14	results	result	NOUN
ejde-320	289	15	in	in	ADP
ejde-320	289	16	∫	∫	PROPN
ejde-320	289	17	w2[∇u]2a	w2[∇u]2a	PROPN
ejde-320	289	18	≤	≤	PROPN
ejde-320	289	19	c	c	PROPN
ejde-320	289	20	(	(	PUNCT
ejde-320	289	21	sup	sup	PROPN
ejde-320	289	22	b	b	NOUN
ejde-320	289	23	|u|	|u|	NOUN
ejde-320	289	24	)	)	PUNCT
ejde-320	289	25	2	2	NUM
ejde-320	289	26	∫	∫	NOUN
ejde-320	290	1	[	[	X
ejde-320	290	2	∇w]2a	∇w]2a	X
ejde-320	291	1	+	+	X
ejde-320	291	2	c	c	X
ejde-320	291	3	(	(	PUNCT
ejde-320	291	4	sup	sup	PROPN
ejde-320	291	5	b	b	NOUN
ejde-320	291	6	|u|	|u|	PROPN
ejde-320	291	7	)	)	PUNCT
ejde-320	291	8	∫	∫	PROPN
ejde-320	292	1	w2|f	w2|f	PROPN
ejde-320	293	1	|	|	ADV
ejde-320	293	2	≤	≤	PROPN
ejde-320	293	3	c	c	PROPN
ejde-320	293	4	max	max	PROPN
ejde-320	293	5	{	{	PUNCT
ejde-320	293	6	(	(	PUNCT
ejde-320	293	7	sup	sup	PROPN
ejde-320	293	8	b	b	NOUN
ejde-320	293	9	|u|	|u|	NOUN
ejde-320	293	10	)	)	PUNCT
ejde-320	293	11	2	2	NUM
ejde-320	293	12	∫	∫	NOUN
ejde-320	294	1	[	[	X
ejde-320	294	2	∇w]2a	∇w]2a	NOUN
ejde-320	294	3	,	,	PUNCT
ejde-320	294	4	(	(	PUNCT
ejde-320	294	5	sup	sup	PROPN
ejde-320	294	6	b	b	NOUN
ejde-320	294	7	|u|	|u|	PROPN
ejde-320	294	8	)	)	PUNCT
ejde-320	294	9	∫	∫	PROPN
ejde-320	294	10	w2|f	w2|f	PROPN
ejde-320	295	1	|	|	NOUN
ejde-320	295	2	}	}	PUNCT
ejde-320	295	3	=	=	PUNCT
ejde-320	295	4	c	c	X
ejde-320	295	5	max	max	PROPN
ejde-320	295	6	{	{	PUNCT
ejde-320	295	7	(	(	PUNCT
ejde-320	295	8	sup	sup	PROPN
ejde-320	295	9	b	b	NOUN
ejde-320	295	10	|u|	|u|	NOUN
ejde-320	295	11	)	)	PUNCT
ejde-320	295	12	2	2	NUM
ejde-320	295	13	∫	∫	NOUN
ejde-320	296	1	[	[	X
ejde-320	296	2	∇w]2a	∇w]2a	NOUN
ejde-320	296	3	,	,	PUNCT
ejde-320	296	4	(	(	PUNCT
ejde-320	296	5	sup	sup	PROPN
ejde-320	296	6	b	b	NOUN
ejde-320	296	7	|u|	|u|	PROPN
ejde-320	296	8	)	)	PUNCT
ejde-320	296	9	∫	∫	PROPN
ejde-320	296	10	w2f	w2f	VERB
ejde-320	296	11	}	}	PUNCT
ejde-320	296	12	,	,	PUNCT
ejde-320	296	13	where	where	SCONJ
ejde-320	296	14	the	the	DET
ejde-320	296	15	last	last	ADJ
ejde-320	296	16	equality	equality	NOUN
ejde-320	296	17	follows	follow	VERB
ejde-320	296	18	from	from	ADP
ejde-320	296	19	the	the	DET
ejde-320	296	20	assumption	assumption	NOUN
ejde-320	296	21	that	that	SCONJ
ejde-320	296	22	f	f	PROPN
ejde-320	296	23	is	be	AUX
ejde-320	296	24	non	non	ADJ
ejde-320	296	25	-	-	ADJ
ejde-320	296	26	negative	negative	ADJ
ejde-320	296	27	.	.	PUNCT
ejde-320	297	1	we	we	PRON
ejde-320	297	2	claim	claim	VERB
ejde-320	297	3	that	that	SCONJ
ejde-320	297	4	comparing	compare	VERB
ejde-320	297	5	∫	∫	PROPN
ejde-320	297	6	w2[∇u]2a	w2[∇u]2a	PROPN
ejde-320	297	7	to	to	ADP
ejde-320	297	8	either	either	DET
ejde-320	297	9	term	term	NOUN
ejde-320	297	10	inside	inside	ADP
ejde-320	297	11	the	the	DET
ejde-320	297	12	maximum	maximum	ADJ
ejde-320	297	13	results	result	NOUN
ejde-320	297	14	in	in	ADP
ejde-320	297	15	equivalent	equivalent	ADJ
ejde-320	297	16	inequalities	inequality	NOUN
ejde-320	297	17	.	.	PUNCT
ejde-320	298	1	first	first	ADV
ejde-320	298	2	assume	assume	VERB
ejde-320	298	3	that	that	SCONJ
ejde-320	298	4	the	the	DET
ejde-320	298	5	first	first	ADJ
ejde-320	298	6	term	term	NOUN
ejde-320	298	7	,	,	PUNCT
ejde-320	298	8	(	(	PUNCT
ejde-320	298	9	supb	supb	VERB
ejde-320	298	10	|u|	|u|	PROPN
ejde-320	298	11	)	)	PUNCT
ejde-320	298	12	2	2	NUM
ejde-320	298	13	∫	∫	NOUN
ejde-320	299	1	[	[	X
ejde-320	299	2	∇w]2a	∇w]2a	NOUN
ejde-320	299	3	,	,	PUNCT
ejde-320	299	4	dominates	dominate	VERB
ejde-320	299	5	.	.	PUNCT
ejde-320	300	1	then	then	ADV
ejde-320	300	2	combining	combine	VERB
ejde-320	300	3	the	the	DET
ejde-320	300	4	above	above	ADJ
ejde-320	300	5	inequality	inequality	NOUN
ejde-320	300	6	with	with	ADP
ejde-320	300	7	(	(	PUNCT
ejde-320	300	8	4.2	4.2	NUM
ejde-320	300	9	)	)	PUNCT
ejde-320	300	10	gives∫	gives∫	NOUN
ejde-320	300	11	w2f	w2f	VERB
ejde-320	300	12	≤	≤	PUNCT
ejde-320	300	13	c	c	NOUN
ejde-320	300	14	(	(	PUNCT
ejde-320	300	15	sup	sup	PROPN
ejde-320	300	16	b	b	NOUN
ejde-320	300	17	|u|	|u|	ADV
ejde-320	300	18	)	)	PUNCT
ejde-320	300	19	∫	∫	PROPN
ejde-320	301	1	[	[	X
ejde-320	301	2	∇w]2a	∇w]2a	X
ejde-320	301	3	=	=	SYM
ejde-320	301	4	c	c	X
ejde-320	301	5	(	(	PUNCT
ejde-320	301	6	sup	sup	PROPN
ejde-320	301	7	b	b	PROPN
ejde-320	301	8	|u|	|u|	PROPN
ejde-320	301	9	)	)	PUNCT
ejde-320	302	1	‖∇aw‖2l2	‖∇aw‖2l2	PROPN
ejde-320	302	2	≤	≤	ADJ
ejde-320	302	3	c‖f‖lϕ̃‖∇aw‖2l2	c‖f‖lϕ̃‖∇aw‖2l2	PROPN
ejde-320	302	4	,	,	PUNCT
ejde-320	302	5	where	where	SCONJ
ejde-320	302	6	the	the	DET
ejde-320	302	7	final	final	ADJ
ejde-320	302	8	inequality	inequality	NOUN
ejde-320	302	9	follows	follow	VERB
ejde-320	302	10	from	from	ADP
ejde-320	302	11	the	the	DET
ejde-320	302	12	global	global	ADJ
ejde-320	302	13	boundedness	boundedness	PROPN
ejde-320	302	14	estimate	estimate	NOUN
ejde-320	302	15	.	.	PUNCT
ejde-320	303	1	on	on	ADP
ejde-320	303	2	the	the	DET
ejde-320	303	3	other	other	ADJ
ejde-320	303	4	hand	hand	NOUN
ejde-320	303	5	,	,	PUNCT
ejde-320	303	6	if	if	SCONJ
ejde-320	303	7	supb	supb	NOUN
ejde-320	303	8	|u|	|u|	PROPN
ejde-320	303	9	∫	∫	NOUN
ejde-320	303	10	w2f	w2f	PROPN
ejde-320	303	11	dominates	dominate	VERB
ejde-320	303	12	,	,	PUNCT
ejde-320	303	13	then	then	ADV
ejde-320	303	14	(	(	PUNCT
ejde-320	303	15	4.2	4.2	NUM
ejde-320	303	16	)	)	PUNCT
ejde-320	303	17	gives∫	gives∫	NOUN
ejde-320	303	18	w2f	w2f	VERB
ejde-320	303	19	≤	≤	PUNCT
ejde-320	303	20	c	c	NOUN
ejde-320	303	21	(	(	PUNCT
ejde-320	303	22	sup	sup	PROPN
ejde-320	303	23	b	b	NOUN
ejde-320	303	24	|u|	|u|	PROPN
ejde-320	303	25	∫	∫	NOUN
ejde-320	303	26	w2f	w2f	VERB
ejde-320	303	27	)	)	PUNCT
ejde-320	303	28	1/2(∫	1/2(∫	NUM
ejde-320	304	1	[	[	NOUN
ejde-320	304	2	∇w]2a	∇w]2a	NOUN
ejde-320	304	3	)	)	PUNCT
ejde-320	304	4	1/2	1/2	NUM
ejde-320	304	5	.	.	PUNCT
ejde-320	305	1	combining	combine	VERB
ejde-320	305	2	with	with	ADP
ejde-320	305	3	the	the	DET
ejde-320	305	4	global	global	ADJ
ejde-320	305	5	boundedness	boundedness	PROPN
ejde-320	305	6	estimate	estimate	NOUN
ejde-320	305	7	,	,	PUNCT
ejde-320	305	8	this	this	DET
ejde-320	305	9	becomes∫	becomes∫	NOUN
ejde-320	305	10	w2f	w2f	VERB
ejde-320	305	11	≤	≤	NUM
ejde-320	305	12	‖f‖lϕ̃	‖f‖lϕ̃	ADP
ejde-320	305	13	∫	∫	PROPN
ejde-320	306	1	[	[	X
ejde-320	306	2	∇w]2a	∇w]2a	X
ejde-320	306	3	=	=	SYM
ejde-320	306	4	c‖f‖lϕ̃‖∇aw‖2l2	c‖f‖lϕ̃‖∇aw‖2l2	PROPN
ejde-320	306	5	,	,	PUNCT
ejde-320	306	6	(	(	PUNCT
ejde-320	306	7	4.4	4.4	NUM
ejde-320	306	8	)	)	PUNCT
ejde-320	306	9	which	which	PRON
ejde-320	306	10	is	be	AUX
ejde-320	306	11	the	the	DET
ejde-320	306	12	same	same	ADJ
ejde-320	306	13	estimate	estimate	NOUN
ejde-320	306	14	as	as	ADP
ejde-320	306	15	above	above	ADV
ejde-320	306	16	.	.	PUNCT
ejde-320	307	1	using	use	VERB
ejde-320	307	2	the	the	DET
ejde-320	307	3	equivalent	equivalent	ADJ
ejde-320	307	4	definition	definition	NOUN
ejde-320	307	5	of	of	ADP
ejde-320	307	6	the	the	DET
ejde-320	307	7	orlicz	orlicz	ADJ
ejde-320	307	8	norm	norm	NOUN
ejde-320	307	9	(	(	PUNCT
ejde-320	307	10	2.7	2.7	NUM
ejde-320	307	11	)	)	PUNCT
ejde-320	307	12	and	and	CCONJ
ejde-320	307	13	lemma	lemma	PROPN
ejde-320	307	14	4.1	4.1	NUM
ejde-320	307	15	,	,	PUNCT
ejde-320	307	16	we	we	PRON
ejde-320	307	17	have	have	VERB
ejde-320	307	18	|w2|lϕ	|w2|lϕ	PROPN
ejde-320	307	19	=	=	PUNCT
ejde-320	307	20	sup	sup	PROPN
ejde-320	307	21	{	{	PUNCT
ejde-320	307	22	∫	∫	PROPN
ejde-320	307	23	b	b	PROPN
ejde-320	307	24	w2f	w2f	PROPN
ejde-320	307	25	:	:	PUNCT
ejde-320	307	26	∫	∫	PROPN
ejde-320	307	27	b	b	PROPN
ejde-320	307	28	ϕ̃(f	ϕ̃(f	PROPN
ejde-320	307	29	)	)	PUNCT
ejde-320	307	30	≤	≤	NUM
ejde-320	307	31	1	1	NUM
ejde-320	307	32	and	and	CCONJ
ejde-320	307	33	f	f	PROPN
ejde-320	307	34	≥	≥	NOUN
ejde-320	307	35	0	0	NUM
ejde-320	307	36	}	}	PUNCT
ejde-320	307	37	=	=	SYM
ejde-320	307	38	sup	sup	NOUN
ejde-320	307	39	{	{	PUNCT
ejde-320	307	40	∫	∫	PROPN
ejde-320	307	41	b	b	PROPN
ejde-320	307	42	w2f	w2f	PROPN
ejde-320	307	43	:	:	PUNCT
ejde-320	307	44	‖f‖lϕ̃	‖f‖lϕ̃	VERB
ejde-320	307	45	≤	≤	ADV
ejde-320	307	46	1	1	NUM
ejde-320	307	47	and	and	CCONJ
ejde-320	307	48	f	f	PROPN
ejde-320	307	49	≥	≥	NUM
ejde-320	307	50	0	0	NUM
ejde-320	307	51	}	}	PUNCT
ejde-320	307	52	.	.	PUNCT
ejde-320	308	1	combining	combine	VERB
ejde-320	308	2	with	with	ADP
ejde-320	308	3	(	(	PUNCT
ejde-320	308	4	4.4	4.4	NUM
ejde-320	308	5	)	)	PUNCT
ejde-320	308	6	and	and	CCONJ
ejde-320	308	7	(	(	PUNCT
ejde-320	308	8	2.3	2.3	NUM
ejde-320	308	9	)	)	PUNCT
ejde-320	308	10	gives	give	VERB
ejde-320	308	11	‖w2‖lϕ	‖w2‖lϕ	PUNCT
ejde-320	308	12	≤	≤	NUM
ejde-320	308	13	c‖∇aw‖2l2	c‖∇aw‖2l2	PROPN
ejde-320	308	14	.	.	PUNCT
ejde-320	309	1	to	to	PART
ejde-320	309	2	obtain	obtain	VERB
ejde-320	309	3	the	the	DET
ejde-320	309	4	desired	desire	VERB
ejde-320	309	5	(	(	PUNCT
ejde-320	309	6	φ	φ	PROPN
ejde-320	309	7	,	,	PUNCT
ejde-320	309	8	2	2	NUM
ejde-320	309	9	)	)	PUNCT
ejde-320	309	10	orlicz	orlicz	ADJ
ejde-320	309	11	-	-	PUNCT
ejde-320	309	12	sobolev	sobolev	NOUN
ejde-320	309	13	inequality	inequality	NOUN
ejde-320	309	14	it	it	PRON
ejde-320	309	15	remains	remain	VERB
ejde-320	309	16	to	to	PART
ejde-320	309	17	show	show	VERB
ejde-320	309	18	that	that	SCONJ
ejde-320	309	19	‖w‖2lφ	‖w‖2lφ	ADJ
ejde-320	309	20	≤	≤	NUM
ejde-320	309	21	c‖w	c‖w	VERB
ejde-320	309	22	2‖lϕ	2‖lϕ	NOUN
ejde-320	309	23	,	,	PUNCT
ejde-320	309	24	which	which	PRON
ejde-320	309	25	follows	follow	VERB
ejde-320	309	26	immediately	immediately	ADV
ejde-320	309	27	from	from	ADP
ejde-320	309	28	the	the	DET
ejde-320	309	29	second	second	ADJ
ejde-320	309	30	inequality	inequality	NOUN
ejde-320	309	31	in	in	ADP
ejde-320	309	32	lemma	lemma	PROPN
ejde-320	309	33	3.6	3.6	NUM
ejde-320	309	34	.	.	PUNCT
ejde-320	310	1	hence	hence	ADV
ejde-320	310	2	,	,	PUNCT
ejde-320	310	3	‖w‖2lφ	‖w‖2lφ	ADJ
ejde-320	310	4	≤	≤	NUM
ejde-320	310	5	4‖w2‖lϕ	4‖w2‖lϕ	NUM
ejde-320	310	6	≤	≤	NUM
ejde-320	310	7	c‖∇aw‖2l2	c‖∇aw‖2l2	PROPN
ejde-320	310	8	.	.	PUNCT
ejde-320	311	1	by	by	ADP
ejde-320	311	2	the	the	DET
ejde-320	311	3	density	density	NOUN
ejde-320	311	4	of	of	ADP
ejde-320	311	5	lip0(b	lip0(b	NOUN
ejde-320	311	6	)	)	PUNCT
ejde-320	311	7	in	in	ADP
ejde-320	311	8	w	w	PROPN
ejde-320	311	9	1,2	1,2	NUM
ejde-320	311	10	0	0	NUM
ejde-320	311	11	(	(	PUNCT
ejde-320	311	12	b	b	X
ejde-320	311	13	)	)	PUNCT
ejde-320	311	14	we	we	PRON
ejde-320	311	15	obtain	obtain	VERB
ejde-320	311	16	the	the	DET
ejde-320	311	17	desired	desire	VERB
ejde-320	311	18	orlicz	orlicz	ADJ
ejde-320	311	19	-	-	PUNCT
ejde-320	311	20	sobolev	sobolev	NOUN
ejde-320	311	21	inequality	inequality	NOUN
ejde-320	311	22	,	,	PUNCT
ejde-320	311	23	‖v‖lφ(b	‖v‖lφ(b	VERB
ejde-320	311	24	,	,	PUNCT
ejde-320	311	25	dµ	dµ	ADJ
ejde-320	311	26	)	)	PUNCT
ejde-320	311	27	≤	≤	NOUN
ejde-320	311	28	‖∇av‖l2(b	‖∇av‖l2(b	NUM
ejde-320	311	29	,	,	PUNCT
ejde-320	311	30	dµ	dµ	PROPN
ejde-320	311	31	)	)	PUNCT
ejde-320	311	32	for	for	ADP
ejde-320	311	33	all	all	PRON
ejde-320	311	34	v	v	ADP
ejde-320	311	35	∈	∈	NOUN
ejde-320	311	36	(	(	PUNCT
ejde-320	311	37	w	w	PROPN
ejde-320	311	38	1,2	1,2	NUM
ejde-320	311	39	a	a	PRON
ejde-320	311	40	)	)	PUNCT
ejde-320	311	41	0	0	NUM
ejde-320	311	42	�	�	PROPN
ejde-320	311	43	14	14	NUM
ejde-320	311	44	u.	u.	PROPN
ejde-320	311	45	hafeez	hafeez	PROPN
ejde-320	311	46	,	,	PUNCT
ejde-320	311	47	t.	t.	PROPN
ejde-320	311	48	lavier	lavier	PROPN
ejde-320	311	49	,	,	PUNCT
ejde-320	311	50	l.	l.	PROPN
ejde-320	311	51	williams	williams	PROPN
ejde-320	311	52	,	,	PUNCT
ejde-320	311	53	l.	l.	PROPN
ejde-320	311	54	korobenko	korobenko	PROPN
ejde-320	311	55	ejde-2021/82	ejde-2021/82	PROPN
ejde-320	311	56	5	5	NUM
ejde-320	311	57	.	.	NOUN
ejde-320	311	58	sharpness	sharpness	NOUN
ejde-320	311	59	in	in	ADP
ejde-320	311	60	this	this	DET
ejde-320	311	61	section	section	NOUN
ejde-320	311	62	,	,	PUNCT
ejde-320	311	63	we	we	PRON
ejde-320	311	64	demonstrate	demonstrate	VERB
ejde-320	311	65	a	a	DET
ejde-320	311	66	weak	weak	ADJ
ejde-320	311	67	degree	degree	NOUN
ejde-320	311	68	of	of	ADP
ejde-320	311	69	sharpness	sharpness	NOUN
ejde-320	311	70	of	of	ADP
ejde-320	311	71	our	our	PRON
ejde-320	311	72	results	result	NOUN
ejde-320	311	73	.	.	PUNCT
ejde-320	312	1	more	more	ADV
ejde-320	312	2	precisely	precisely	ADV
ejde-320	312	3	,	,	PUNCT
ejde-320	312	4	we	we	PRON
ejde-320	312	5	show	show	VERB
ejde-320	312	6	that	that	SCONJ
ejde-320	312	7	even	even	ADV
ejde-320	312	8	though	though	SCONJ
ejde-320	312	9	the	the	DET
ejde-320	312	10	requirement	requirement	NOUN
ejde-320	312	11	on	on	ADP
ejde-320	312	12	the	the	DET
ejde-320	312	13	right	right	ADJ
ejde-320	312	14	hand	hand	NOUN
ejde-320	312	15	side	side	NOUN
ejde-320	312	16	function	function	NOUN
ejde-320	312	17	f	f	PROPN
ejde-320	312	18	in	in	ADP
ejde-320	312	19	the	the	DET
ejde-320	312	20	sufficiency	sufficiency	NOUN
ejde-320	312	21	result	result	NOUN
ejde-320	312	22	,	,	PUNCT
ejde-320	312	23	theorem	theorem	VERB
ejde-320	312	24	1.1	1.1	NUM
ejde-320	312	25	,	,	PUNCT
ejde-320	312	26	is	be	AUX
ejde-320	312	27	stronger	strong	ADJ
ejde-320	312	28	then	then	ADV
ejde-320	312	29	the	the	DET
ejde-320	312	30	one	one	NOUN
ejde-320	312	31	in	in	ADP
ejde-320	312	32	theorem	theorem	ADJ
ejde-320	312	33	1.2	1.2	NUM
ejde-320	312	34	,	,	PUNCT
ejde-320	312	35	it	it	PRON
ejde-320	312	36	can	can	AUX
ejde-320	312	37	not	not	PART
ejde-320	312	38	be	be	AUX
ejde-320	312	39	significantly	significantly	ADV
ejde-320	312	40	relaxed	relax	VERB
ejde-320	312	41	.	.	PUNCT
ejde-320	313	1	namely	namely	ADV
ejde-320	313	2	,	,	PUNCT
ejde-320	313	3	there	there	PRON
ejde-320	313	4	exist	exist	VERB
ejde-320	313	5	an	an	DET
ejde-320	313	6	operator	operator	NOUN
ejde-320	313	7	a	a	PRON
ejde-320	313	8	and	and	CCONJ
ejde-320	313	9	a	a	DET
ejde-320	313	10	function	function	NOUN
ejde-320	313	11	u	u	NOUN
ejde-320	313	12	∈	∈	PROPN
ejde-320	313	13	(	(	PUNCT
ejde-320	313	14	w	w	PROPN
ejde-320	313	15	1,2	1,2	NUM
ejde-320	313	16	a	a	PRON
ejde-320	313	17	)	)	PUNCT
ejde-320	313	18	0	0	NUM
ejde-320	313	19	such	such	ADJ
ejde-320	313	20	that	that	SCONJ
ejde-320	313	21	(	(	PUNCT
ejde-320	313	22	1	1	X
ejde-320	313	23	)	)	PUNCT
ejde-320	313	24	a	a	DET
ejde-320	313	25	(	(	PUNCT
ejde-320	313	26	ψ	ψ	NOUN
ejde-320	313	27	,	,	PUNCT
ejde-320	313	28	2	2	NUM
ejde-320	313	29	)	)	PUNCT
ejde-320	313	30	orlicz	orlicz	ADJ
ejde-320	313	31	-	-	PUNCT
ejde-320	313	32	sobolev	sobolev	NOUN
ejde-320	313	33	inequality	inequality	NOUN
ejde-320	313	34	holds	hold	VERB
ejde-320	313	35	in	in	ADP
ejde-320	313	36	a	a	DET
ejde-320	313	37	subunit	subunit	NOUN
ejde-320	313	38	metric	metric	PROPN
ejde-320	313	39	ball	ball	PROPN
ejde-320	313	40	b	b	PROPN
ejde-320	313	41	with	with	ADP
ejde-320	313	42	ψ(t	ψ(t	PROPN
ejde-320	313	43	)	)	PUNCT
ejde-320	313	44	=	=	SYM
ejde-320	313	45	t2(ln	t2(ln	PROPN
ejde-320	313	46	t)n	t)n	NOUN
ejde-320	313	47	,	,	PUNCT
ejde-320	313	48	n	n	CCONJ
ejde-320	313	49	>	>	X
ejde-320	313	50	1	1	NUM
ejde-320	313	51	,	,	PUNCT
ejde-320	313	52	for	for	ADP
ejde-320	313	53	all	all	DET
ejde-320	313	54	t	t	NOUN
ejde-320	313	55	>	>	X
ejde-320	313	56	1	1	NUM
ejde-320	313	57	;	;	PUNCT
ejde-320	313	58	(	(	PUNCT
ejde-320	313	59	2	2	X
ejde-320	313	60	)	)	PUNCT
ejde-320	313	61	lu	lu	PROPN
ejde-320	313	62	∈	∈	PROPN
ejde-320	313	63	lφ̃m	lφ̃m	NOUN
ejde-320	313	64	with	with	ADP
ejde-320	313	65	φm	φm	PROPN
ejde-320	313	66	(	(	PUNCT
ejde-320	313	67	t	t	PROPN
ejde-320	313	68	)	)	PUNCT
ejde-320	314	1	=	=	SYM
ejde-320	314	2	t(ln	t(ln	PROPN
ejde-320	314	3	t)m	t)m	NOUN
ejde-320	314	4	,	,	PUNCT
ejde-320	314	5	m	m	VERB
ejde-320	314	6	>	>	X
ejde-320	314	7	2	2	NUM
ejde-320	314	8	+	+	NUM
ejde-320	314	9	2n	2n	NUM
ejde-320	314	10	,	,	PUNCT
ejde-320	314	11	for	for	ADP
ejde-320	314	12	all	all	DET
ejde-320	314	13	t	t	NOUN
ejde-320	314	14	>	>	X
ejde-320	314	15	1	1	NUM
ejde-320	314	16	;	;	PUNCT
ejde-320	314	17	(	(	PUNCT
ejde-320	314	18	3	3	X
ejde-320	314	19	)	)	PUNCT
ejde-320	314	20	u	u	NOUN
ejde-320	314	21	is	be	AUX
ejde-320	314	22	unbounded	unbounded	ADJ
ejde-320	314	23	at	at	ADP
ejde-320	314	24	the	the	DET
ejde-320	314	25	origin	origin	NOUN
ejde-320	314	26	.	.	PUNCT
ejde-320	315	1	to	to	PART
ejde-320	315	2	set	set	VERB
ejde-320	315	3	the	the	DET
ejde-320	315	4	stage	stage	NOUN
ejde-320	315	5	,	,	PUNCT
ejde-320	315	6	we	we	PRON
ejde-320	315	7	first	first	ADV
ejde-320	315	8	provide	provide	VERB
ejde-320	315	9	similar	similar	ADJ
ejde-320	315	10	constructions	construction	NOUN
ejde-320	315	11	in	in	ADP
ejde-320	315	12	the	the	DET
ejde-320	315	13	case	case	NOUN
ejde-320	315	14	of	of	ADP
ejde-320	315	15	the	the	DET
ejde-320	315	16	laplacian	laplacian	ADJ
ejde-320	315	17	operator	operator	NOUN
ejde-320	315	18	,	,	PUNCT
ejde-320	315	19	and	and	CCONJ
ejde-320	315	20	a	a	DET
ejde-320	315	21	finitely	finitely	ADV
ejde-320	315	22	degenerate	degenerate	ADJ
ejde-320	315	23	elliptic	elliptic	ADJ
ejde-320	315	24	operator	operator	NOUN
ejde-320	315	25	.	.	PUNCT
ejde-320	316	1	5.1	5.1	NUM
ejde-320	316	2	.	.	PUNCT
ejde-320	317	1	laplacian	laplacian	ADJ
ejde-320	317	2	counterexample	counterexample	PROPN
ejde-320	317	3	.	.	PUNCT
ejde-320	318	1	recall	recall	VERB
ejde-320	318	2	that	that	SCONJ
ejde-320	318	3	in	in	ADP
ejde-320	318	4	general	general	ADJ
ejde-320	318	5	we	we	PRON
ejde-320	318	6	are	be	AUX
ejde-320	318	7	concerned	concerned	ADJ
ejde-320	318	8	with	with	ADP
ejde-320	318	9	the	the	DET
ejde-320	318	10	following	following	ADJ
ejde-320	318	11	divergence	divergence	NOUN
ejde-320	318	12	form	form	NOUN
ejde-320	318	13	operator	operator	NOUN
ejde-320	318	14	lu	lu	NOUN
ejde-320	318	15	=	=	PUNCT
ejde-320	318	16	∇	∇	X
ejde-320	318	17	·	·	PUNCT
ejde-320	318	18	a∇u	a∇u	X
ejde-320	318	19	.	.	PUNCT
ejde-320	319	1	now	now	ADV
ejde-320	319	2	consider	consider	VERB
ejde-320	319	3	the	the	DET
ejde-320	319	4	two	two	NUM
ejde-320	319	5	dimensional	dimensional	ADJ
ejde-320	319	6	case	case	NOUN
ejde-320	319	7	of	of	ADP
ejde-320	319	8	r2	r2	PROPN
ejde-320	319	9	and	and	CCONJ
ejde-320	319	10	let	let	VERB
ejde-320	319	11	a	a	DET
ejde-320	319	12	=	=	X
ejde-320	319	13	(	(	PUNCT
ejde-320	319	14	1	1	NUM
ejde-320	319	15	0	0	NUM
ejde-320	319	16	0	0	NUM
ejde-320	319	17	1	1	NUM
ejde-320	319	18	)	)	PUNCT
ejde-320	319	19	,	,	PUNCT
ejde-320	319	20	so	so	ADV
ejde-320	319	21	l	l	NOUN
ejde-320	319	22	=	=	SYM
ejde-320	319	23	∆	∆	PROPN
ejde-320	319	24	,	,	PUNCT
ejde-320	319	25	the	the	DET
ejde-320	319	26	laplace	laplace	NOUN
ejde-320	319	27	operator	operator	NOUN
ejde-320	319	28	.	.	PUNCT
ejde-320	320	1	for	for	ADP
ejde-320	320	2	a	a	DET
ejde-320	320	3	generic	generic	ADJ
ejde-320	320	4	u	u	NOUN
ejde-320	320	5	changing	change	VERB
ejde-320	320	6	to	to	ADP
ejde-320	320	7	polar	polar	ADJ
ejde-320	320	8	coordinates	coordinate	NOUN
ejde-320	320	9	gives	give	VERB
ejde-320	320	10	∆u	∆u	PROPN
ejde-320	320	11	=	=	SYM
ejde-320	320	12	1	1	NUM
ejde-320	320	13	r	r	NOUN
ejde-320	320	14	∂	∂	NOUN
ejde-320	320	15	∂r	∂r	NOUN
ejde-320	320	16	(	(	PUNCT
ejde-320	320	17	r	r	NOUN
ejde-320	320	18	∂u	∂u	PROPN
ejde-320	320	19	∂r	∂r	ADJ
ejde-320	320	20	)	)	PUNCT
ejde-320	321	1	+	+	CCONJ
ejde-320	321	2	1	1	NUM
ejde-320	321	3	r2	r2	NOUN
ejde-320	321	4	∂2u	∂2u	X
ejde-320	321	5	∂θ2	∂θ2	PROPN
ejde-320	321	6	.	.	PUNCT
ejde-320	322	1	now	now	ADV
ejde-320	322	2	we	we	PRON
ejde-320	322	3	choose	choose	VERB
ejde-320	322	4	a	a	DET
ejde-320	322	5	weak	weak	ADJ
ejde-320	322	6	solution	solution	NOUN
ejde-320	322	7	u	u	NOUN
ejde-320	322	8	that	that	PRON
ejde-320	322	9	is	be	AUX
ejde-320	322	10	unbounded	unbounded	ADJ
ejde-320	322	11	at	at	ADP
ejde-320	322	12	the	the	DET
ejde-320	322	13	origin	origin	NOUN
ejde-320	322	14	.	.	PUNCT
ejde-320	323	1	the	the	DET
ejde-320	323	2	power	power	NOUN
ejde-320	323	3	α	α	PROPN
ejde-320	323	4	helps	help	VERB
ejde-320	323	5	us	we	PRON
ejde-320	323	6	control	control	VERB
ejde-320	323	7	the	the	DET
ejde-320	323	8	integrability	integrability	NOUN
ejde-320	323	9	of	of	ADP
ejde-320	323	10	this	this	DET
ejde-320	323	11	unbounded	unbounded	ADJ
ejde-320	323	12	function	function	NOUN
ejde-320	323	13	.	.	PUNCT
ejde-320	324	1	define	define	VERB
ejde-320	324	2	u	u	NOUN
ejde-320	324	3	=	=	PUNCT
ejde-320	324	4	(	(	PUNCT
ejde-320	324	5	ln	ln	NOUN
ejde-320	324	6	1	1	NUM
ejde-320	324	7	r	r	NOUN
ejde-320	324	8	)	)	PUNCT
ejde-320	324	9	α	α	NOUN
ejde-320	324	10	,	,	PUNCT
ejde-320	324	11	where	where	SCONJ
ejde-320	324	12	0	0	X
ejde-320	324	13	<	<	X
ejde-320	324	14	α	α	X
ejde-320	324	15	<	<	X
ejde-320	324	16	1/2	1/2	NUM
ejde-320	324	17	,	,	PUNCT
ejde-320	324	18	so	so	SCONJ
ejde-320	324	19	one	one	PRON
ejde-320	324	20	can	can	AUX
ejde-320	324	21	check	check	VERB
ejde-320	324	22	that	that	SCONJ
ejde-320	324	23	u	u	PRON
ejde-320	324	24	∈w	∈w	NOUN
ejde-320	324	25	1,2(b(0	1,2(b(0	NUM
ejde-320	324	26	,	,	PUNCT
ejde-320	324	27	1/2	1/2	NUM
ejde-320	324	28	)	)	PUNCT
ejde-320	324	29	)	)	PUNCT
ejde-320	324	30	.	.	PUNCT
ejde-320	325	1	since	since	SCONJ
ejde-320	325	2	this	this	DET
ejde-320	325	3	function	function	NOUN
ejde-320	325	4	does	do	AUX
ejde-320	325	5	not	not	PART
ejde-320	325	6	depend	depend	VERB
ejde-320	325	7	on	on	ADP
ejde-320	325	8	θ	θ	PROPN
ejde-320	325	9	,	,	PUNCT
ejde-320	325	10	we	we	PRON
ejde-320	325	11	have	have	VERB
ejde-320	325	12	lu	lu	NOUN
ejde-320	325	13	=	=	PUNCT
ejde-320	325	14	∆u	∆u	PROPN
ejde-320	325	15	=	=	SYM
ejde-320	325	16	1	1	NUM
ejde-320	325	17	r2	r2	NOUN
ejde-320	325	18	α(α−	α(α−	NOUN
ejde-320	325	19	1	1	NUM
ejde-320	325	20	)	)	PUNCT
ejde-320	325	21	ln	ln	NOUN
ejde-320	325	22	(	(	PUNCT
ejde-320	325	23	1	1	NUM
ejde-320	325	24	r	r	NOUN
ejde-320	325	25	)	)	PUNCT
ejde-320	325	26	α−2	α−2	NOUN
ejde-320	325	27	.	.	PUNCT
ejde-320	326	1	thus	thus	ADV
ejde-320	326	2	with	with	ADP
ejde-320	326	3	f	f	PROPN
ejde-320	326	4	:	:	PUNCT
ejde-320	326	5	=	=	SYM
ejde-320	326	6	1	1	NUM
ejde-320	326	7	r2	r2	NOUN
ejde-320	326	8	α(α−	α(α−	NOUN
ejde-320	326	9	1	1	NUM
ejde-320	326	10	)	)	PUNCT
ejde-320	326	11	(	(	PUNCT
ejde-320	326	12	ln	ln	NOUN
ejde-320	326	13	1	1	NUM
ejde-320	326	14	r	r	NOUN
ejde-320	326	15	)	)	PUNCT
ejde-320	326	16	α−2	α−2	NOUN
ejde-320	326	17	,	,	PUNCT
ejde-320	326	18	u	u	NOUN
ejde-320	326	19	is	be	AUX
ejde-320	326	20	a	a	DET
ejde-320	326	21	weak	weak	ADJ
ejde-320	326	22	solution	solution	NOUN
ejde-320	326	23	to	to	ADP
ejde-320	326	24	lu	lu	PROPN
ejde-320	326	25	=	=	SYM
ejde-320	326	26	f	f	PROPN
ejde-320	326	27	which	which	PRON
ejde-320	326	28	is	be	AUX
ejde-320	326	29	unbounded	unbounded	ADJ
ejde-320	326	30	at	at	ADP
ejde-320	326	31	the	the	DET
ejde-320	326	32	origin	origin	NOUN
ejde-320	326	33	.	.	PUNCT
ejde-320	327	1	we	we	PRON
ejde-320	327	2	now	now	ADV
ejde-320	327	3	calculate	calculate	VERB
ejde-320	327	4	the	the	DET
ejde-320	327	5	lq	lq	NOUN
ejde-320	327	6	norm	norm	NOUN
ejde-320	327	7	of	of	ADP
ejde-320	327	8	f	f	PROPN
ejde-320	327	9	in	in	ADP
ejde-320	327	10	the	the	DET
ejde-320	327	11	ball	ball	NOUN
ejde-320	327	12	b	b	PROPN
ejde-320	327	13	=	=	PUNCT
ejde-320	327	14	b(0	b(0	PROPN
ejde-320	327	15	,	,	PUNCT
ejde-320	327	16	1/2	1/2	NUM
ejde-320	327	17	)	)	PUNCT
ejde-320	327	18	‖f‖lq(b	‖f‖lq(b	NUM
ejde-320	327	19	)	)	PUNCT
ejde-320	327	20	=	=	SYM
ejde-320	328	1	∫	∫	PROPN
ejde-320	328	2	2π	2π	PROPN
ejde-320	328	3	0	0	NUM
ejde-320	329	1	∫	∫	NUM
ejde-320	329	2	1/2	1/2	NUM
ejde-320	329	3	0	0	NUM
ejde-320	329	4	|f(r)|qr	|f(r)|qr	PROPN
ejde-320	329	5	dr	dr	PROPN
ejde-320	329	6	dθ	dθ	PROPN
ejde-320	329	7	=	=	PROPN
ejde-320	329	8	2πα(α−	2πα(α−	NUM
ejde-320	329	9	1	1	NUM
ejde-320	329	10	)	)	PUNCT
ejde-320	329	11	∫	∫	PROPN
ejde-320	330	1	1/2	1/2	NUM
ejde-320	330	2	0	0	NUM
ejde-320	330	3	∣∣	∣∣	NUM
ejde-320	330	4	1	1	NUM
ejde-320	330	5	r2	r2	NOUN
ejde-320	330	6	(	(	PUNCT
ejde-320	330	7	ln	ln	NOUN
ejde-320	330	8	1	1	NUM
ejde-320	330	9	r	r	NOUN
ejde-320	330	10	)	)	PUNCT
ejde-320	330	11	α−2∣∣q	α−2∣∣q	PROPN
ejde-320	330	12	r	r	NOUN
ejde-320	330	13	dr	dr	PROPN
ejde-320	330	14	≈	≈	PROPN
ejde-320	330	15	∫	∫	PROPN
ejde-320	330	16	1/2	1/2	NUM
ejde-320	330	17	0	0	NUM
ejde-320	331	1	(	(	PUNCT
ejde-320	331	2	ln	ln	NOUN
ejde-320	331	3	1	1	NUM
ejde-320	331	4	r	r	NOUN
ejde-320	331	5	)	)	PUNCT
ejde-320	331	6	q(α−2	q(α−2	PART
ejde-320	331	7	)	)	PUNCT
ejde-320	331	8	dr	dr	PROPN
ejde-320	331	9	r2q−1	r2q−1	PROPN
ejde-320	331	10	.	.	PUNCT
ejde-320	332	1	the	the	DET
ejde-320	332	2	integral	integral	ADJ
ejde-320	332	3	on	on	ADP
ejde-320	332	4	the	the	DET
ejde-320	332	5	right	right	NOUN
ejde-320	332	6	is	be	AUX
ejde-320	332	7	finite	finite	NOUN
ejde-320	332	8	provided	provide	VERB
ejde-320	332	9	q	q	NOUN
ejde-320	332	10	<	<	X
ejde-320	332	11	1	1	NUM
ejde-320	332	12	or	or	CCONJ
ejde-320	332	13	q	q	NOUN
ejde-320	332	14	=	=	SYM
ejde-320	332	15	1	1	NUM
ejde-320	332	16	and	and	CCONJ
ejde-320	332	17	α	α	PRON
ejde-320	332	18	<	<	X
ejde-320	332	19	1	1	NUM
ejde-320	332	20	.	.	PUNCT
ejde-320	333	1	in	in	ADP
ejde-320	333	2	particular	particular	ADJ
ejde-320	333	3	,	,	PUNCT
ejde-320	333	4	u	u	NOUN
ejde-320	333	5	=	=	NOUN
ejde-320	333	6	ln(1	ln(1	NOUN
ejde-320	333	7	/	/	SYM
ejde-320	333	8	r)1/4	r)1/4	PROPN
ejde-320	333	9	is	be	AUX
ejde-320	333	10	an	an	DET
ejde-320	333	11	unbounded	unbounded	ADJ
ejde-320	333	12	weak	weak	ADJ
ejde-320	333	13	solution	solution	NOUN
ejde-320	333	14	to	to	ADP
ejde-320	333	15	∆u	∆u	PROPN
ejde-320	333	16	=	=	SYM
ejde-320	333	17	f	f	PROPN
ejde-320	333	18	with	with	ADP
ejde-320	333	19	f	f	PROPN
ejde-320	333	20	∈	∈	PROPN
ejde-320	333	21	lq(b	lq(b	PRON
ejde-320	333	22	)	)	PUNCT
ejde-320	333	23	,	,	PUNCT
ejde-320	333	24	q	q	NOUN
ejde-320	333	25	=	=	SYM
ejde-320	333	26	1	1	NUM
ejde-320	333	27	=	=	SYM
ejde-320	333	28	n/2	n/2	NOUN
ejde-320	333	29	.	.	PUNCT
ejde-320	334	1	on	on	ADP
ejde-320	334	2	the	the	DET
ejde-320	334	3	other	other	ADJ
ejde-320	334	4	hand	hand	NOUN
ejde-320	334	5	,	,	PUNCT
ejde-320	334	6	if	if	SCONJ
ejde-320	334	7	u	u	NOUN
ejde-320	334	8	is	be	AUX
ejde-320	334	9	a	a	DET
ejde-320	334	10	weak	weak	ADJ
ejde-320	334	11	solution	solution	NOUN
ejde-320	334	12	to	to	ADP
ejde-320	334	13	lu	lu	PROPN
ejde-320	334	14	=	=	SYM
ejde-320	334	15	f	f	PROPN
ejde-320	334	16	and	and	CCONJ
ejde-320	334	17	f	f	PROPN
ejde-320	334	18	∈	∈	PROPN
ejde-320	334	19	lq(b	lq(b	PRON
ejde-320	334	20	)	)	PUNCT
ejde-320	334	21	with	with	ADP
ejde-320	334	22	q	q	ADJ
ejde-320	334	23	>	>	X
ejde-320	334	24	n/2	n/2	PROPN
ejde-320	334	25	,	,	PUNCT
ejde-320	334	26	[	[	X
ejde-320	334	27	4	4	NUM
ejde-320	334	28	,	,	PUNCT
ejde-320	334	29	theorem	theorem	VERB
ejde-320	334	30	8.16	8.16	NUM
ejde-320	334	31	]	]	PUNCT
ejde-320	334	32	gives	give	VERB
ejde-320	334	33	that	that	DET
ejde-320	334	34	sup	sup	PROPN
ejde-320	334	35	b	b	PROPN
ejde-320	334	36	|u|	|u|	ADJ
ejde-320	334	37	≤	≤	NUM
ejde-320	334	38	sup	sup	NOUN
ejde-320	334	39	∂b	∂b	PROPN
ejde-320	334	40	|u|+	|u|+	NOUN
ejde-320	334	41	c‖f‖q	c‖f‖q	PROPN
ejde-320	334	42	<	<	NOUN
ejde-320	334	43	∞.	∞.	PROPN
ejde-320	334	44	ejde-2021/82	ejde-2021/82	VERB
ejde-320	334	45	orlicz	orlicz	ADJ
ejde-320	334	46	-	-	PUNCT
ejde-320	334	47	sobolev	sobolev	NOUN
ejde-320	334	48	inequalities	inequality	NOUN
ejde-320	334	49	and	and	CCONJ
ejde-320	334	50	the	the	DET
ejde-320	334	51	dirichlet	dirichlet	PROPN
ejde-320	334	52	problem	problem	NOUN
ejde-320	334	53	15	15	NUM
ejde-320	334	54	furthermore	furthermore	ADV
ejde-320	334	55	,	,	PUNCT
ejde-320	334	56	for	for	ADP
ejde-320	334	57	the	the	DET
ejde-320	334	58	laplace	laplace	NOUN
ejde-320	334	59	operator	operator	NOUN
ejde-320	334	60	,	,	PUNCT
ejde-320	334	61	the	the	DET
ejde-320	334	62	associated	associated	ADJ
ejde-320	334	63	subunit	subunit	NOUN
ejde-320	334	64	metric	metric	ADJ
ejde-320	334	65	space	space	NOUN
ejde-320	334	66	coincides	coincide	VERB
ejde-320	334	67	with	with	ADP
ejde-320	334	68	the	the	DET
ejde-320	334	69	euclidean	euclidean	PROPN
ejde-320	334	70	rn	rn	PROPN
ejde-320	334	71	,	,	PUNCT
ejde-320	334	72	and	and	CCONJ
ejde-320	334	73	we	we	PRON
ejde-320	334	74	have	have	VERB
ejde-320	334	75	the	the	DET
ejde-320	334	76	following	follow	VERB
ejde-320	334	77	sobolev	sobolev	NOUN
ejde-320	334	78	inequality	inequality	NOUN
ejde-320	334	79	(	(	PUNCT
ejde-320	334	80	1	1	NUM
ejde-320	334	81	|b|	|b|	PROPN
ejde-320	334	82	∫	∫	PROPN
ejde-320	334	83	b	b	PROPN
ejde-320	334	84	|w|2σ	|w|2σ	NOUN
ejde-320	334	85	)	)	PUNCT
ejde-320	334	86	1	1	NUM
ejde-320	334	87	2σ	2σ	NOUN
ejde-320	335	1	≤	≤	NUM
ejde-320	335	2	cr	cr	NOUN
ejde-320	336	1	(	(	PUNCT
ejde-320	336	2	1	1	NUM
ejde-320	336	3	|b|	|b|	PROPN
ejde-320	336	4	∫	∫	PROPN
ejde-320	336	5	b	b	X
ejde-320	336	6	|∇aw|2	|∇aw|2	NOUN
ejde-320	336	7	)	)	PUNCT
ejde-320	336	8	1/2	1/2	NUM
ejde-320	337	1	+	+	NUM
ejde-320	337	2	c	c	NOUN
ejde-320	337	3	(	(	PUNCT
ejde-320	337	4	1	1	NUM
ejde-320	337	5	|b|	|b|	PROPN
ejde-320	337	6	∫	∫	PROPN
ejde-320	337	7	b	b	PROPN
ejde-320	337	8	|w|2	|w|2	PROPN
ejde-320	337	9	)	)	PUNCT
ejde-320	337	10	1/2	1/2	NUM
ejde-320	337	11	(	(	PUNCT
ejde-320	337	12	5.1	5.1	NUM
ejde-320	337	13	)	)	PUNCT
ejde-320	337	14	for	for	ADP
ejde-320	337	15	σ	σ	PROPN
ejde-320	337	16	≤	≤	PROPN
ejde-320	337	17	n	n	PRON
ejde-320	337	18	n−2	n−2	PROPN
ejde-320	337	19	and	and	CCONJ
ejde-320	337	20	so	so	ADV
ejde-320	337	21	σ′	σ′	PROPN
ejde-320	337	22	=	=	SYM
ejde-320	337	23	n/2	n/2	PRON
ejde-320	337	24	is	be	AUX
ejde-320	337	25	the	the	DET
ejde-320	337	26	dual	dual	ADJ
ejde-320	337	27	of	of	ADP
ejde-320	337	28	σ	σ	PROPN
ejde-320	337	29	.	.	PROPN
ejde-320	338	1	5.2	5.2	NUM
ejde-320	338	2	.	.	PUNCT
ejde-320	339	1	degenerate	degenerate	ADJ
ejde-320	339	2	counterexamples	counterexample	NOUN
ejde-320	339	3	.	.	PUNCT
ejde-320	340	1	the	the	DET
ejde-320	340	2	following	follow	VERB
ejde-320	340	3	two	two	NUM
ejde-320	340	4	examples	example	NOUN
ejde-320	340	5	are	be	AUX
ejde-320	340	6	based	base	VERB
ejde-320	340	7	on	on	ADP
ejde-320	340	8	the	the	DET
ejde-320	340	9	examples	example	NOUN
ejde-320	340	10	constructed	construct	VERB
ejde-320	340	11	in	in	ADP
ejde-320	340	12	[	[	X
ejde-320	340	13	6	6	NUM
ejde-320	340	14	,	,	PUNCT
ejde-320	340	15	section	section	NOUN
ejde-320	340	16	3	3	NUM
ejde-320	340	17	chapter	chapter	NOUN
ejde-320	340	18	9	9	NUM
ejde-320	340	19	]	]	PUNCT
ejde-320	340	20	.	.	PUNCT
ejde-320	341	1	let	let	VERB
ejde-320	341	2	l	l	NOUN
ejde-320	341	3	=	=	X
ejde-320	341	4	∇	∇	X
ejde-320	341	5	·	·	PUNCT
ejde-320	341	6	a∇	a∇	NOUN
ejde-320	341	7	with	with	ADP
ejde-320	341	8	a	a	DET
ejde-320	341	9	=	=	X
ejde-320	341	10	(	(	PUNCT
ejde-320	341	11	1	1	NUM
ejde-320	341	12	0	0	NUM
ejde-320	341	13	0	0	NUM
ejde-320	341	14	g(x)2	g(x)2	PROPN
ejde-320	341	15	)	)	PUNCT
ejde-320	341	16	,	,	PUNCT
ejde-320	341	17	where	where	SCONJ
ejde-320	341	18	g(0	g(0	NOUN
ejde-320	341	19	)	)	PUNCT
ejde-320	341	20	=	=	SYM
ejde-320	341	21	0	0	NUM
ejde-320	341	22	,	,	PUNCT
ejde-320	341	23	g	g	PROPN
ejde-320	341	24	is	be	AUX
ejde-320	341	25	positive	positive	ADJ
ejde-320	341	26	away	away	ADV
ejde-320	341	27	from	from	ADP
ejde-320	341	28	the	the	DET
ejde-320	341	29	origin	origin	NOUN
ejde-320	341	30	,	,	PUNCT
ejde-320	341	31	and	and	CCONJ
ejde-320	341	32	g	g	NOUN
ejde-320	341	33	=	=	NOUN
ejde-320	341	34	ψ′	ψ′	PROPN
ejde-320	341	35	where	where	SCONJ
ejde-320	341	36	ψ	ψ	NOUN
ejde-320	341	37	is	be	AUX
ejde-320	341	38	smooth	smooth	ADJ
ejde-320	341	39	,	,	PUNCT
ejde-320	341	40	even	even	ADV
ejde-320	341	41	,	,	PUNCT
ejde-320	341	42	strictly	strictly	ADV
ejde-320	341	43	convex	convex	VERB
ejde-320	341	44	on	on	ADP
ejde-320	341	45	r	r	NOUN
ejde-320	341	46	and	and	CCONJ
ejde-320	341	47	ψ(0	ψ(0	PROPN
ejde-320	341	48	)	)	PUNCT
ejde-320	341	49	=	=	SYM
ejde-320	342	1	0	0	X
ejde-320	342	2	.	.	PUNCT
ejde-320	343	1	moreover	moreover	ADV
ejde-320	343	2	,	,	PUNCT
ejde-320	343	3	we	we	PRON
ejde-320	343	4	will	will	AUX
ejde-320	343	5	assume	assume	VERB
ejde-320	343	6	that	that	SCONJ
ejde-320	343	7	g(x)/x→	g(x)/x→	NOUN
ejde-320	343	8	0	0	NUM
ejde-320	343	9	and	and	CCONJ
ejde-320	343	10	ψ(x)/x2	ψ(x)/x2	PROPN
ejde-320	343	11	→	→	SYM
ejde-320	343	12	0	0	PUNCT
ejde-320	343	13	as	as	ADP
ejde-320	343	14	x	x	X
ejde-320	343	15	→	→	SYM
ejde-320	343	16	0	0	NUM
ejde-320	343	17	.	.	PUNCT
ejde-320	344	1	since	since	SCONJ
ejde-320	344	2	the	the	DET
ejde-320	344	3	operator	operator	NOUN
ejde-320	344	4	l	l	NOUN
ejde-320	344	5	is	be	AUX
ejde-320	344	6	elliptic	elliptic	ADJ
ejde-320	344	7	away	away	ADV
ejde-320	344	8	from	from	ADP
ejde-320	344	9	the	the	DET
ejde-320	344	10	yaxis	yaxis	NOUN
ejde-320	344	11	,	,	PUNCT
ejde-320	344	12	and	and	CCONJ
ejde-320	344	13	translation	translation	NOUN
ejde-320	344	14	invariant	invariant	VERB
ejde-320	344	15	with	with	ADP
ejde-320	344	16	respect	respect	NOUN
ejde-320	344	17	to	to	ADP
ejde-320	344	18	the	the	DET
ejde-320	344	19	y	y	PROPN
ejde-320	344	20	variable	variable	NOUN
ejde-320	344	21	,	,	PUNCT
ejde-320	344	22	we	we	PRON
ejde-320	344	23	may	may	AUX
ejde-320	344	24	restrict	restrict	VERB
ejde-320	344	25	our	our	PRON
ejde-320	344	26	attention	attention	NOUN
ejde-320	344	27	to	to	ADP
ejde-320	344	28	the	the	DET
ejde-320	344	29	ball	ball	NOUN
ejde-320	344	30	b	b	PROPN
ejde-320	344	31	=	=	PUNCT
ejde-320	344	32	b(0	b(0	PROPN
ejde-320	344	33	,	,	PUNCT
ejde-320	344	34	ρ	ρ	NOUN
ejde-320	344	35	)	)	PUNCT
ejde-320	344	36	centered	center	VERB
ejde-320	344	37	at	at	ADP
ejde-320	344	38	the	the	DET
ejde-320	344	39	origin	origin	NOUN
ejde-320	344	40	,	,	PUNCT
ejde-320	344	41	with	with	ADP
ejde-320	344	42	radius	radius	NOUN
ejde-320	344	43	ρ	ρ	PROPN
ejde-320	344	44	sufficiently	sufficiently	ADV
ejde-320	344	45	small	small	ADJ
ejde-320	344	46	.	.	PUNCT
ejde-320	345	1	define	define	VERB
ejde-320	345	2	the	the	DET
ejde-320	345	3	function	function	NOUN
ejde-320	345	4	u	u	NOUN
ejde-320	345	5	by	by	ADP
ejde-320	345	6	u(x	u(x	NOUN
ejde-320	345	7	,	,	PUNCT
ejde-320	345	8	y	y	NOUN
ejde-320	345	9	)	)	PUNCT
ejde-320	345	10	:	:	PUNCT
ejde-320	346	1	=	=	SYM
ejde-320	346	2	χ	χ	X
ejde-320	346	3	(	(	PUNCT
ejde-320	346	4	y	y	PROPN
ejde-320	346	5	ψ(x	ψ(x	PROPN
ejde-320	346	6	)	)	PUNCT
ejde-320	346	7	)	)	PUNCT
ejde-320	347	1	ln	ln	ADV
ejde-320	347	2	1	1	NUM
ejde-320	347	3	x	x	SYM
ejde-320	347	4	,	,	PUNCT
ejde-320	347	5	(	(	PUNCT
ejde-320	347	6	5.2	5.2	NUM
ejde-320	347	7	)	)	PUNCT
ejde-320	347	8	where	where	SCONJ
ejde-320	347	9	χ(s	χ(s	NOUN
ejde-320	347	10	)	)	PUNCT
ejde-320	347	11	is	be	AUX
ejde-320	347	12	a	a	DET
ejde-320	347	13	smooth	smooth	ADJ
ejde-320	347	14	odd	odd	ADJ
ejde-320	347	15	function	function	NOUN
ejde-320	347	16	on	on	ADP
ejde-320	347	17	r	r	NOUN
ejde-320	347	18	such	such	ADJ
ejde-320	347	19	that	that	DET
ejde-320	347	20	χ(s	χ(s	NOUN
ejde-320	347	21	)	)	PUNCT
ejde-320	347	22	=	=	SYM
ejde-320	347	23	1	1	NUM
ejde-320	347	24	for	for	ADP
ejde-320	347	25	s	s	NOUN
ejde-320	347	26	∈	∈	PROPN
ejde-320	348	1	[	[	X
ejde-320	348	2	−1	−1	NOUN
ejde-320	348	3	,	,	PUNCT
ejde-320	348	4	1	1	NUM
ejde-320	348	5	]	]	PUNCT
ejde-320	348	6	and	and	CCONJ
ejde-320	348	7	χ(s	χ(s	NOUN
ejde-320	348	8	)	)	PUNCT
ejde-320	349	1	=	=	SYM
ejde-320	349	2	0	0	NUM
ejde-320	350	1	for	for	ADP
ejde-320	350	2	s	s	NOUN
ejde-320	350	3	∈	∈	PROPN
ejde-320	350	4	r	r	NOUN
ejde-320	350	5	r	r	NOUN
ejde-320	350	6	[	[	X
ejde-320	350	7	−2	−2	X
ejde-320	350	8	,	,	PUNCT
ejde-320	350	9	2	2	NUM
ejde-320	350	10	]	]	PUNCT
ejde-320	350	11	.	.	PUNCT
ejde-320	351	1	first	first	ADV
ejde-320	351	2	note	note	VERB
ejde-320	351	3	that	that	SCONJ
ejde-320	351	4	the	the	DET
ejde-320	351	5	function	function	NOUN
ejde-320	351	6	u	u	NOUN
ejde-320	351	7	is	be	AUX
ejde-320	351	8	supported	support	VERB
ejde-320	351	9	in	in	ADP
ejde-320	351	10	the	the	DET
ejde-320	351	11	narrow	narrow	ADJ
ejde-320	351	12	region	region	NOUN
ejde-320	351	13	along	along	ADP
ejde-320	351	14	the	the	DET
ejde-320	351	15	x	x	NOUN
ejde-320	351	16	-	-	NOUN
ejde-320	351	17	axis	axis	ADJ
ejde-320	351	18	,	,	PUNCT
ejde-320	351	19	where	where	SCONJ
ejde-320	351	20	|y|	|y|	VERB
ejde-320	351	21	≤	≤	ADJ
ejde-320	351	22	2ψ(x	2ψ(x	NUM
ejde-320	351	23	)	)	PUNCT
ejde-320	351	24	.	.	PUNCT
ejde-320	352	1	next	next	ADV
ejde-320	352	2	we	we	PRON
ejde-320	352	3	calculate	calculate	VERB
ejde-320	352	4	uy	uy	PROPN
ejde-320	352	5	=	=	SYM
ejde-320	352	6	χ′	χ′	PROPN
ejde-320	352	7	(	(	PUNCT
ejde-320	352	8	y	y	PROPN
ejde-320	352	9	ψ(x	ψ(x	PROPN
ejde-320	352	10	)	)	PUNCT
ejde-320	352	11	)	)	PUNCT
ejde-320	352	12	1	1	NUM
ejde-320	352	13	ψ(x	ψ(x	NOUN
ejde-320	352	14	)	)	PUNCT
ejde-320	352	15	ln	ln	NOUN
ejde-320	353	1	1	1	NUM
ejde-320	353	2	x	x	NOUN
ejde-320	353	3	,	,	PUNCT
ejde-320	353	4	uyy	uyy	PROPN
ejde-320	353	5	=	=	PUNCT
ejde-320	353	6	χ′′	χ′′	PROPN
ejde-320	353	7	(	(	PUNCT
ejde-320	353	8	y	y	PROPN
ejde-320	353	9	ψ(x	ψ(x	PROPN
ejde-320	353	10	)	)	PUNCT
ejde-320	353	11	)	)	PUNCT
ejde-320	354	1	1	1	NUM
ejde-320	354	2	ψ(x)2	ψ(x)2	NOUN
ejde-320	354	3	ln	ln	ADJ
ejde-320	354	4	1	1	NUM
ejde-320	354	5	x	x	NOUN
ejde-320	354	6	,	,	PUNCT
ejde-320	354	7	ux	ux	PROPN
ejde-320	354	8	=	=	SYM
ejde-320	354	9	χ′	χ′	PROPN
ejde-320	354	10	(	(	PUNCT
ejde-320	354	11	y	y	PROPN
ejde-320	354	12	ψ(x	ψ(x	PROPN
ejde-320	354	13	)	)	PUNCT
ejde-320	354	14	)	)	PUNCT
ejde-320	354	15	(	(	PUNCT
ejde-320	354	16	−yψ′(x	−yψ′(x	X
ejde-320	354	17	)	)	PUNCT
ejde-320	354	18	ψ(x)2	ψ(x)2	PROPN
ejde-320	354	19	)	)	PUNCT
ejde-320	355	1	ln	ln	ADV
ejde-320	356	1	1	1	NUM
ejde-320	356	2	x	x	SYM
ejde-320	356	3	−	−	PROPN
ejde-320	356	4	1	1	NUM
ejde-320	356	5	x	x	SYM
ejde-320	356	6	χ	χ	X
ejde-320	356	7	(	(	PUNCT
ejde-320	356	8	y	y	PROPN
ejde-320	356	9	ψ(x	ψ(x	PROPN
ejde-320	356	10	)	)	PUNCT
ejde-320	356	11	)	)	PUNCT
ejde-320	356	12	,	,	PUNCT
ejde-320	356	13	uxx	uxx	X
ejde-320	356	14	=	=	PUNCT
ejde-320	357	1	χ′′	χ′′	PROPN
ejde-320	357	2	(	(	PUNCT
ejde-320	357	3	y	y	PROPN
ejde-320	357	4	ψ(x	ψ(x	PROPN
ejde-320	357	5	)	)	PUNCT
ejde-320	357	6	)	)	PUNCT
ejde-320	357	7	(	(	PUNCT
ejde-320	357	8	yψ′(x	yψ′(x	PROPN
ejde-320	357	9	)	)	PUNCT
ejde-320	357	10	ψ(x)2	ψ(x)2	PROPN
ejde-320	357	11	)	)	PUNCT
ejde-320	357	12	2	2	NUM
ejde-320	357	13	ln	ln	NOUN
ejde-320	357	14	1	1	NUM
ejde-320	357	15	x	x	SYM
ejde-320	357	16	+	+	CCONJ
ejde-320	357	17	χ′	χ′	PROPN
ejde-320	357	18	(	(	PUNCT
ejde-320	357	19	y	y	PROPN
ejde-320	357	20	ψ(x	ψ(x	PROPN
ejde-320	357	21	)	)	PUNCT
ejde-320	357	22	)	)	PUNCT
ejde-320	358	1	(	(	PUNCT
ejde-320	358	2	2yψ′(x)2	2yψ′(x)2	NUM
ejde-320	358	3	ψ(x)3	ψ(x)3	NOUN
ejde-320	358	4	−	−	NOUN
ejde-320	358	5	yψ′′(x	yψ′′(x	NOUN
ejde-320	358	6	)	)	PUNCT
ejde-320	358	7	ψ(x)2	ψ(x)2	PROPN
ejde-320	358	8	)	)	PUNCT
ejde-320	359	1	ln	ln	ADV
ejde-320	360	1	1	1	NUM
ejde-320	360	2	x	x	SYM
ejde-320	360	3	+	+	NUM
ejde-320	360	4	2	2	NUM
ejde-320	360	5	x	x	SYM
ejde-320	360	6	χ′	χ′	PROPN
ejde-320	360	7	(	(	PUNCT
ejde-320	360	8	y	y	PROPN
ejde-320	360	9	ψ(x	ψ(x	PROPN
ejde-320	360	10	)	)	PUNCT
ejde-320	360	11	)	)	PUNCT
ejde-320	360	12	(	(	PUNCT
ejde-320	360	13	yψ′(x	yψ′(x	PROPN
ejde-320	360	14	)	)	PUNCT
ejde-320	360	15	ψ(x)2	ψ(x)2	PROPN
ejde-320	360	16	)	)	PUNCT
ejde-320	361	1	+	+	CCONJ
ejde-320	361	2	1	1	NUM
ejde-320	361	3	x2	x2	SYM
ejde-320	361	4	χ	χ	X
ejde-320	361	5	(	(	PUNCT
ejde-320	361	6	y	y	PROPN
ejde-320	361	7	ψ(x	ψ(x	PROPN
ejde-320	361	8	)	)	PUNCT
ejde-320	361	9	)	)	PUNCT
ejde-320	361	10	.	.	PUNCT
ejde-320	362	1	to	to	PART
ejde-320	362	2	make	make	VERB
ejde-320	362	3	further	further	ADJ
ejde-320	362	4	estimates	estimate	NOUN
ejde-320	362	5	,	,	PUNCT
ejde-320	362	6	we	we	PRON
ejde-320	362	7	will	will	AUX
ejde-320	362	8	write	write	VERB
ejde-320	362	9	a	a	DET
ejde-320	362	10	≈	≈	PROPN
ejde-320	362	11	b	b	PROPN
ejde-320	362	12	for	for	ADP
ejde-320	362	13	any	any	DET
ejde-320	362	14	two	two	NUM
ejde-320	362	15	given	give	VERB
ejde-320	362	16	functions	function	NOUN
ejde-320	362	17	a	a	PRON
ejde-320	362	18	and	and	CCONJ
ejde-320	362	19	b	b	NOUN
ejde-320	362	20	to	to	PART
ejde-320	362	21	imply	imply	VERB
ejde-320	362	22	that	that	SCONJ
ejde-320	362	23	the	the	DET
ejde-320	362	24	two	two	NUM
ejde-320	362	25	inequalities	inequality	NOUN
ejde-320	362	26	c1a	c1a	VERB
ejde-320	362	27	≤	≤	NUM
ejde-320	362	28	b	b	NOUN
ejde-320	362	29	≤	≤	NOUN
ejde-320	362	30	c2a	c2a	PROPN
ejde-320	362	31	hold	hold	VERB
ejde-320	362	32	for	for	ADP
ejde-320	362	33	all	all	DET
ejde-320	362	34	elements	element	NOUN
ejde-320	362	35	of	of	ADP
ejde-320	362	36	the	the	DET
ejde-320	362	37	domain	domain	NOUN
ejde-320	362	38	of	of	ADP
ejde-320	362	39	a	a	PRON
ejde-320	362	40	and	and	CCONJ
ejde-320	362	41	b	b	NOUN
ejde-320	362	42	and	and	CCONJ
ejde-320	362	43	for	for	ADP
ejde-320	362	44	some	some	DET
ejde-320	362	45	constants	constant	NOUN
ejde-320	362	46	c1	c1	NOUN
ejde-320	362	47	,	,	PUNCT
ejde-320	362	48	c2	c2	PROPN
ejde-320	362	49	>	>	X
ejde-320	362	50	0	0	X
ejde-320	362	51	.	.	PUNCT
ejde-320	363	1	define	define	VERB
ejde-320	363	2	f(x	f(x	PROPN
ejde-320	363	3	,	,	PUNCT
ejde-320	363	4	y	y	PROPN
ejde-320	363	5	)	)	PUNCT
ejde-320	363	6	:	:	PUNCT
ejde-320	364	1	=	=	SYM
ejde-320	364	2	lu	lu	NOUN
ejde-320	364	3	=	=	SYM
ejde-320	364	4	uxx	uxx	PROPN
ejde-320	364	5	+	+	CCONJ
ejde-320	364	6	g(x)2uyy	g(x)2uyy	PROPN
ejde-320	364	7	,	,	PUNCT
ejde-320	364	8	using	use	VERB
ejde-320	364	9	the	the	DET
ejde-320	364	10	properties	property	NOUN
ejde-320	364	11	of	of	ADP
ejde-320	364	12	g	g	NOUN
ejde-320	364	13	,	,	PUNCT
ejde-320	364	14	ψ	ψ	ADP
ejde-320	364	15	,	,	PUNCT
ejde-320	364	16	and	and	CCONJ
ejde-320	364	17	χ	χ	X
ejde-320	364	18	,	,	PUNCT
ejde-320	364	19	we	we	PRON
ejde-320	364	20	then	then	ADV
ejde-320	364	21	have	have	VERB
ejde-320	364	22	|f(x	|f(x	PROPN
ejde-320	364	23	,	,	PUNCT
ejde-320	364	24	y)|	y)|	PROPN
ejde-320	364	25	≈	≈	PROPN
ejde-320	364	26	1	1	NUM
ejde-320	365	1	x2	x2	NOUN
ejde-320	366	1	+	+	CCONJ
ejde-320	366	2	ln	ln	ADJ
ejde-320	366	3	1	1	NUM
ejde-320	366	4	x	x	SYM
ejde-320	366	5	(	(	PUNCT
ejde-320	366	6	frac|ψ′′(x)|ψ(x	frac|ψ′′(x)|ψ(x	PROPN
ejde-320	366	7	)	)	PUNCT
ejde-320	366	8	)	)	PUNCT
ejde-320	367	1	+	+	CCONJ
ejde-320	367	2	ln	ln	ADJ
ejde-320	367	3	1	1	NUM
ejde-320	367	4	x	x	SYM
ejde-320	367	5	(	(	PUNCT
ejde-320	367	6	ψ′(x	ψ′(x	NOUN
ejde-320	367	7	)	)	PUNCT
ejde-320	367	8	ψ(x	ψ(x	NOUN
ejde-320	367	9	)	)	PUNCT
ejde-320	367	10	)	)	PUNCT
ejde-320	368	1	2	2	NUM
ejde-320	369	1	+	+	CCONJ
ejde-320	369	2	1	1	NUM
ejde-320	369	3	x	x	SYM
ejde-320	369	4	(	(	PUNCT
ejde-320	369	5	ψ′(x	ψ′(x	NOUN
ejde-320	369	6	)	)	PUNCT
ejde-320	369	7	ψ(x	ψ(x	NOUN
ejde-320	369	8	)	)	PUNCT
ejde-320	369	9	)	)	PUNCT
ejde-320	369	10	,	,	PUNCT
ejde-320	369	11	(	(	PUNCT
ejde-320	369	12	5.3	5.3	NUM
ejde-320	369	13	)	)	PUNCT
ejde-320	369	14	and	and	CCONJ
ejde-320	369	15	f	f	PROPN
ejde-320	369	16	is	be	AUX
ejde-320	369	17	supported	support	VERB
ejde-320	369	18	in	in	ADP
ejde-320	369	19	|y|	|y|	PROPN
ejde-320	369	20	≤	≤	ADJ
ejde-320	369	21	2ψ(x	2ψ(x	NUM
ejde-320	369	22	)	)	PUNCT
ejde-320	369	23	.	.	PUNCT
ejde-320	370	1	16	16	NUM
ejde-320	370	2	u.	u.	PROPN
ejde-320	370	3	hafeez	hafeez	PROPN
ejde-320	370	4	,	,	PUNCT
ejde-320	370	5	t.	t.	PROPN
ejde-320	370	6	lavier	lavier	PROPN
ejde-320	370	7	,	,	PUNCT
ejde-320	370	8	l.	l.	PROPN
ejde-320	370	9	williams	williams	PROPN
ejde-320	370	10	,	,	PUNCT
ejde-320	370	11	l.	l.	PROPN
ejde-320	370	12	korobenko	korobenko	PROPN
ejde-320	370	13	ejde-2021/82	ejde-2021/82	ADJ
ejde-320	370	14	finite	finite	ADJ
ejde-320	370	15	vanishing	vanish	VERB
ejde-320	370	16	.	.	PUNCT
ejde-320	371	1	fix	fix	VERB
ejde-320	371	2	m	m	PROPN
ejde-320	371	3	≥	≥	NOUN
ejde-320	371	4	1	1	NUM
ejde-320	371	5	and	and	CCONJ
ejde-320	371	6	let	let	VERB
ejde-320	371	7	ψ(x	ψ(x	NOUN
ejde-320	371	8	)	)	PUNCT
ejde-320	371	9	=	=	SYM
ejde-320	372	1	1	1	NUM
ejde-320	372	2	m+	m+	NUM
ejde-320	372	3	1	1	NUM
ejde-320	372	4	xm+1	xm+1	PROPN
ejde-320	372	5	.	.	PUNCT
ejde-320	372	6	differentiating	differentiate	VERB
ejde-320	372	7	gives	give	VERB
ejde-320	372	8	g(x	g(x	NOUN
ejde-320	372	9	)	)	PUNCT
ejde-320	372	10	=	=	PUNCT
ejde-320	373	1	ψ′(x	ψ′(x	X
ejde-320	373	2	)	)	PUNCT
ejde-320	373	3	=	=	SYM
ejde-320	373	4	xm	xm	PROPN
ejde-320	373	5	,	,	PUNCT
ejde-320	373	6	ψ′′(x	ψ′′(x	NOUN
ejde-320	373	7	)	)	PUNCT
ejde-320	373	8	=	=	SYM
ejde-320	374	1	mxm−1	mxm−1	PROPN
ejde-320	374	2	,	,	PUNCT
ejde-320	374	3	which	which	PRON
ejde-320	374	4	combined	combine	VERB
ejde-320	374	5	with	with	ADP
ejde-320	374	6	(	(	PUNCT
ejde-320	374	7	5.3	5.3	NUM
ejde-320	374	8	)	)	PUNCT
ejde-320	374	9	implies	imply	VERB
ejde-320	374	10	|f(x	|f(x	PROPN
ejde-320	374	11	,	,	PUNCT
ejde-320	374	12	y)|	y)|	PROPN
ejde-320	374	13	≈	≈	PROPN
ejde-320	375	1	1	1	NUM
ejde-320	376	1	x2	x2	NOUN
ejde-320	376	2	ln	ln	ADJ
ejde-320	376	3	1	1	NUM
ejde-320	376	4	x	x	X
ejde-320	376	5	.	.	PUNCT
ejde-320	377	1	recall	recall	NOUN
ejde-320	377	2	that	that	SCONJ
ejde-320	377	3	f	f	PROPN
ejde-320	377	4	is	be	AUX
ejde-320	377	5	supported	support	VERB
ejde-320	377	6	where	where	SCONJ
ejde-320	377	7	|y|	|y|	ADJ
ejde-320	377	8	≤	≤	ADJ
ejde-320	377	9	2ψ(x	2ψ(x	NUM
ejde-320	377	10	)	)	PUNCT
ejde-320	377	11	,	,	PUNCT
ejde-320	377	12	so	so	SCONJ
ejde-320	377	13	we	we	PRON
ejde-320	377	14	can	can	AUX
ejde-320	377	15	estimate	estimate	VERB
ejde-320	377	16	the	the	DET
ejde-320	377	17	lq	lq	NOUN
ejde-320	377	18	norm	norm	NOUN
ejde-320	377	19	in	in	ADP
ejde-320	377	20	the	the	DET
ejde-320	377	21	ball	ball	NOUN
ejde-320	377	22	b∫	b∫	PROPN
ejde-320	377	23	b	b	PROPN
ejde-320	377	24	|f(x	|f(x	PROPN
ejde-320	377	25	,	,	PUNCT
ejde-320	377	26	y)|q	y)|q	PROPN
ejde-320	377	27	dx	dx	PROPN
ejde-320	377	28	dy	dy	PROPN
ejde-320	377	29	.	.	PUNCT
ejde-320	378	1	∫	∫	PROPN
ejde-320	378	2	ρ	ρ	PROPN
ejde-320	378	3	0	0	NUM
ejde-320	378	4	1	1	NUM
ejde-320	378	5	x2q	x2q	NUM
ejde-320	378	6	(	(	PUNCT
ejde-320	378	7	ln	ln	NOUN
ejde-320	378	8	1	1	NUM
ejde-320	378	9	x	x	X
ejde-320	378	10	)	)	PUNCT
ejde-320	378	11	q	q	NOUN
ejde-320	378	12	ψ(x)dx	ψ(x)dx	NUM
ejde-320	379	1	≈	≈	PROPN
ejde-320	379	2	∫	∫	PROPN
ejde-320	379	3	ρ	ρ	PROPN
ejde-320	379	4	0	0	NUM
ejde-320	379	5	1	1	NUM
ejde-320	379	6	x2q−m−1	x2q−m−1	PROPN
ejde-320	379	7	(	(	PUNCT
ejde-320	379	8	ln	ln	NOUN
ejde-320	379	9	1	1	NUM
ejde-320	379	10	x	x	X
ejde-320	379	11	)	)	PUNCT
ejde-320	379	12	q	q	PROPN
ejde-320	379	13	dx	dx	PROPN
ejde-320	379	14	.	.	PUNCT
ejde-320	380	1	the	the	DET
ejde-320	380	2	right	right	ADJ
ejde-320	380	3	hand	hand	NOUN
ejde-320	380	4	side	side	NOUN
ejde-320	380	5	is	be	AUX
ejde-320	380	6	finite	finite	ADJ
ejde-320	380	7	if	if	SCONJ
ejde-320	380	8	and	and	CCONJ
ejde-320	380	9	only	only	ADV
ejde-320	380	10	if	if	SCONJ
ejde-320	380	11	q	q	X
ejde-320	380	12	<	<	X
ejde-320	380	13	m+2	m+2	X
ejde-320	380	14	2	2	NUM
ejde-320	380	15	.	.	PUNCT
ejde-320	381	1	we	we	PRON
ejde-320	381	2	now	now	ADV
ejde-320	381	3	verify	verify	VERB
ejde-320	381	4	that	that	SCONJ
ejde-320	381	5	the	the	DET
ejde-320	381	6	function	function	NOUN
ejde-320	381	7	u	u	NOUN
ejde-320	381	8	belongs	belong	VERB
ejde-320	381	9	to	to	ADP
ejde-320	381	10	the	the	DET
ejde-320	381	11	sobolev	sobolev	NOUN
ejde-320	381	12	space	space	NOUN
ejde-320	381	13	w	w	PROPN
ejde-320	381	14	1,2	1,2	NUM
ejde-320	381	15	a	a	DET
ejde-320	381	16	(	(	PUNCT
ejde-320	381	17	b	b	NOUN
ejde-320	381	18	)	)	PUNCT
ejde-320	381	19	,	,	PUNCT
ejde-320	381	20	i.e.	i.e.	X
ejde-320	381	21	u	u	X
ejde-320	381	22	∈	∈	PROPN
ejde-320	381	23	l2(b	l2(b	NOUN
ejde-320	381	24	)	)	PUNCT
ejde-320	381	25	and∫	and∫	NOUN
ejde-320	381	26	b	b	PROPN
ejde-320	381	27	|∇au|2	|∇au|2	NOUN
ejde-320	381	28	dx	dx	PROPN
ejde-320	381	29	dy	dy	X
ejde-320	381	30	=	=	SYM
ejde-320	381	31	∫	∫	PROPN
ejde-320	381	32	b	b	PROPN
ejde-320	381	33	(	(	PUNCT
ejde-320	381	34	|ux|2	|ux|2	PUNCT
ejde-320	381	35	+	+	X
ejde-320	381	36	g(x)2|uy|2	g(x)2|uy|2	NUM
ejde-320	381	37	)	)	PUNCT
ejde-320	381	38	dx	dx	PROPN
ejde-320	381	39	dy	dy	NOUN
ejde-320	381	40	<	<	X
ejde-320	381	41	∞.	∞.	PROPN
ejde-320	381	42	using	use	VERB
ejde-320	381	43	the	the	DET
ejde-320	381	44	expressions	expression	NOUN
ejde-320	381	45	for	for	ADP
ejde-320	381	46	ux	ux	PROPN
ejde-320	381	47	and	and	CCONJ
ejde-320	381	48	uy	uy	PROPN
ejde-320	381	49	and	and	CCONJ
ejde-320	381	50	the	the	DET
ejde-320	381	51	estimates	estimate	NOUN
ejde-320	381	52	for	for	ADP
ejde-320	381	53	ψ′	ψ′	NUM
ejde-320	381	54	and	and	CCONJ
ejde-320	381	55	ψ′′	ψ′′	NOUN
ejde-320	381	56	we	we	PRON
ejde-320	381	57	have	have	VERB
ejde-320	381	58	for	for	ADP
ejde-320	381	59	|y|	|y|	PROPN
ejde-320	381	60	≤	≤	ADJ
ejde-320	381	61	2ψ(x	2ψ(x	NUM
ejde-320	381	62	)	)	PUNCT
ejde-320	381	63	,	,	PUNCT
ejde-320	381	64	|ux|2	|ux|2	PUNCT
ejde-320	382	1	+	+	CCONJ
ejde-320	382	2	g(x)2|uy|2	g(x)2|uy|2	ADJ
ejde-320	382	3	≈	≈	PROPN
ejde-320	382	4	1	1	NUM
ejde-320	382	5	x2	x2	NOUN
ejde-320	382	6	(	(	PUNCT
ejde-320	382	7	ln	ln	NOUN
ejde-320	382	8	1	1	NUM
ejde-320	382	9	x	x	SYM
ejde-320	382	10	)	)	PUNCT
ejde-320	382	11	2	2	NUM
ejde-320	382	12	+	+	NUM
ejde-320	382	13	g(x)2	g(x)2	PROPN
ejde-320	382	14	ψ(x)2	ψ(x)2	VERB
ejde-320	382	15	(	(	PUNCT
ejde-320	382	16	ln	ln	NOUN
ejde-320	382	17	1	1	NUM
ejde-320	382	18	x	x	SYM
ejde-320	382	19	)	)	PUNCT
ejde-320	382	20	2	2	NUM
ejde-320	382	21	≈	≈	PROPN
ejde-320	382	22	1	1	NUM
ejde-320	382	23	x2	x2	NOUN
ejde-320	382	24	(	(	PUNCT
ejde-320	382	25	ln	ln	NOUN
ejde-320	382	26	1	1	NUM
ejde-320	382	27	x	x	SYM
ejde-320	382	28	)	)	PUNCT
ejde-320	382	29	2	2	NUM
ejde-320	382	30	,	,	PUNCT
ejde-320	382	31	where	where	SCONJ
ejde-320	382	32	for	for	ADP
ejde-320	382	33	the	the	DET
ejde-320	382	34	last	last	ADJ
ejde-320	382	35	equality	equality	NOUN
ejde-320	382	36	we	we	PRON
ejde-320	382	37	used	use	VERB
ejde-320	382	38	g	g	PROPN
ejde-320	382	39	=	=	SYM
ejde-320	382	40	ψ′.	ψ′.	PROPN
ejde-320	382	41	altogether	altogether	ADV
ejde-320	382	42	we	we	PRON
ejde-320	382	43	obtain∫	obtain∫	VERB
ejde-320	383	1	b	b	NUM
ejde-320	383	2	|∇au|2	|∇au|2	X
ejde-320	383	3	dx	dx	PROPN
ejde-320	383	4	dy	dy	PROPN
ejde-320	383	5	≈	≈	PROPN
ejde-320	383	6	∫	∫	PROPN
ejde-320	383	7	ρ	ρ	NOUN
ejde-320	383	8	0	0	NUM
ejde-320	383	9	1	1	NUM
ejde-320	383	10	x2	x2	NOUN
ejde-320	383	11	(	(	PUNCT
ejde-320	383	12	ln	ln	NOUN
ejde-320	383	13	1	1	NUM
ejde-320	383	14	x	x	SYM
ejde-320	383	15	)	)	PUNCT
ejde-320	383	16	2	2	NUM
ejde-320	383	17	ψ(x)dx	ψ(x)dx	NUM
ejde-320	383	18	≈	≈	PROPN
ejde-320	383	19	∫	∫	PROPN
ejde-320	383	20	ρ	ρ	NOUN
ejde-320	383	21	0	0	NUM
ejde-320	383	22	1	1	NUM
ejde-320	383	23	x1−m	x1−m	PROPN
ejde-320	383	24	(	(	PUNCT
ejde-320	383	25	ln	ln	NOUN
ejde-320	383	26	1	1	NUM
ejde-320	383	27	x	x	SYM
ejde-320	383	28	)	)	PUNCT
ejde-320	383	29	2	2	NUM
ejde-320	383	30	dx	dx	PROPN
ejde-320	383	31	,	,	PUNCT
ejde-320	383	32	which	which	PRON
ejde-320	383	33	is	be	AUX
ejde-320	383	34	finite	finite	ADJ
ejde-320	383	35	for	for	ADP
ejde-320	383	36	all	all	DET
ejde-320	383	37	m	m	NOUN
ejde-320	383	38	>	>	X
ejde-320	383	39	0	0	X
ejde-320	383	40	.	.	PUNCT
ejde-320	384	1	it	it	PRON
ejde-320	384	2	is	be	AUX
ejde-320	384	3	easy	easy	ADJ
ejde-320	384	4	to	to	PART
ejde-320	384	5	see	see	VERB
ejde-320	384	6	that	that	SCONJ
ejde-320	384	7	u	u	PROPN
ejde-320	384	8	∈	∈	PROPN
ejde-320	384	9	l2(b	l2(b	NOUN
ejde-320	384	10	)	)	PUNCT
ejde-320	384	11	,	,	PUNCT
ejde-320	384	12	so	so	SCONJ
ejde-320	384	13	that	that	SCONJ
ejde-320	384	14	u	u	PRON
ejde-320	384	15	∈w	∈w	VERB
ejde-320	384	16	1,2	1,2	NUM
ejde-320	384	17	a	a	DET
ejde-320	384	18	(	(	PUNCT
ejde-320	384	19	b	b	NOUN
ejde-320	384	20	)	)	PUNCT
ejde-320	384	21	.	.	PUNCT
ejde-320	385	1	moreover	moreover	ADV
ejde-320	385	2	,	,	PUNCT
ejde-320	385	3	we	we	PRON
ejde-320	385	4	have	have	VERB
ejde-320	385	5	u(x	u(x	NOUN
ejde-320	385	6	,	,	PUNCT
ejde-320	385	7	ψ(x	ψ(x	NOUN
ejde-320	385	8	)	)	PUNCT
ejde-320	385	9	)	)	PUNCT
ejde-320	386	1	=	=	PUNCT
ejde-320	386	2	ln(1	ln(1	NOUN
ejde-320	386	3	/	/	SYM
ejde-320	386	4	x	x	NOUN
ejde-320	386	5	)	)	PUNCT
ejde-320	386	6	,	,	PUNCT
ejde-320	386	7	so	so	CCONJ
ejde-320	386	8	u	u	NOUN
ejde-320	386	9	is	be	AUX
ejde-320	386	10	unbounded	unbounded	ADJ
ejde-320	386	11	at	at	ADP
ejde-320	386	12	the	the	DET
ejde-320	386	13	origin	origin	NOUN
ejde-320	386	14	.	.	PUNCT
ejde-320	387	1	thus	thus	ADV
ejde-320	387	2	,	,	PUNCT
ejde-320	387	3	for	for	ADP
ejde-320	387	4	any	any	DET
ejde-320	387	5	q	q	NOUN
ejde-320	387	6	<	<	X
ejde-320	387	7	m+2	m+2	SYM
ejde-320	387	8	2	2	NUM
ejde-320	387	9	we	we	PRON
ejde-320	387	10	obtain	obtain	VERB
ejde-320	387	11	that	that	SCONJ
ejde-320	387	12	u	u	NOUN
ejde-320	387	13	is	be	AUX
ejde-320	387	14	an	an	DET
ejde-320	387	15	unbounded	unbounded	ADJ
ejde-320	387	16	weak	weak	ADJ
ejde-320	387	17	solution	solution	NOUN
ejde-320	387	18	to	to	ADP
ejde-320	387	19	lu	lu	PROPN
ejde-320	387	20	=	=	SYM
ejde-320	387	21	f	f	PROPN
ejde-320	387	22	with	with	ADP
ejde-320	387	23	f	f	PROPN
ejde-320	387	24	∈	∈	PROPN
ejde-320	387	25	lq(b	lq(b	PRON
ejde-320	387	26	)	)	PUNCT
ejde-320	387	27	.	.	PUNCT
ejde-320	388	1	on	on	ADP
ejde-320	388	2	the	the	DET
ejde-320	388	3	other	other	ADJ
ejde-320	388	4	hand	hand	NOUN
ejde-320	388	5	[	[	X
ejde-320	388	6	11	11	NUM
ejde-320	388	7	,	,	PUNCT
ejde-320	388	8	proposition	proposition	NOUN
ejde-320	388	9	74	74	NUM
ejde-320	388	10	]	]	PUNCT
ejde-320	388	11	implies	imply	VERB
ejde-320	388	12	the	the	DET
ejde-320	388	13	sobolev	sobolev	NOUN
ejde-320	388	14	inequality	inequality	NOUN
ejde-320	388	15	(	(	PUNCT
ejde-320	388	16	1	1	NUM
ejde-320	388	17	|b|	|b|	PROPN
ejde-320	388	18	∫	∫	PROPN
ejde-320	389	1	b	b	PROPN
ejde-320	389	2	|w|2σ	|w|2σ	NOUN
ejde-320	389	3	)	)	PUNCT
ejde-320	389	4	1	1	NUM
ejde-320	389	5	2σ	2σ	NOUN
ejde-320	389	6	≤	≤	NUM
ejde-320	389	7	cr	cr	NOUN
ejde-320	390	1	(	(	PUNCT
ejde-320	390	2	1	1	NUM
ejde-320	390	3	|b|	|b|	PROPN
ejde-320	390	4	∫	∫	PROPN
ejde-320	390	5	b	b	X
ejde-320	390	6	|∇aw|2	|∇aw|2	NOUN
ejde-320	390	7	)	)	PUNCT
ejde-320	390	8	1/2	1/2	NUM
ejde-320	391	1	+	+	NUM
ejde-320	391	2	c	c	NOUN
ejde-320	391	3	(	(	PUNCT
ejde-320	391	4	1	1	NUM
ejde-320	391	5	|b|	|b|	PROPN
ejde-320	391	6	∫	∫	PROPN
ejde-320	391	7	b	b	PROPN
ejde-320	391	8	|w|2	|w|2	PROPN
ejde-320	391	9	)	)	PUNCT
ejde-320	391	10	1/2	1/2	NUM
ejde-320	391	11	(	(	PUNCT
ejde-320	391	12	5.4	5.4	NUM
ejde-320	391	13	)	)	PUNCT
ejde-320	391	14	for	for	ADP
ejde-320	391	15	all	all	DET
ejde-320	391	16	w	w	NOUN
ejde-320	391	17	∈w	∈w	PROPN
ejde-320	391	18	1,2	1,2	NUM
ejde-320	391	19	0	0	NUM
ejde-320	391	20	(	(	PUNCT
ejde-320	391	21	b	b	NOUN
ejde-320	391	22	)	)	PUNCT
ejde-320	391	23	,	,	PUNCT
ejde-320	391	24	where	where	SCONJ
ejde-320	391	25	σ′	σ′	VERB
ejde-320	391	26	=	=	SYM
ejde-320	391	27	(	(	PUNCT
ejde-320	391	28	m+	m+	NUM
ejde-320	391	29	2)/2	2)/2	NUM
ejde-320	391	30	.	.	PUNCT
ejde-320	392	1	[	[	X
ejde-320	392	2	11	11	NUM
ejde-320	392	3	,	,	PUNCT
ejde-320	392	4	theorem	theorem	VERB
ejde-320	392	5	8	8	NUM
ejde-320	392	6	]	]	PUNCT
ejde-320	392	7	then	then	ADV
ejde-320	392	8	implies	imply	VERB
ejde-320	392	9	that	that	SCONJ
ejde-320	392	10	if	if	SCONJ
ejde-320	392	11	u	u	NOUN
ejde-320	392	12	is	be	AUX
ejde-320	392	13	a	a	DET
ejde-320	392	14	weak	weak	ADJ
ejde-320	392	15	solution	solution	NOUN
ejde-320	392	16	to	to	ADP
ejde-320	392	17	lu	lu	PROPN
ejde-320	392	18	=	=	SYM
ejde-320	392	19	f	f	PROPN
ejde-320	392	20	and	and	CCONJ
ejde-320	392	21	f	f	PROPN
ejde-320	392	22	∈	∈	PROPN
ejde-320	392	23	lq(b	lq(b	PRON
ejde-320	392	24	)	)	PUNCT
ejde-320	392	25	with	with	ADP
ejde-320	392	26	q	q	PROPN
ejde-320	392	27	>	>	X
ejde-320	392	28	m+2	m+2	NOUN
ejde-320	392	29	2	2	NUM
ejde-320	392	30	,	,	PUNCT
ejde-320	392	31	it	it	PRON
ejde-320	392	32	is	be	AUX
ejde-320	392	33	locally	locally	ADV
ejde-320	392	34	bounded	bound	VERB
ejde-320	392	35	.	.	PUNCT
ejde-320	393	1	infinite	infinite	ADJ
ejde-320	393	2	vanishing	vanishing	NOUN
ejde-320	393	3	.	.	PUNCT
ejde-320	394	1	we	we	PRON
ejde-320	394	2	now	now	ADV
ejde-320	394	3	consider	consider	VERB
ejde-320	394	4	the	the	DET
ejde-320	394	5	case	case	NOUN
ejde-320	394	6	when	when	SCONJ
ejde-320	394	7	the	the	DET
ejde-320	394	8	function	function	NOUN
ejde-320	394	9	g	g	NOUN
ejde-320	394	10	,	,	PUNCT
ejde-320	394	11	and	and	CCONJ
ejde-320	394	12	therefore	therefore	ADV
ejde-320	394	13	ψ	ψ	X
ejde-320	394	14	,	,	PUNCT
ejde-320	394	15	vanishes	vanish	VERB
ejde-320	394	16	to	to	PART
ejde-320	394	17	infinite	infinite	VERB
ejde-320	394	18	order	order	NOUN
ejde-320	394	19	at	at	ADP
ejde-320	394	20	the	the	DET
ejde-320	394	21	origin	origin	NOUN
ejde-320	394	22	.	.	PUNCT
ejde-320	395	1	namely	namely	ADV
ejde-320	395	2	,	,	PUNCT
ejde-320	395	3	fix	fix	VERB
ejde-320	395	4	α	α	PRON
ejde-320	395	5	>	>	X
ejde-320	395	6	0	0	PUNCT
ejde-320	395	7	and	and	CCONJ
ejde-320	395	8	define	define	VERB
ejde-320	395	9	ψ(x	ψ(x	NOUN
ejde-320	395	10	)	)	PUNCT
ejde-320	395	11	:	:	PUNCT
ejde-320	396	1	=	=	SYM
ejde-320	396	2	xα+1e−	xα+1e−	PUNCT
ejde-320	396	3	1	1	NUM
ejde-320	396	4	xα	xα	INTJ
ejde-320	396	5	,	,	PUNCT
ejde-320	396	6	so	so	SCONJ
ejde-320	396	7	that	that	SCONJ
ejde-320	396	8	g(x	g(x	NOUN
ejde-320	396	9	)	)	PUNCT
ejde-320	396	10	=	=	PUNCT
ejde-320	397	1	ψ′(x	ψ′(x	X
ejde-320	397	2	)	)	PUNCT
ejde-320	397	3	=	=	SYM
ejde-320	398	1	αe−	αe−	X
ejde-320	398	2	1	1	NUM
ejde-320	398	3	xα	xα	ADP
ejde-320	398	4	+	+	CCONJ
ejde-320	398	5	(	(	PUNCT
ejde-320	398	6	α+	α+	X
ejde-320	398	7	1)xαe−	1)xαe−	NOUN
ejde-320	398	8	1	1	NUM
ejde-320	398	9	xα	xα	PUNCT
ejde-320	399	1	≈	≈	NUM
ejde-320	399	2	e−	e−	PROPN
ejde-320	399	3	1	1	NUM
ejde-320	399	4	xσ	xσ	NOUN
ejde-320	399	5	,	,	PUNCT
ejde-320	399	6	and	and	CCONJ
ejde-320	399	7	(	(	PUNCT
ejde-320	399	8	ψ′(x	ψ′(x	NOUN
ejde-320	399	9	)	)	PUNCT
ejde-320	399	10	ψ(x	ψ(x	NOUN
ejde-320	399	11	)	)	PUNCT
ejde-320	399	12	)	)	PUNCT
ejde-320	399	13	2	2	NUM
ejde-320	399	14	≈	≈	NUM
ejde-320	399	15	ψ′′(x	ψ′′(x	NOUN
ejde-320	399	16	)	)	PUNCT
ejde-320	399	17	ψ(x	ψ(x	PROPN
ejde-320	399	18	)	)	PUNCT
ejde-320	400	1	≈	≈	PROPN
ejde-320	400	2	1	1	NUM
ejde-320	400	3	x2α+2	x2α+2	PROPN
ejde-320	400	4	.	.	PUNCT
ejde-320	401	1	ejde-2021/82	ejde-2021/82	ADJ
ejde-320	401	2	orlicz	orlicz	NUM
ejde-320	401	3	-	-	PUNCT
ejde-320	401	4	sobolev	sobolev	NOUN
ejde-320	401	5	inequalities	inequality	NOUN
ejde-320	401	6	and	and	CCONJ
ejde-320	401	7	the	the	DET
ejde-320	401	8	dirichlet	dirichlet	PROPN
ejde-320	401	9	problem	problem	NOUN
ejde-320	401	10	17	17	NUM
ejde-320	401	11	combining	combine	VERB
ejde-320	401	12	this	this	PRON
ejde-320	401	13	with	with	ADP
ejde-320	401	14	(	(	PUNCT
ejde-320	401	15	5.3	5.3	NUM
ejde-320	401	16	)	)	PUNCT
ejde-320	401	17	gives	give	VERB
ejde-320	401	18	|f(x	|f(x	PROPN
ejde-320	401	19	,	,	PUNCT
ejde-320	401	20	y)|	y)|	PROPN
ejde-320	401	21	≈	≈	PROPN
ejde-320	401	22	1	1	NUM
ejde-320	402	1	x2α+2	x2α+2	NOUN
ejde-320	402	2	ln	ln	ADJ
ejde-320	402	3	1	1	NUM
ejde-320	402	4	x	x	NOUN
ejde-320	402	5	,	,	PUNCT
ejde-320	402	6	and	and	CCONJ
ejde-320	402	7	f	f	PROPN
ejde-320	402	8	is	be	AUX
ejde-320	402	9	supported	support	VERB
ejde-320	402	10	in	in	ADP
ejde-320	402	11	|y|	|y|	PROPN
ejde-320	402	12	≤	≤	ADJ
ejde-320	402	13	2ψ(x	2ψ(x	NUM
ejde-320	402	14	)	)	PUNCT
ejde-320	402	15	.	.	PUNCT
ejde-320	403	1	note	note	VERB
ejde-320	403	2	that	that	SCONJ
ejde-320	403	3	f	f	PROPN
ejde-320	403	4	does	do	AUX
ejde-320	403	5	not	not	PART
ejde-320	403	6	belong	belong	VERB
ejde-320	403	7	to	to	ADP
ejde-320	403	8	l∞(b	l∞(b	PROPN
ejde-320	403	9	)	)	PUNCT
ejde-320	403	10	since	since	SCONJ
ejde-320	403	11	it	it	PRON
ejde-320	403	12	is	be	AUX
ejde-320	403	13	unbounded	unbounded	ADJ
ejde-320	403	14	at	at	ADP
ejde-320	403	15	the	the	DET
ejde-320	403	16	origin	origin	NOUN
ejde-320	403	17	,	,	PUNCT
ejde-320	403	18	so	so	SCONJ
ejde-320	403	19	we	we	PRON
ejde-320	403	20	look	look	VERB
ejde-320	403	21	for	for	ADP
ejde-320	403	22	an	an	DET
ejde-320	403	23	appropriate	appropriate	ADJ
ejde-320	403	24	orlicz	orlicz	ADJ
ejde-320	403	25	space	space	NOUN
ejde-320	403	26	for	for	ADP
ejde-320	403	27	the	the	DET
ejde-320	403	28	function	function	NOUN
ejde-320	403	29	f	f	PROPN
ejde-320	403	30	.	.	PUNCT
ejde-320	404	1	recall	recall	VERB
ejde-320	404	2	the	the	DET
ejde-320	404	3	family	family	NOUN
ejde-320	404	4	of	of	ADP
ejde-320	404	5	young	young	ADJ
ejde-320	404	6	functions	function	NOUN
ejde-320	404	7	φn	φn	ADP
ejde-320	404	8	given	give	VERB
ejde-320	404	9	in	in	ADP
ejde-320	404	10	definition	definition	NOUN
ejde-320	404	11	2.9	2.9	NUM
ejde-320	404	12	.	.	PUNCT
ejde-320	405	1	the	the	DET
ejde-320	405	2	following	follow	VERB
ejde-320	405	3	orlicz	orlicz	PROPN
ejde-320	405	4	-	-	PUNCT
ejde-320	405	5	sobolev	sobolev	NOUN
ejde-320	405	6	inequality	inequality	NOUN
ejde-320	405	7	has	have	AUX
ejde-320	405	8	been	be	AUX
ejde-320	405	9	shown	show	VERB
ejde-320	405	10	in	in	ADP
ejde-320	405	11	[	[	X
ejde-320	405	12	7	7	NUM
ejde-320	405	13	]	]	SYM
ejde-320	405	14	‖w‖lφ(b	‖w‖lφ(b	VERB
ejde-320	405	15	)	)	PUNCT
ejde-320	405	16	≤	≤	NUM
ejde-320	405	17	c‖∇aw‖l1(b	c‖∇aw‖l1(b	PROPN
ejde-320	405	18	)	)	PUNCT
ejde-320	405	19	for	for	ADP
ejde-320	405	20	w	w	PROPN
ejde-320	405	21	∈	∈	PROPN
ejde-320	405	22	(	(	PUNCT
ejde-320	405	23	w	w	PROPN
ejde-320	405	24	1,1	1,1	NUM
ejde-320	405	25	a	a	PRON
ejde-320	405	26	)	)	PUNCT
ejde-320	405	27	0	0	PUNCT
ejde-320	406	1	(	(	PUNCT
ejde-320	406	2	b	b	NOUN
ejde-320	406	3	)	)	PUNCT
ejde-320	406	4	,	,	PUNCT
ejde-320	406	5	if	if	SCONJ
ejde-320	406	6	φ	φ	PROPN
ejde-320	406	7	=	=	SYM
ejde-320	406	8	φn	φn	PROPN
ejde-320	406	9	,	,	PUNCT
ejde-320	406	10	n	n	CCONJ
ejde-320	406	11	≥	≥	NOUN
ejde-320	406	12	1	1	NUM
ejde-320	406	13	,	,	PUNCT
ejde-320	406	14	and	and	CCONJ
ejde-320	406	15	αn	αn	NOUN
ejde-320	406	16	<	<	X
ejde-320	406	17	1	1	X
ejde-320	406	18	.	.	PUNCT
ejde-320	407	1	here	here	ADV
ejde-320	407	2	,	,	PUNCT
ejde-320	407	3	b	b	PROPN
ejde-320	407	4	is	be	AUX
ejde-320	407	5	a	a	DET
ejde-320	407	6	sufficiently	sufficiently	ADV
ejde-320	407	7	small	small	ADJ
ejde-320	407	8	subunit	subunit	NOUN
ejde-320	407	9	metric	metric	ADJ
ejde-320	407	10	ball	ball	NOUN
ejde-320	407	11	centered	center	VERB
ejde-320	407	12	at	at	ADP
ejde-320	407	13	the	the	DET
ejde-320	407	14	origin	origin	NOUN
ejde-320	407	15	.	.	PUNCT
ejde-320	408	1	letting	let	VERB
ejde-320	408	2	w	w	NOUN
ejde-320	408	3	=	=	PUNCT
ejde-320	408	4	v2	v2	PROPN
ejde-320	408	5	and	and	CCONJ
ejde-320	408	6	using	use	VERB
ejde-320	408	7	cauchy	cauchy	PROPN
ejde-320	408	8	-	-	PUNCT
ejde-320	408	9	schwartz	schwartz	PROPN
ejde-320	408	10	inequality	inequality	NOUN
ejde-320	408	11	and	and	CCONJ
ejde-320	408	12	lemma	lemma	PROPN
ejde-320	408	13	3.6	3.6	NUM
ejde-320	408	14	we	we	PRON
ejde-320	408	15	then	then	ADV
ejde-320	408	16	obtain	obtain	VERB
ejde-320	408	17	‖v‖lψ(b	‖v‖lψ(b	NOUN
ejde-320	408	18	)	)	PUNCT
ejde-320	408	19	≤	≤	NUM
ejde-320	408	20	c‖∇av‖l2(b	c‖∇av‖l2(b	NOUN
ejde-320	408	21	)	)	PUNCT
ejde-320	409	1	+	+	CCONJ
ejde-320	409	2	c‖v‖l2(b	c‖v‖l2(b	NOUN
ejde-320	409	3	)	)	PUNCT
ejde-320	409	4	for	for	ADP
ejde-320	409	5	v	v	NOUN
ejde-320	409	6	∈	∈	NOUN
ejde-320	409	7	(	(	PUNCT
ejde-320	409	8	w	w	PROPN
ejde-320	409	9	1,2	1,2	NUM
ejde-320	409	10	a	a	PRON
ejde-320	409	11	)	)	PUNCT
ejde-320	409	12	0	0	PUNCT
ejde-320	410	1	(	(	PUNCT
ejde-320	410	2	b	b	NOUN
ejde-320	410	3	)	)	PUNCT
ejde-320	410	4	,	,	PUNCT
ejde-320	410	5	(	(	PUNCT
ejde-320	410	6	5.5	5.5	NUM
ejde-320	410	7	)	)	PUNCT
ejde-320	410	8	with	with	ADP
ejde-320	410	9	ψ	ψ	PRON
ejde-320	410	10	defined	define	VERB
ejde-320	410	11	by	by	ADP
ejde-320	410	12	ψ(t	ψ(t	PROPN
ejde-320	410	13	)	)	PUNCT
ejde-320	410	14	=	=	SYM
ejde-320	410	15	φ(t2	φ(t2	NOUN
ejde-320	410	16	)	)	PUNCT
ejde-320	410	17	,	,	PUNCT
ejde-320	410	18	i.e.	i.e.	X
ejde-320	410	19	ψ(t	ψ(t	PROPN
ejde-320	410	20	)	)	PUNCT
ejde-320	410	21	≈	≈	PROPN
ejde-320	410	22	t2(ln	t2(ln	PROPN
ejde-320	410	23	t)n	t)n	NOUN
ejde-320	410	24	for	for	ADP
ejde-320	410	25	all	all	DET
ejde-320	410	26	t	t	PROPN
ejde-320	410	27	>	>	X
ejde-320	410	28	1	1	NUM
ejde-320	410	29	,	,	PUNCT
ejde-320	410	30	provided	provide	VERB
ejde-320	410	31	nα	nα	PRON
ejde-320	410	32	<	<	X
ejde-320	410	33	1	1	NUM
ejde-320	410	34	.	.	PUNCT
ejde-320	410	35	to	to	PART
ejde-320	410	36	make	make	VERB
ejde-320	410	37	analogy	analogy	NOUN
ejde-320	410	38	to	to	ADP
ejde-320	410	39	the	the	DET
ejde-320	410	40	finite	finite	ADJ
ejde-320	410	41	type	type	NOUN
ejde-320	410	42	case	case	NOUN
ejde-320	410	43	note	note	VERB
ejde-320	410	44	that	that	SCONJ
ejde-320	410	45	(	(	PUNCT
ejde-320	410	46	5.4	5.4	NUM
ejde-320	410	47	)	)	PUNCT
ejde-320	410	48	is	be	AUX
ejde-320	410	49	(	(	PUNCT
ejde-320	410	50	5.5	5.5	NUM
ejde-320	410	51	)	)	PUNCT
ejde-320	410	52	with	with	ADP
ejde-320	410	53	ψ(t	ψ(t	PROPN
ejde-320	410	54	)	)	PUNCT
ejde-320	410	55	=	=	SYM
ejde-320	410	56	t2σ	t2σ	NOUN
ejde-320	410	57	,	,	PUNCT
ejde-320	410	58	or	or	CCONJ
ejde-320	410	59	φ(t	φ(t	NOUN
ejde-320	410	60	)	)	PUNCT
ejde-320	410	61	=	=	SYM
ejde-320	410	62	tσ	tσ	PROPN
ejde-320	410	63	.	.	PUNCT
ejde-320	411	1	thus	thus	ADV
ejde-320	411	2	,	,	PUNCT
ejde-320	411	3	just	just	ADV
ejde-320	411	4	as	as	ADP
ejde-320	411	5	in	in	ADP
ejde-320	411	6	the	the	DET
ejde-320	411	7	finite	finite	ADJ
ejde-320	411	8	type	type	NOUN
ejde-320	411	9	case	case	NOUN
ejde-320	411	10	(	(	PUNCT
ejde-320	411	11	or	or	CCONJ
ejde-320	411	12	elliptic	elliptic	ADJ
ejde-320	411	13	case	case	NOUN
ejde-320	411	14	)	)	PUNCT
ejde-320	411	15	,	,	PUNCT
ejde-320	411	16	we	we	PRON
ejde-320	411	17	expect	expect	VERB
ejde-320	411	18	all	all	DET
ejde-320	411	19	weak	weak	ADJ
ejde-320	411	20	solutions	solution	NOUN
ejde-320	411	21	to	to	PART
ejde-320	411	22	lu	lu	VERB
ejde-320	411	23	=	=	PUNCT
ejde-320	411	24	f	f	PROPN
ejde-320	411	25	to	to	PART
ejde-320	411	26	be	be	AUX
ejde-320	411	27	bounded	bound	VERB
ejde-320	411	28	provided	provide	VERB
ejde-320	411	29	f	f	PROPN
ejde-320	411	30	belongs	belong	VERB
ejde-320	411	31	to	to	ADP
ejde-320	411	32	a	a	DET
ejde-320	411	33	slightly	slightly	ADV
ejde-320	411	34	smaller	small	ADJ
ejde-320	411	35	space	space	NOUN
ejde-320	411	36	than	than	ADP
ejde-320	411	37	the	the	DET
ejde-320	411	38	dual	dual	NOUN
ejde-320	411	39	of	of	ADP
ejde-320	411	40	lφ	lφ	NOUN
ejde-320	411	41	,	,	PUNCT
ejde-320	411	42	which	which	PRON
ejde-320	411	43	is	be	AUX
ejde-320	411	44	lφ̃.	lφ̃.	NOUN
ejde-320	411	45	on	on	ADP
ejde-320	411	46	the	the	DET
ejde-320	411	47	other	other	ADJ
ejde-320	411	48	hand	hand	NOUN
ejde-320	411	49	,	,	PUNCT
ejde-320	411	50	if	if	SCONJ
ejde-320	411	51	f	f	PROPN
ejde-320	411	52	is	be	AUX
ejde-320	411	53	in	in	ADP
ejde-320	411	54	a	a	DET
ejde-320	411	55	slightly	slightly	ADV
ejde-320	411	56	bigger	big	ADJ
ejde-320	411	57	space	space	NOUN
ejde-320	411	58	than	than	ADP
ejde-320	411	59	lφ̃	lφ̃	PROPN
ejde-320	411	60	we	we	PRON
ejde-320	411	61	expect	expect	VERB
ejde-320	411	62	there	there	PRON
ejde-320	411	63	to	to	PART
ejde-320	411	64	exist	exist	VERB
ejde-320	411	65	an	an	DET
ejde-320	411	66	unbounded	unbounded	ADJ
ejde-320	411	67	weak	weak	ADJ
ejde-320	411	68	solution	solution	NOUN
ejde-320	411	69	to	to	ADP
ejde-320	411	70	lu	lu	PROPN
ejde-320	411	71	=	=	SYM
ejde-320	411	72	f	f	PROPN
ejde-320	411	73	.	.	PUNCT
ejde-320	412	1	we	we	PRON
ejde-320	412	2	have	have	AUX
ejde-320	412	3	already	already	ADV
ejde-320	412	4	shown	show	VERB
ejde-320	412	5	in	in	ADP
ejde-320	412	6	theorem	theorem	ADJ
ejde-320	412	7	3.7	3.7	NUM
ejde-320	412	8	that	that	PRON
ejde-320	412	9	every	every	DET
ejde-320	412	10	weak	weak	ADJ
ejde-320	412	11	solution	solution	NOUN
ejde-320	412	12	u	u	NOUN
ejde-320	412	13	∈	∈	PROPN
ejde-320	412	14	(	(	PUNCT
ejde-320	412	15	w	w	PROPN
ejde-320	412	16	1,2	1,2	NUM
ejde-320	412	17	a	a	PRON
ejde-320	412	18	)	)	PUNCT
ejde-320	412	19	0	0	PUNCT
ejde-320	413	1	(	(	PUNCT
ejde-320	413	2	b	b	NOUN
ejde-320	413	3	)	)	PUNCT
ejde-320	413	4	to	to	PART
ejde-320	413	5	lu	lu	VERB
ejde-320	413	6	=	=	SYM
ejde-320	413	7	f	f	PROPN
ejde-320	413	8	with	with	ADP
ejde-320	413	9	f	f	PROPN
ejde-320	413	10	∈	∈	PROPN
ejde-320	413	11	l∞(b	l∞(b	PROPN
ejde-320	413	12	)	)	PUNCT
ejde-320	413	13	(	(	PUNCT
ejde-320	413	14	lφ̃(b	lφ̃(b	X
ejde-320	413	15	)	)	PUNCT
ejde-320	413	16	is	be	AUX
ejde-320	413	17	bounded	bound	VERB
ejde-320	413	18	.	.	PUNCT
ejde-320	414	1	on	on	ADP
ejde-320	414	2	the	the	DET
ejde-320	414	3	other	other	ADJ
ejde-320	414	4	hand	hand	NOUN
ejde-320	414	5	,	,	PUNCT
ejde-320	414	6	let	let	VERB
ejde-320	414	7	u	u	PRON
ejde-320	414	8	be	be	AUX
ejde-320	414	9	defined	define	VERB
ejde-320	414	10	by	by	ADP
ejde-320	414	11	(	(	PUNCT
ejde-320	414	12	5.2	5.2	NUM
ejde-320	414	13	)	)	PUNCT
ejde-320	414	14	,	,	PUNCT
ejde-320	414	15	and	and	CCONJ
ejde-320	414	16	one	one	PRON
ejde-320	414	17	can	can	AUX
ejde-320	414	18	verify	verify	VERB
ejde-320	414	19	that	that	SCONJ
ejde-320	414	20	u	u	PRON
ejde-320	414	21	∈w	∈w	VERB
ejde-320	414	22	1,2	1,2	NUM
ejde-320	414	23	a	a	DET
ejde-320	414	24	(	(	PUNCT
ejde-320	414	25	b	b	NOUN
ejde-320	414	26	)	)	PUNCT
ejde-320	414	27	.	.	PUNCT
ejde-320	415	1	recall	recall	VERB
ejde-320	415	2	that	that	PRON
ejde-320	415	3	with	with	ADP
ejde-320	415	4	f	f	PROPN
ejde-320	415	5	=	=	SYM
ejde-320	415	6	lu	lu	PROPN
ejde-320	415	7	we	we	PRON
ejde-320	415	8	have	have	VERB
ejde-320	415	9	(	(	PUNCT
ejde-320	415	10	5.3	5.3	NUM
ejde-320	415	11	)	)	PUNCT
ejde-320	415	12	,	,	PUNCT
ejde-320	415	13	i.e.	i.e.	X
ejde-320	415	14	|f(x	|f(x	PROPN
ejde-320	415	15	,	,	PUNCT
ejde-320	415	16	y)|	y)|	PROPN
ejde-320	415	17	≈	≈	PROPN
ejde-320	415	18	1	1	NUM
ejde-320	416	1	x2α+2	x2α+2	NOUN
ejde-320	416	2	ln	ln	ADJ
ejde-320	416	3	1	1	NUM
ejde-320	416	4	x	x	PUNCT
ejde-320	416	5	supported	support	VERB
ejde-320	416	6	in	in	ADP
ejde-320	416	7	|y|	|y|	PROPN
ejde-320	416	8	≤	≤	ADJ
ejde-320	416	9	2ψ(x	2ψ(x	NUM
ejde-320	416	10	)	)	PUNCT
ejde-320	416	11	.	.	PUNCT
ejde-320	417	1	we	we	PRON
ejde-320	417	2	now	now	ADV
ejde-320	417	3	would	would	AUX
ejde-320	417	4	like	like	VERB
ejde-320	417	5	to	to	PART
ejde-320	417	6	find	find	VERB
ejde-320	417	7	a	a	DET
ejde-320	417	8	young	young	ADJ
ejde-320	417	9	function	function	NOUN
ejde-320	417	10	θ	θ	PROPN
ejde-320	417	11	so	so	SCONJ
ejde-320	417	12	that	that	SCONJ
ejde-320	417	13	f	f	PROPN
ejde-320	417	14	∈	∈	PROPN
ejde-320	417	15	lθ(b	lθ(b	X
ejde-320	417	16	)	)	PUNCT
ejde-320	417	17	and	and	CCONJ
ejde-320	417	18	we	we	PRON
ejde-320	417	19	expect	expect	VERB
ejde-320	417	20	lθ(b	lθ(b	NOUN
ejde-320	417	21	)	)	PUNCT
ejde-320	417	22	to	to	PART
ejde-320	417	23	be	be	AUX
ejde-320	417	24	larger	large	ADJ
ejde-320	417	25	than	than	ADP
ejde-320	417	26	lφ̃(b	lφ̃(b	NOUN
ejde-320	417	27	)	)	PUNCT
ejde-320	417	28	(	(	PUNCT
ejde-320	417	29	analogous	analogous	ADJ
ejde-320	417	30	to	to	ADP
ejde-320	417	31	q	q	X
ejde-320	417	32	<	<	X
ejde-320	417	33	σ′	σ′	PROPN
ejde-320	417	34	)	)	PUNCT
ejde-320	417	35	.	.	PUNCT
ejde-320	418	1	let	let	VERB
ejde-320	418	2	θ	θ	NOUN
ejde-320	418	3	=	=	PUNCT
ejde-320	418	4	φ̃m	φ̃m	PROPN
ejde-320	418	5	,	,	PUNCT
ejde-320	418	6	m	m	VERB
ejde-320	418	7	≥	≥	NOUN
ejde-320	418	8	1	1	NUM
ejde-320	418	9	,	,	PUNCT
ejde-320	418	10	using	use	VERB
ejde-320	418	11	the	the	DET
ejde-320	418	12	estimates	estimate	NOUN
ejde-320	418	13	from	from	ADP
ejde-320	418	14	[	[	X
ejde-320	418	15	7	7	X
ejde-320	418	16	]	]	X
ejde-320	418	17	we	we	PRON
ejde-320	418	18	have	have	VERB
ejde-320	418	19	θ(s	θ(s	NOUN
ejde-320	418	20	)	)	PUNCT
ejde-320	418	21	≤ms1−	≤ms1−	NOUN
ejde-320	418	22	1	1	NUM
ejde-320	418	23	m	m	NOUN
ejde-320	418	24	es	es	X
ejde-320	418	25	1	1	NUM
ejde-320	418	26	m	m	NOUN
ejde-320	418	27	for	for	ADP
ejde-320	418	28	s	s	PRON
ejde-320	418	29	≥	≥	X
ejde-320	418	30	(	(	PUNCT
ejde-320	418	31	2m)m	2m)m	PROPN
ejde-320	418	32	.	.	PUNCT
ejde-320	419	1	therefore,∫	therefore,∫	X
ejde-320	420	1	b	b	X
ejde-320	420	2	θ(f(x	θ(f(x	PROPN
ejde-320	420	3	,	,	PUNCT
ejde-320	420	4	y	y	NOUN
ejde-320	420	5	)	)	PUNCT
ejde-320	420	6	)	)	PUNCT
ejde-320	421	1	dx	dx	PROPN
ejde-320	422	1	dy	dy	NOUN
ejde-320	422	2	≈	≈	PROPN
ejde-320	422	3	∫	∫	PROPN
ejde-320	423	1	ρ	ρ	NOUN
ejde-320	423	2	0	0	NUM
ejde-320	423	3	θ	θ	NOUN
ejde-320	423	4	(	(	PUNCT
ejde-320	423	5	1	1	X
ejde-320	423	6	x2α+2	x2α+2	PRON
ejde-320	423	7	ln	ln	ADJ
ejde-320	423	8	1	1	NUM
ejde-320	423	9	x	x	X
ejde-320	423	10	)	)	PUNCT
ejde-320	423	11	ψ(x)dx	ψ(x)dx	PROPN
ejde-320	424	1	≈	≈	PROPN
ejde-320	424	2	∫	∫	PROPN
ejde-320	424	3	ρ	ρ	PROPN
ejde-320	424	4	0	0	PROPN
ejde-320	425	1	(	(	PUNCT
ejde-320	425	2	ln	ln	NOUN
ejde-320	425	3	1	1	NUM
ejde-320	425	4	x	x	X
ejde-320	425	5	)	)	PUNCT
ejde-320	425	6	1−	1−	NUM
ejde-320	425	7	1	1	NUM
ejde-320	425	8	m	m	NOUN
ejde-320	425	9	1	1	NUM
ejde-320	425	10	x(2α+2)(1−1	x(2α+2)(1−1	ADP
ejde-320	425	11	/	/	SYM
ejde-320	425	12	m	m	NOUN
ejde-320	425	13	)	)	PUNCT
ejde-320	425	14	exp	exp	NOUN
ejde-320	425	15	{	{	PUNCT
ejde-320	425	16	(	(	PUNCT
ejde-320	425	17	ln	ln	NOUN
ejde-320	425	18	1	1	NUM
ejde-320	425	19	x	x	SYM
ejde-320	425	20	)	)	PUNCT
ejde-320	425	21	1	1	NUM
ejde-320	425	22	m	m	NOUN
ejde-320	425	23	1	1	NUM
ejde-320	425	24	x(2α+2)/m	x(2α+2)/m	PROPN
ejde-320	425	25	−	−	PROPN
ejde-320	425	26	1	1	NUM
ejde-320	425	27	xα	xα	ADJ
ejde-320	425	28	}	}	PUNCT
ejde-320	425	29	dx	dx	PROPN
ejde-320	425	30	.	.	PUNCT
ejde-320	426	1	in	in	ADP
ejde-320	426	2	order	order	NOUN
ejde-320	426	3	for	for	SCONJ
ejde-320	426	4	this	this	DET
ejde-320	426	5	integral	integral	ADJ
ejde-320	426	6	to	to	PART
ejde-320	426	7	be	be	AUX
ejde-320	426	8	finite	finite	ADJ
ejde-320	426	9	we	we	PRON
ejde-320	426	10	must	must	AUX
ejde-320	426	11	require	require	VERB
ejde-320	426	12	2α+	2α+	NUM
ejde-320	426	13	2	2	NUM
ejde-320	426	14	m	m	NOUN
ejde-320	426	15	<	<	X
ejde-320	426	16	α	α	NOUN
ejde-320	426	17	,	,	PUNCT
ejde-320	426	18	which	which	PRON
ejde-320	426	19	implies	imply	VERB
ejde-320	426	20	m	m	VERB
ejde-320	426	21	>	>	X
ejde-320	426	22	2	2	NUM
ejde-320	427	1	+	+	CCONJ
ejde-320	427	2	2	2	NUM
ejde-320	427	3	α	α	NOUN
ejde-320	427	4	>	>	X
ejde-320	427	5	2	2	NUM
ejde-320	427	6	+	+	CCONJ
ejde-320	427	7	2n	2n	NUM
ejde-320	427	8	,	,	PUNCT
ejde-320	427	9	18	18	NUM
ejde-320	427	10	u.	u.	PROPN
ejde-320	427	11	hafeez	hafeez	PROPN
ejde-320	427	12	,	,	PUNCT
ejde-320	427	13	t.	t.	PROPN
ejde-320	427	14	lavier	lavier	PROPN
ejde-320	427	15	,	,	PUNCT
ejde-320	427	16	l.	l.	PROPN
ejde-320	427	17	williams	williams	PROPN
ejde-320	427	18	,	,	PUNCT
ejde-320	427	19	l.	l.	PROPN
ejde-320	427	20	korobenko	korobenko	PROPN
ejde-320	427	21	ejde-2021/82	ejde-2021/82	PROPN
ejde-320	427	22	since	since	SCONJ
ejde-320	427	23	αn	αn	NOUN
ejde-320	427	24	<	<	X
ejde-320	427	25	1	1	NUM
ejde-320	427	26	.	.	PUNCT
ejde-320	428	1	thus	thus	ADV
ejde-320	428	2	f	f	PROPN
ejde-320	428	3	∈	∈	PROPN
ejde-320	428	4	lφ̃m	lφ̃m	NOUN
ejde-320	428	5	(	(	PUNCT
ejde-320	428	6	b	b	NOUN
ejde-320	428	7	)	)	PUNCT
ejde-320	428	8	for	for	ADP
ejde-320	428	9	m	m	PROPN
ejde-320	428	10	>	>	X
ejde-320	428	11	2	2	NUM
ejde-320	428	12	+	+	NUM
ejde-320	428	13	2n	2n	NUM
ejde-320	428	14	,	,	PUNCT
ejde-320	428	15	and	and	CCONJ
ejde-320	428	16	since	since	SCONJ
ejde-320	428	17	m	m	PROPN
ejde-320	428	18	>	>	X
ejde-320	428	19	n	n	CCONJ
ejde-320	428	20	(	(	PUNCT
ejde-320	428	21	i.e.	i.e.	X
ejde-320	428	22	lφm	lφm	X
ejde-320	428	23	(	(	PUNCT
ejde-320	428	24	lφn	lφn	PROPN
ejde-320	428	25	)	)	PUNCT
ejde-320	428	26	we	we	PRON
ejde-320	428	27	have	have	VERB
ejde-320	428	28	lφ̃n	lφ̃n	PROPN
ejde-320	428	29	(	(	PUNCT
ejde-320	428	30	lφ̃m	lφ̃m	NOUN
ejde-320	428	31	as	as	SCONJ
ejde-320	428	32	expected	expect	VERB
ejde-320	428	33	.	.	PUNCT
ejde-320	429	1	therefore	therefore	ADV
ejde-320	429	2	,	,	PUNCT
ejde-320	429	3	there	there	PRON
ejde-320	429	4	exists	exist	VERB
ejde-320	429	5	an	an	DET
ejde-320	429	6	unbounded	unbounded	ADJ
ejde-320	429	7	weak	weak	ADJ
ejde-320	429	8	solution	solution	NOUN
ejde-320	429	9	to	to	ADP
ejde-320	429	10	lu	lu	PROPN
ejde-320	429	11	=	=	SYM
ejde-320	429	12	f	f	PROPN
ejde-320	429	13	with	with	ADP
ejde-320	429	14	f	f	PROPN
ejde-320	429	15	∈	∈	PROPN
ejde-320	429	16	lφ̃m	lφ̃m	NOUN
ejde-320	429	17	(	(	PUNCT
ejde-320	429	18	b	b	NOUN
ejde-320	429	19	)	)	PUNCT
ejde-320	429	20	.	.	PUNCT
ejde-320	430	1	we	we	PRON
ejde-320	430	2	showed	show	VERB
ejde-320	430	3	that	that	SCONJ
ejde-320	430	4	given	give	VERB
ejde-320	430	5	weak	weak	ADJ
ejde-320	430	6	solutions	solution	NOUN
ejde-320	430	7	,	,	PUNCT
ejde-320	430	8	u	u	PROPN
ejde-320	430	9	∈	∈	PROPN
ejde-320	430	10	(	(	PUNCT
ejde-320	430	11	w	w	PROPN
ejde-320	430	12	1,2	1,2	NUM
ejde-320	430	13	a	a	PRON
ejde-320	430	14	)	)	PUNCT
ejde-320	430	15	0	0	NUM
ejde-320	430	16	,	,	PUNCT
ejde-320	430	17	to	to	ADP
ejde-320	430	18	eq	eq	NOUN
ejde-320	430	19	.	.	PROPN
ejde-320	430	20	1.1	1.1	NUM
ejde-320	430	21	with	with	ADP
ejde-320	430	22	supb	supb	NOUN
ejde-320	430	23	|u|	|u|	PROPN
ejde-320	430	24	≤	≤	PROPN
ejde-320	430	25	‖f‖lψ(b	‖f‖lψ(b	PROPN
ejde-320	430	26	)	)	PUNCT
ejde-320	430	27	,	,	PUNCT
ejde-320	430	28	then	then	ADV
ejde-320	430	29	the	the	DET
ejde-320	430	30	following	follow	VERB
ejde-320	430	31	orlicz	orlicz	ADJ
ejde-320	430	32	-	-	PUNCT
ejde-320	430	33	sobolev	sobolev	NOUN
ejde-320	430	34	inequality	inequality	NOUN
ejde-320	430	35	holds	hold	VERB
ejde-320	430	36	:	:	PUNCT
ejde-320	430	37	‖u‖lφ(b	‖u‖lφ(b	ADJ
ejde-320	430	38	)	)	PUNCT
ejde-320	430	39	≤	≤	NOUN
ejde-320	430	40	c‖∇au‖l2(b	c‖∇au‖l2(b	PROPN
ejde-320	430	41	)	)	PUNCT
ejde-320	430	42	.	.	PUNCT
ejde-320	431	1	we	we	PRON
ejde-320	431	2	furthermore	furthermore	ADV
ejde-320	431	3	showed	show	VERB
ejde-320	431	4	that	that	SCONJ
ejde-320	431	5	given	give	VERB
ejde-320	431	6	this	this	DET
ejde-320	431	7	inequality	inequality	NOUN
ejde-320	431	8	on	on	ADP
ejde-320	431	9	all	all	DET
ejde-320	431	10	u	u	NOUN
ejde-320	431	11	∈	∈	PROPN
ejde-320	431	12	(	(	PUNCT
ejde-320	431	13	w	w	PROPN
ejde-320	431	14	1,2	1,2	NUM
ejde-320	431	15	a	a	PRON
ejde-320	431	16	)	)	PUNCT
ejde-320	431	17	0	0	NUM
ejde-320	431	18	,	,	PUNCT
ejde-320	431	19	we	we	PRON
ejde-320	431	20	are	be	AUX
ejde-320	431	21	guaranteed	guarantee	VERB
ejde-320	431	22	the	the	DET
ejde-320	431	23	existence	existence	NOUN
ejde-320	431	24	of	of	ADP
ejde-320	431	25	unique	unique	ADJ
ejde-320	431	26	weak	weak	ADJ
ejde-320	431	27	solutions	solution	NOUN
ejde-320	431	28	to	to	ADP
ejde-320	431	29	eq	eq	NOUN
ejde-320	431	30	.	.	PROPN
ejde-320	431	31	1.1	1.1	NUM
ejde-320	431	32	.	.	PUNCT
ejde-320	432	1	however	however	ADV
ejde-320	432	2	,	,	PUNCT
ejde-320	432	3	as	as	SCONJ
ejde-320	432	4	demonstrated	demonstrate	VERB
ejde-320	432	5	in	in	ADP
ejde-320	432	6	section	section	NOUN
ejde-320	432	7	5	5	NUM
ejde-320	432	8	,	,	PUNCT
ejde-320	432	9	this	this	DET
ejde-320	432	10	same	same	ADJ
ejde-320	432	11	inequality	inequality	NOUN
ejde-320	432	12	does	do	AUX
ejde-320	432	13	not	not	PART
ejde-320	432	14	ensure	ensure	VERB
ejde-320	432	15	that	that	PRON
ejde-320	432	16	supb	supb	VERB
ejde-320	432	17	|u|	|u|	ADV
ejde-320	432	18	≤	≤	PROPN
ejde-320	432	19	‖f‖lψ(b	‖f‖lψ(b	PROPN
ejde-320	432	20	)	)	PUNCT
ejde-320	432	21	.	.	PUNCT
ejde-320	433	1	instead	instead	ADV
ejde-320	433	2	,	,	PUNCT
ejde-320	433	3	we	we	PRON
ejde-320	433	4	proved	prove	VERB
ejde-320	433	5	in	in	ADP
ejde-320	433	6	section	section	NOUN
ejde-320	433	7	3.2	3.2	NUM
ejde-320	433	8	,	,	PUNCT
ejde-320	433	9	the	the	DET
ejde-320	433	10	(	(	PUNCT
ejde-320	433	11	φ	φ	PROPN
ejde-320	433	12	,	,	PUNCT
ejde-320	433	13	2	2	NUM
ejde-320	433	14	)	)	PUNCT
ejde-320	433	15	orlicz	orlicz	ADJ
ejde-320	433	16	-	-	PUNCT
ejde-320	433	17	sobolev	sobolev	NOUN
ejde-320	433	18	inequality	inequality	NOUN
ejde-320	433	19	implies	imply	VERB
ejde-320	433	20	the	the	DET
ejde-320	433	21	global	global	ADJ
ejde-320	433	22	bound	bind	VERB
ejde-320	433	23	supb	supb	NOUN
ejde-320	433	24	|u|	|u|	PROPN
ejde-320	433	25	≤	≤	NUM
ejde-320	433	26	‖f‖l∞(b	‖f‖l∞(b	PROPN
ejde-320	433	27	)	)	PUNCT
ejde-320	433	28	.	.	PUNCT
ejde-320	434	1	thus	thus	ADV
ejde-320	434	2	,	,	PUNCT
ejde-320	434	3	our	our	PRON
ejde-320	434	4	necessary	necessary	ADJ
ejde-320	434	5	and	and	CCONJ
ejde-320	434	6	sufficient	sufficient	ADJ
ejde-320	434	7	conditions	condition	NOUN
ejde-320	434	8	miss	miss	VERB
ejde-320	434	9	each	each	DET
ejde-320	434	10	other	other	ADJ
ejde-320	434	11	.	.	PUNCT
ejde-320	435	1	counterexamples	counterexample	NOUN
ejde-320	435	2	presented	present	VERB
ejde-320	435	3	in	in	ADP
ejde-320	435	4	section	section	NOUN
ejde-320	435	5	5	5	NUM
ejde-320	435	6	demonstrate	demonstrate	VERB
ejde-320	435	7	that	that	SCONJ
ejde-320	435	8	this	this	PRON
ejde-320	435	9	miss	miss	VERB
ejde-320	435	10	in	in	ADP
ejde-320	435	11	not	not	PART
ejde-320	435	12	merely	merely	ADV
ejde-320	435	13	a	a	DET
ejde-320	435	14	limitation	limitation	NOUN
ejde-320	435	15	of	of	ADP
ejde-320	435	16	our	our	PRON
ejde-320	435	17	proof	proof	ADJ
ejde-320	435	18	techniques	technique	NOUN
ejde-320	435	19	but	but	CCONJ
ejde-320	435	20	rather	rather	ADV
ejde-320	435	21	,	,	PUNCT
ejde-320	435	22	a	a	DET
ejde-320	435	23	true	true	ADJ
ejde-320	435	24	phenomenon	phenomenon	NOUN
ejde-320	435	25	within	within	ADP
ejde-320	435	26	the	the	DET
ejde-320	435	27	field	field	NOUN
ejde-320	435	28	of	of	ADP
ejde-320	435	29	degenerate	degenerate	ADJ
ejde-320	435	30	elliptic	elliptic	ADJ
ejde-320	435	31	partial	partial	ADJ
ejde-320	435	32	differential	differential	NOUN
ejde-320	435	33	equations	equation	NOUN
ejde-320	435	34	.	.	PUNCT
ejde-320	436	1	acknowledgements	acknowledgement	NOUN
ejde-320	436	2	.	.	PUNCT
ejde-320	437	1	the	the	DET
ejde-320	437	2	first	first	ADJ
ejde-320	437	3	three	three	NUM
ejde-320	437	4	authors	author	NOUN
ejde-320	437	5	would	would	AUX
ejde-320	437	6	like	like	VERB
ejde-320	437	7	to	to	PART
ejde-320	437	8	first	first	ADV
ejde-320	437	9	and	and	CCONJ
ejde-320	437	10	foremost	foremost	ADV
ejde-320	437	11	thank	thank	VERB
ejde-320	437	12	luda	luda	PROPN
ejde-320	437	13	korobenko	korobenko	ADJ
ejde-320	437	14	.	.	PUNCT
ejde-320	438	1	without	without	ADP
ejde-320	438	2	her	her	PRON
ejde-320	438	3	incredible	incredible	ADJ
ejde-320	438	4	kindness	kindness	NOUN
ejde-320	438	5	,	,	PUNCT
ejde-320	438	6	patience	patience	NOUN
ejde-320	438	7	,	,	PUNCT
ejde-320	438	8	and	and	CCONJ
ejde-320	438	9	generosity	generosity	NOUN
ejde-320	438	10	this	this	DET
ejde-320	438	11	research	research	NOUN
ejde-320	438	12	would	would	AUX
ejde-320	438	13	never	never	ADV
ejde-320	438	14	have	have	AUX
ejde-320	438	15	taken	take	VERB
ejde-320	438	16	place	place	NOUN
ejde-320	438	17	,	,	PUNCT
ejde-320	438	18	and	and	CCONJ
ejde-320	438	19	our	our	PRON
ejde-320	438	20	summers	summer	NOUN
ejde-320	438	21	would	would	AUX
ejde-320	438	22	have	have	AUX
ejde-320	438	23	been	be	AUX
ejde-320	438	24	much	much	ADV
ejde-320	438	25	more	more	ADV
ejde-320	438	26	dull	dull	ADJ
ejde-320	438	27	.	.	PUNCT
ejde-320	439	1	references	reference	NOUN
ejde-320	439	2	[	[	X
ejde-320	439	3	1	1	X
ejde-320	439	4	]	]	PUNCT
ejde-320	439	5	e.	e.	PROPN
ejde-320	439	6	de	de	PROPN
ejde-320	439	7	giorgi	giorgi	PROPN
ejde-320	439	8	;	;	PUNCT
ejde-320	439	9	sulla	sulla	PROPN
ejde-320	439	10	differenziabilit‘a	differenziabilit‘a	PROPN
ejde-320	439	11	e	e	PROPN
ejde-320	439	12	l’analiticit‘a	l’analiticit‘a	NOUN
ejde-320	439	13	delle	delle	NOUN
ejde-320	439	14	estremaili	estremaili	NOUN
ejde-320	439	15	degli	degli	PROPN
ejde-320	439	16	integrali	integrali	PROPN
ejde-320	439	17	multipli	multipli	PROPN
ejde-320	439	18	regolari	regolari	PROPN
ejde-320	439	19	,	,	PUNCT
ejde-320	439	20	mem	mem	PROPN
ejde-320	439	21	.	.	PUNCT
ejde-320	439	22	accad	accad	PROPN
ejde-320	439	23	.	.	PUNCT
ejde-320	440	1	sci	sci	PROPN
ejde-320	440	2	.	.	PUNCT
ejde-320	440	3	torino	torino	PROPN
ejde-320	440	4	.	.	PUNCT
ejde-320	441	1	cl	cl	NOUN
ejde-320	441	2	.	.	PUNCT
ejde-320	442	1	sci	sci	PROPN
ejde-320	442	2	.	.	PROPN
ejde-320	442	3	fis	fis	PROPN
ejde-320	442	4	.	.	PUNCT
ejde-320	442	5	math	math	PROPN
ejde-320	442	6	.	.	PUNCT
ejde-320	443	1	nat	nat	PROPN
ejde-320	443	2	.	.	PROPN
ejde-320	443	3	,	,	PUNCT
ejde-320	443	4	3	3	NUM
ejde-320	443	5	(	(	PUNCT
ejde-320	443	6	1957	1957	NUM
ejde-320	443	7	)	)	PUNCT
ejde-320	443	8	,	,	PUNCT
ejde-320	443	9	25–43	25–43	NUM
ejde-320	443	10	.	.	PUNCT
ejde-320	444	1	[	[	X
ejde-320	444	2	2	2	X
ejde-320	444	3	]	]	PUNCT
ejde-320	444	4	e.	e.	PROPN
ejde-320	444	5	b.	b.	PROPN
ejde-320	444	6	fabes	fabes	PROPN
ejde-320	444	7	,	,	PUNCT
ejde-320	444	8	c.	c.	PROPN
ejde-320	444	9	e.	e.	PROPN
ejde-320	444	10	kenig	kenig	PROPN
ejde-320	444	11	,	,	PUNCT
ejde-320	444	12	r.	r.	PROPN
ejde-320	444	13	p.	p.	PROPN
ejde-320	444	14	serapioni	serapioni	NOUN
ejde-320	444	15	;	;	PUNCT
ejde-320	444	16	the	the	DET
ejde-320	444	17	local	local	ADJ
ejde-320	444	18	regularity	regularity	NOUN
ejde-320	444	19	of	of	ADP
ejde-320	444	20	solutions	solution	NOUN
ejde-320	444	21	of	of	ADP
ejde-320	444	22	degenerate	degenerate	ADJ
ejde-320	444	23	elliptic	elliptic	ADJ
ejde-320	444	24	equations	equation	NOUN
ejde-320	444	25	,	,	PUNCT
ejde-320	444	26	comm	comm	NOUN
ejde-320	444	27	.	.	PUNCT
ejde-320	445	1	partial	partial	ADJ
ejde-320	445	2	differential	differential	NOUN
ejde-320	445	3	equations	equation	NOUN
ejde-320	445	4	,	,	PUNCT
ejde-320	445	5	7	7	NUM
ejde-320	445	6	(	(	PUNCT
ejde-320	445	7	1982	1982	NUM
ejde-320	445	8	)	)	PUNCT
ejde-320	445	9	,	,	PUNCT
ejde-320	445	10	no	no	INTJ
ejde-320	445	11	.	.	NOUN
ejde-320	445	12	1	1	NUM
ejde-320	445	13	,	,	PUNCT
ejde-320	445	14	77–116	77–116	NOUN
ejde-320	445	15	.	.	PUNCT
ejde-320	446	1	[	[	X
ejde-320	446	2	3	3	X
ejde-320	446	3	]	]	X
ejde-320	446	4	b.	b.	PROPN
ejde-320	446	5	franchi	franchi	PROPN
ejde-320	446	6	,	,	PUNCT
ejde-320	446	7	e.	e.	PROPN
ejde-320	446	8	lanconelli	lanconelli	PROPN
ejde-320	446	9	;	;	PUNCT
ejde-320	446	10	hölder	hölder	NOUN
ejde-320	446	11	regularity	regularity	NOUN
ejde-320	446	12	theorem	theorem	NOUN
ejde-320	446	13	for	for	ADP
ejde-320	446	14	a	a	DET
ejde-320	446	15	class	class	NOUN
ejde-320	446	16	of	of	ADP
ejde-320	446	17	linear	linear	PROPN
ejde-320	446	18	nonuniformly	nonuniformly	PROPN
ejde-320	446	19	elliptic	elliptic	ADJ
ejde-320	446	20	operators	operator	NOUN
ejde-320	446	21	with	with	ADP
ejde-320	446	22	measurable	measurable	ADJ
ejde-320	446	23	coefficients	coefficient	NOUN
ejde-320	446	24	,	,	PUNCT
ejde-320	446	25	ann	ann	PROPN
ejde-320	446	26	.	.	PROPN
ejde-320	446	27	sc	sc	PROPN
ejde-320	446	28	.	.	PROPN
ejde-320	446	29	norm	norm	PROPN
ejde-320	446	30	.	.	PUNCT
ejde-320	447	1	super	super	ADJ
ejde-320	447	2	.	.	PUNCT
ejde-320	447	3	pisa	pisa	PROPN
ejde-320	447	4	,	,	PUNCT
ejde-320	447	5	cl	cl	NOUN
ejde-320	447	6	.	.	PUNCT
ejde-320	448	1	sci	sci	PROPN
ejde-320	448	2	.	.	PROPN
ejde-320	448	3	,	,	PUNCT
ejde-320	448	4	iv	iv	X
ejde-320	448	5	.	.	PUNCT
ejde-320	448	6	ser	ser	PROPN
ejde-320	448	7	.	.	PROPN
ejde-320	449	1	10	10	NUM
ejde-320	449	2	(	(	PUNCT
ejde-320	449	3	1983	1983	NUM
ejde-320	449	4	)	)	PUNCT
ejde-320	449	5	,	,	PUNCT
ejde-320	449	6	523	523	NUM
ejde-320	449	7	-	-	SYM
ejde-320	449	8	541	541	NUM
ejde-320	449	9	.	.	PUNCT
ejde-320	450	1	[	[	X
ejde-320	450	2	4	4	X
ejde-320	450	3	]	]	X
ejde-320	450	4	d.	d.	PROPN
ejde-320	450	5	gilbarg	gilbarg	PROPN
ejde-320	450	6	,	,	PUNCT
ejde-320	450	7	n.	n.	PROPN
ejde-320	450	8	s.	s.	PROPN
ejde-320	450	9	trudinger	trudinger	PROPN
ejde-320	450	10	;	;	PUNCT
ejde-320	450	11	elliptic	elliptic	ADJ
ejde-320	450	12	partial	partial	ADJ
ejde-320	450	13	differential	differential	ADJ
ejde-320	450	14	equations	equation	NOUN
ejde-320	450	15	of	of	ADP
ejde-320	450	16	second	second	ADJ
ejde-320	450	17	order	order	NOUN
ejde-320	450	18	,	,	PUNCT
ejde-320	450	19	revised	revise	VERB
ejde-320	450	20	3rd	3rd	ADJ
ejde-320	450	21	printing	printing	NOUN
ejde-320	450	22	,	,	PUNCT
ejde-320	450	23	1998	1998	NUM
ejde-320	450	24	,	,	PUNCT
ejde-320	450	25	springer	springer	NOUN
ejde-320	450	26	-	-	PUNCT
ejde-320	450	27	verlag	verlag	NOUN
ejde-320	450	28	.	.	PUNCT
ejde-320	451	1	[	[	X
ejde-320	451	2	5	5	X
ejde-320	451	3	]	]	PUNCT
ejde-320	451	4	l.	l.	PROPN
ejde-320	451	5	korobenko	korobenko	PROPN
ejde-320	451	6	,	,	PUNCT
ejde-320	451	7	d.	d.	PROPN
ejde-320	451	8	maldonado	maldonado	PROPN
ejde-320	451	9	,	,	PUNCT
ejde-320	451	10	c.	c.	PROPN
ejde-320	451	11	rios	rio	NOUN
ejde-320	451	12	;	;	PUNCT
ejde-320	451	13	from	from	ADP
ejde-320	451	14	sobolev	sobolev	PROPN
ejde-320	451	15	inequality	inequality	NOUN
ejde-320	451	16	to	to	ADP
ejde-320	451	17	doubling	doubling	NOUN
ejde-320	451	18	,	,	PUNCT
ejde-320	451	19	proceedings	proceeding	NOUN
ejde-320	451	20	of	of	ADP
ejde-320	451	21	the	the	DET
ejde-320	451	22	ams	am	NOUN
ejde-320	451	23	143	143	NUM
ejde-320	451	24	(	(	PUNCT
ejde-320	451	25	9	9	NUM
ejde-320	451	26	)	)	PUNCT
ejde-320	451	27	,	,	PUNCT
ejde-320	451	28	(	(	PUNCT
ejde-320	451	29	2015	2015	NUM
ejde-320	451	30	)	)	PUNCT
ejde-320	451	31	,	,	PUNCT
ejde-320	451	32	4017–4028	4017–4028	NOUN
ejde-320	451	33	.	.	PUNCT
ejde-320	452	1	[	[	X
ejde-320	452	2	6	6	NUM
ejde-320	452	3	]	]	PUNCT
ejde-320	452	4	l.	l.	PROPN
ejde-320	452	5	korobenko	korobenko	PROPN
ejde-320	452	6	,	,	PUNCT
ejde-320	452	7	c.	c.	PROPN
ejde-320	452	8	rios	rios	PROPN
ejde-320	452	9	,	,	PUNCT
ejde-320	452	10	e.	e.	PROPN
ejde-320	452	11	sawyer	sawyer	PROPN
ejde-320	452	12	,	,	PUNCT
ejde-320	452	13	r.	r.	PROPN
ejde-320	452	14	shen	shen	PROPN
ejde-320	452	15	;	;	PUNCT
ejde-320	452	16	local	local	ADJ
ejde-320	452	17	boundedness	boundedness	NOUN
ejde-320	452	18	,	,	PUNCT
ejde-320	452	19	maximum	maximum	ADJ
ejde-320	452	20	principles	principle	NOUN
ejde-320	452	21	,	,	PUNCT
ejde-320	452	22	and	and	CCONJ
ejde-320	452	23	continuity	continuity	NOUN
ejde-320	452	24	of	of	ADP
ejde-320	452	25	solutions	solution	NOUN
ejde-320	452	26	to	to	PART
ejde-320	452	27	infinitely	infinitely	ADV
ejde-320	452	28	degenerate	degenerate	ADJ
ejde-320	452	29	elliptic	elliptic	ADJ
ejde-320	452	30	equations	equation	NOUN
ejde-320	452	31	,	,	PUNCT
ejde-320	452	32	arxiv:1506.09203	arxiv:1506.09203	NOUN
ejde-320	452	33	,	,	PUNCT
ejde-320	452	34	(	(	PUNCT
ejde-320	452	35	2016	2016	NUM
ejde-320	452	36	)	)	PUNCT
ejde-320	452	37	.	.	PUNCT
ejde-320	453	1	[	[	X
ejde-320	453	2	7	7	X
ejde-320	453	3	]	]	X
ejde-320	453	4	l.	l.	PROPN
ejde-320	453	5	korobenko	korobenko	PROPN
ejde-320	453	6	,	,	PUNCT
ejde-320	453	7	c.	c.	PROPN
ejde-320	453	8	rios	rios	PROPN
ejde-320	453	9	,	,	PUNCT
ejde-320	453	10	e.	e.	PROPN
ejde-320	453	11	sawyer	sawyer	PROPN
ejde-320	453	12	,	,	PUNCT
ejde-320	453	13	r.	r.	PROPN
ejde-320	453	14	shen	shen	PROPN
ejde-320	453	15	;	;	PUNCT
ejde-320	453	16	local	local	ADJ
ejde-320	453	17	boundedness	boundedness	NOUN
ejde-320	453	18	,	,	PUNCT
ejde-320	453	19	maximum	maximum	ADJ
ejde-320	453	20	principles	principle	NOUN
ejde-320	453	21	,	,	PUNCT
ejde-320	453	22	and	and	CCONJ
ejde-320	453	23	continuity	continuity	NOUN
ejde-320	453	24	of	of	ADP
ejde-320	453	25	solutions	solution	NOUN
ejde-320	453	26	to	to	PART
ejde-320	453	27	infinitely	infinitely	ADV
ejde-320	453	28	degenerate	degenerate	ADJ
ejde-320	453	29	elliptic	elliptic	ADJ
ejde-320	453	30	equations	equation	NOUN
ejde-320	453	31	,	,	PUNCT
ejde-320	453	32	memoirs	memoir	NOUN
ejde-320	453	33	of	of	ADP
ejde-320	453	34	the	the	DET
ejde-320	453	35	ams	am	NOUN
ejde-320	453	36	,	,	PUNCT
ejde-320	453	37	269	269	NUM
ejde-320	453	38	(	(	PUNCT
ejde-320	453	39	2021	2021	NUM
ejde-320	453	40	)	)	PUNCT
ejde-320	453	41	,	,	PUNCT
ejde-320	453	42	no	no	INTJ
ejde-320	453	43	.	.	NOUN
ejde-320	453	44	1311	1311	NUM
ejde-320	453	45	.	.	PUNCT
ejde-320	454	1	[	[	X
ejde-320	454	2	8	8	NUM
ejde-320	454	3	]	]	X
ejde-320	454	4	c.	c.	NOUN
ejde-320	454	5	léonard	léonard	PROPN
ejde-320	454	6	;	;	PUNCT
ejde-320	454	7	orlicz	orlicz	NOUN
ejde-320	454	8	spaces	space	VERB
ejde-320	454	9	,	,	PUNCT
ejde-320	454	10	(	(	PUNCT
ejde-320	454	11	2007	2007	NUM
ejde-320	454	12	)	)	PUNCT
ejde-320	454	13	.	.	PUNCT
ejde-320	454	14	http://leonard.perso.math.cnrs.fr/papers/leonard-orlicz%20spaces.pdf	http://leonard.perso.math.cnrs.fr/papers/leonard-orlicz%20spaces.pdf	PUNCT
ejde-320	455	1	[	[	X
ejde-320	455	2	9	9	NUM
ejde-320	455	3	]	]	X
ejde-320	455	4	j.	j.	PROPN
ejde-320	455	5	moser	moser	PROPN
ejde-320	455	6	;	;	PUNCT
ejde-320	455	7	on	on	ADP
ejde-320	455	8	harnack	harnack	PROPN
ejde-320	455	9	’s	’s	PART
ejde-320	455	10	theorem	theorem	NOUN
ejde-320	455	11	for	for	ADP
ejde-320	455	12	elliptic	elliptic	ADJ
ejde-320	455	13	differential	differential	ADJ
ejde-320	455	14	equations	equation	NOUN
ejde-320	455	15	,	,	PUNCT
ejde-320	455	16	comm	comm	NOUN
ejde-320	455	17	.	.	PUNCT
ejde-320	456	1	pure	pure	ADJ
ejde-320	456	2	appl	appl	PROPN
ejde-320	456	3	.	.	PUNCT
ejde-320	456	4	math	math	NOUN
ejde-320	456	5	.	.	PUNCT
ejde-320	457	1	14	14	NUM
ejde-320	457	2	(	(	PUNCT
ejde-320	457	3	1961	1961	NUM
ejde-320	457	4	)	)	PUNCT
ejde-320	457	5	,	,	PUNCT
ejde-320	457	6	577–591	577–591	NUM
ejde-320	457	7	.	.	PUNCT
ejde-320	458	1	[	[	X
ejde-320	458	2	10	10	NUM
ejde-320	458	3	]	]	X
ejde-320	458	4	j.	j.	PROPN
ejde-320	458	5	nash	nash	PROPN
ejde-320	458	6	;	;	PUNCT
ejde-320	458	7	continuity	continuity	NOUN
ejde-320	458	8	of	of	ADP
ejde-320	458	9	solutions	solution	NOUN
ejde-320	458	10	of	of	ADP
ejde-320	458	11	parabolic	parabolic	ADJ
ejde-320	458	12	and	and	CCONJ
ejde-320	458	13	elliptic	elliptic	ADJ
ejde-320	458	14	equations	equation	NOUN
ejde-320	458	15	,	,	PUNCT
ejde-320	458	16	amer	amer	PROPN
ejde-320	458	17	.	.	PUNCT
ejde-320	458	18	j.	j.	PROPN
ejde-320	458	19	math	math	PROPN
ejde-320	458	20	.	.	PUNCT
ejde-320	459	1	80	80	NUM
ejde-320	459	2	(	(	PUNCT
ejde-320	459	3	1958	1958	NUM
ejde-320	459	4	)	)	PUNCT
ejde-320	459	5	,	,	PUNCT
ejde-320	459	6	931–954	931–954	NUM
ejde-320	459	7	.	.	PUNCT
ejde-320	460	1	[	[	X
ejde-320	460	2	11	11	NUM
ejde-320	460	3	]	]	PUNCT
ejde-320	460	4	e.	e.	PROPN
ejde-320	460	5	sawyer	sawyer	PROPN
ejde-320	460	6	,	,	PUNCT
ejde-320	460	7	r.	r.	PROPN
ejde-320	460	8	l.	l.	PROPN
ejde-320	460	9	wheeden	wheeden	PROPN
ejde-320	460	10	;	;	PUNCT
ejde-320	460	11	hölder	hölder	VERB
ejde-320	460	12	continuity	continuity	NOUN
ejde-320	460	13	of	of	ADP
ejde-320	460	14	weak	weak	ADJ
ejde-320	460	15	solutions	solution	NOUN
ejde-320	460	16	to	to	ADP
ejde-320	460	17	subelliptic	subelliptic	ADJ
ejde-320	460	18	equations	equation	NOUN
ejde-320	460	19	with	with	ADP
ejde-320	460	20	rough	rough	ADJ
ejde-320	460	21	coefficients	coefficient	NOUN
ejde-320	460	22	,	,	PUNCT
ejde-320	460	23	memoirs	memoir	NOUN
ejde-320	460	24	of	of	ADP
ejde-320	460	25	the	the	DET
ejde-320	460	26	ams	am	NOUN
ejde-320	460	27	,	,	PUNCT
ejde-320	460	28	847	847	NUM
ejde-320	460	29	(	(	PUNCT
ejde-320	460	30	2006	2006	NUM
ejde-320	460	31	)	)	PUNCT
ejde-320	460	32	.	.	PUNCT
ejde-320	461	1	[	[	X
ejde-320	461	2	12	12	NUM
ejde-320	461	3	]	]	X
ejde-320	461	4	w.	w.	PROPN
ejde-320	461	5	ungemach	ungemach	PROPN
ejde-320	461	6	;	;	PUNCT
ejde-320	461	7	sobolev	sobolev	NOUN
ejde-320	461	8	spaces	space	VERB
ejde-320	461	9	with	with	ADP
ejde-320	461	10	application	application	NOUN
ejde-320	461	11	to	to	ADP
ejde-320	461	12	second	second	ADJ
ejde-320	461	13	-	-	PUNCT
ejde-320	461	14	order	order	NOUN
ejde-320	461	15	elliptic	elliptic	ADJ
ejde-320	461	16	pde	pde	NOUN
ejde-320	461	17	,	,	PUNCT
ejde-320	461	18	(	(	PUNCT
ejde-320	461	19	2011	2011	NUM
ejde-320	461	20	)	)	PUNCT
ejde-320	461	21	.	.	PUNCT
ejde-320	462	1	http://www.math.uchicago.edu/∼may/vigre/vigre2011/reupapers/ungemach.pdf	http://www.math.uchicago.edu/∼may/vigre/vigre2011/reupapers/ungemach.pdf	PROPN
ejde-320	462	2	usman	usman	PROPN
ejde-320	462	3	hafeez	hafeez	PROPN
ejde-320	462	4	department	department	PROPN
ejde-320	462	5	of	of	ADP
ejde-320	462	6	mathematics	mathematics	PROPN
ejde-320	462	7	,	,	PUNCT
ejde-320	462	8	reed	reed	PROPN
ejde-320	462	9	college	college	PROPN
ejde-320	462	10	,	,	PUNCT
ejde-320	462	11	portland	portland	PROPN
ejde-320	462	12	,	,	PUNCT
ejde-320	462	13	or	or	CCONJ
ejde-320	462	14	97202	97202	NUM
ejde-320	462	15	,	,	PUNCT
ejde-320	462	16	usa	usa	PROPN
ejde-320	462	17	email	email	NOUN
ejde-320	462	18	address	address	NOUN
ejde-320	462	19	:	:	PUNCT
ejde-320	462	20	usman-20@live.com	usman-20@live.com	X
ejde-320	463	1	ejde-2021/82	ejde-2021/82	ADJ
ejde-320	463	2	orlicz	orlicz	PROPN
ejde-320	463	3	-	-	PUNCT
ejde-320	463	4	sobolev	sobolev	NOUN
ejde-320	463	5	inequalities	inequality	NOUN
ejde-320	463	6	and	and	CCONJ
ejde-320	463	7	the	the	DET
ejde-320	463	8	dirichlet	dirichlet	PROPN
ejde-320	463	9	problem	problem	NOUN
ejde-320	463	10	19	19	NUM
ejde-320	463	11	théo	théo	NOUN
ejde-320	463	12	lavier	lavier	PROPN
ejde-320	463	13	university	university	PROPN
ejde-320	463	14	of	of	ADP
ejde-320	463	15	edinburgh	edinburgh	PROPN
ejde-320	463	16	,	,	PUNCT
ejde-320	463	17	edinburgh	edinburgh	PROPN
ejde-320	463	18	,	,	PUNCT
ejde-320	463	19	uk	uk	PROPN
ejde-320	463	20	email	email	NOUN
ejde-320	463	21	address	address	NOUN
ejde-320	463	22	:	:	PUNCT
ejde-320	463	23	t.p.lavier@sms.ed.ac.uk	t.p.lavier@sms.ed.ac.uk	PROPN
ejde-320	463	24	lucas	lucas	PROPN
ejde-320	463	25	williams	williams	PROPN
ejde-320	463	26	binghamton	binghamton	PROPN
ejde-320	463	27	university	university	PROPN
ejde-320	463	28	,	,	PUNCT
ejde-320	463	29	binghamton	binghamton	PROPN
ejde-320	463	30	,	,	PUNCT
ejde-320	463	31	ny	ny	PROPN
ejde-320	463	32	,	,	PUNCT
ejde-320	463	33	usa	usa	PROPN
ejde-320	463	34	email	email	NOUN
ejde-320	463	35	address	address	NOUN
ejde-320	463	36	:	:	PUNCT
ejde-320	463	37	williaml@math.binghamton.edu	williaml@math.binghamton.edu	NUM
ejde-320	463	38	lyudmila	lyudmila	NOUN
ejde-320	463	39	korobenko	korobenko	PROPN
ejde-320	463	40	reed	reed	PROPN
ejde-320	463	41	college	college	PROPN
ejde-320	463	42	,	,	PUNCT
ejde-320	463	43	portland	portland	PROPN
ejde-320	463	44	,	,	PUNCT
ejde-320	463	45	or	or	CCONJ
ejde-320	463	46	,	,	PUNCT
ejde-320	463	47	usa	usa	PROPN
ejde-320	463	48	email	email	NOUN
ejde-320	463	49	address	address	NOUN
ejde-320	463	50	:	:	PUNCT
ejde-320	463	51	korobenko@reed.edu	korobenko@reed.edu	PROPN
ejde-320	463	52	1	1	X
ejde-320	463	53	.	.	PUNCT
ejde-320	463	54	introduction	introduction	NOUN
ejde-320	463	55	2	2	NUM
ejde-320	463	56	.	.	PUNCT
ejde-320	463	57	preliminaries	preliminary	NOUN
ejde-320	463	58	2.1	2.1	NUM
ejde-320	463	59	.	.	PUNCT
ejde-320	464	1	subunit	subunit	NOUN
ejde-320	464	2	metric	metric	ADJ
ejde-320	464	3	spaces	space	VERB
ejde-320	464	4	2.2	2.2	NUM
ejde-320	464	5	.	.	PUNCT
ejde-320	465	1	orlicz	orlicz	PROPN
ejde-320	465	2	spaces	space	VERB
ejde-320	465	3	3	3	NUM
ejde-320	465	4	.	.	X
ejde-320	465	5	sufficiency	sufficiency	PROPN
ejde-320	465	6	3.1	3.1	NUM
ejde-320	465	7	.	.	PUNCT
ejde-320	466	1	existence	existence	NOUN
ejde-320	466	2	of	of	ADP
ejde-320	466	3	a	a	DET
ejde-320	466	4	unique	unique	ADJ
ejde-320	466	5	weak	weak	ADJ
ejde-320	466	6	solution	solution	NOUN
ejde-320	466	7	3.2	3.2	NUM
ejde-320	466	8	.	.	PUNCT
ejde-320	467	1	global	global	ADJ
ejde-320	467	2	boundedness	boundedness	NOUN
ejde-320	467	3	of	of	ADP
ejde-320	467	4	weak	weak	ADJ
ejde-320	467	5	solutions	solution	NOUN
ejde-320	467	6	4	4	NUM
ejde-320	467	7	.	.	PUNCT
ejde-320	467	8	almost	almost	ADV
ejde-320	467	9	necessity	necessity	NOUN
ejde-320	467	10	5	5	NUM
ejde-320	467	11	.	.	PUNCT
ejde-320	467	12	sharpness	sharpness	NOUN
ejde-320	467	13	5.1	5.1	NUM
ejde-320	467	14	.	.	PUNCT
ejde-320	468	1	laplacian	laplacian	ADJ
ejde-320	468	2	counterexample	counterexample	PROPN
ejde-320	468	3	5.2	5.2	NUM
ejde-320	468	4	.	.	PUNCT
ejde-320	469	1	degenerate	degenerate	ADJ
ejde-320	469	2	counterexamples	counterexample	NOUN
ejde-320	469	3	acknowledgements	acknowledgement	NOUN
ejde-320	469	4	references	reference	NOUN