id	sid	tid	token	lemma	pos
ma-234	1	1	2024	2024	NUM
ma-234	1	2	ada	ada	PROPN
ma-234	1	3	academica	academica	PROPN
ma-234	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-234	1	5	.	.	PUNCT
ma-234	2	1	j.	j.	PROPN
ma-234	2	2	math	math	PROPN
ma-234	2	3	.	.	PUNCT
ma-234	3	1	anal	anal	ADJ
ma-234	3	2	.	.	PUNCT
ma-234	4	1	4	4	NUM
ma-234	4	2	(	(	PUNCT
ma-234	4	3	2024	2024	NUM
ma-234	4	4	)	)	PUNCT
ma-234	4	5	16doi	16doi	NUM
ma-234	4	6	:	:	PUNCT
ma-234	4	7	10.28924	10.28924	NUM
ma-234	4	8	/	/	SYM
ma-234	4	9	ada	ada	PROPN
ma-234	4	10	/	/	SYM
ma-234	4	11	ma.4.16	ma.4.16	PROPN
ma-234	4	12	hardy	hardy	ADJ
ma-234	4	13	-	-	PUNCT
ma-234	4	14	littlewood	littlewood	NOUN
ma-234	4	15	-	-	PUNCT
ma-234	4	16	sobolev	sobolev	NOUN
ma-234	4	17	theorem	theorem	NOUN
ma-234	4	18	for	for	ADP
ma-234	4	19	bourgain	bourgain	NOUN
ma-234	4	20	-	-	PUNCT
ma-234	4	21	morrey	morrey	NOUN
ma-234	4	22	spaces	space	NOUN
ma-234	4	23	and	and	CCONJ
ma-234	4	24	approximation	approximation	NOUN
ma-234	4	25	nouffou	nouffou	PROPN
ma-234	4	26	diarra	diarra	PROPN
ma-234	4	27	laboratoire	laboratoire	PROPN
ma-234	4	28	de	de	PROPN
ma-234	4	29	mathématiques	mathématiques	PROPN
ma-234	4	30	et	et	NOUN
ma-234	4	31	applications	application	NOUN
ma-234	4	32	,	,	PUNCT
ma-234	4	33	ufr	ufr	NOUN
ma-234	4	34	mathématiques	mathématique	NOUN
ma-234	4	35	et	et	PROPN
ma-234	4	36	informatique	informatique	PROPN
ma-234	4	37	,	,	PUNCT
ma-234	4	38	université	université	PROPN
ma-234	4	39	félix	félix	PROPN
ma-234	4	40	houphouët	houphouët	PROPN
ma-234	4	41	boigny	boigny	PROPN
ma-234	4	42	abidjan	abidjan	PROPN
ma-234	4	43	-	-	PROPN
ma-234	4	44	cocody	cocody	NOUN
ma-234	4	45	,	,	PUNCT
ma-234	4	46	22	22	NUM
ma-234	4	47	bp	bp	PROPN
ma-234	4	48	582	582	NUM
ma-234	4	49	abidjan	abidjan	PROPN
ma-234	4	50	22	22	NUM
ma-234	4	51	,	,	PUNCT
ma-234	4	52	côte	côte	NOUN
ma-234	4	53	d’ivoire	d’ivoire	NOUN
ma-234	4	54	correspondence	correspondence	NOUN
ma-234	4	55	:	:	PUNCT
ma-234	4	56	nouffoud@yahoo.fr	nouffoud@yahoo.fr	PROPN
ma-234	4	57	abstract	abstract	NOUN
ma-234	4	58	.	.	PUNCT
ma-234	5	1	in	in	ADP
ma-234	5	2	this	this	DET
ma-234	5	3	paper	paper	NOUN
ma-234	5	4	,	,	PUNCT
ma-234	5	5	we	we	PRON
ma-234	5	6	establish	establish	VERB
ma-234	5	7	an	an	DET
ma-234	5	8	extension	extension	NOUN
ma-234	5	9	of	of	ADP
ma-234	5	10	the	the	DET
ma-234	5	11	hardy	hardy	ADJ
ma-234	5	12	-	-	PUNCT
ma-234	5	13	littlewood	littlewood	NOUN
ma-234	5	14	-	-	PUNCT
ma-234	5	15	sobolev	sobolev	NOUN
ma-234	5	16	theorem	theorem	NOUN
ma-234	5	17	to	to	ADP
ma-234	5	18	thesetting	thesetting	NOUN
ma-234	5	19	of	of	ADP
ma-234	5	20	the	the	DET
ma-234	5	21	bourgain	bourgain	NOUN
ma-234	5	22	-	-	PUNCT
ma-234	5	23	morrey	morrey	NOUN
ma-234	5	24	space	space	NOUN
ma-234	5	25	mα	mα	PROPN
ma-234	5	26	q	q	NOUN
ma-234	5	27	,	,	PUNCT
ma-234	5	28	p(rd	p(rd	ADJ
ma-234	5	29	)	)	PUNCT
ma-234	5	30	(	(	PUNCT
ma-234	5	31	1	1	NUM
ma-234	5	32	≤	≤	NUM
ma-234	5	33	q	q	NOUN
ma-234	5	34	,	,	PUNCT
ma-234	5	35	p	p	X
ma-234	5	36	,	,	PUNCT
ma-234	5	37	α	α	PROPN
ma-234	5	38	≤	≤	NOUN
ma-234	5	39	∞	∞	NUM
ma-234	5	40	)	)	PUNCT
ma-234	5	41	,	,	PUNCT
ma-234	5	42	which	which	DET
ma-234	5	43	theory	theory	NOUN
ma-234	5	44	goes	go	VERB
ma-234	5	45	back	back	ADV
ma-234	5	46	tobourgain	tobourgain	NOUN
ma-234	5	47	in	in	ADP
ma-234	5	48	1991	1991	NUM
ma-234	5	49	.	.	PUNCT
ma-234	6	1	we	we	PRON
ma-234	6	2	also	also	ADV
ma-234	6	3	prove	prove	VERB
ma-234	6	4	that	that	SCONJ
ma-234	6	5	mα	mα	PROPN
ma-234	6	6	q	q	ADJ
ma-234	6	7	,	,	PUNCT
ma-234	6	8	p(rd	p(rd	ADJ
ma-234	6	9	)	)	PUNCT
ma-234	6	10	is	be	AUX
ma-234	6	11	included	include	VERB
ma-234	6	12	in	in	ADP
ma-234	6	13	the	the	DET
ma-234	6	14	closure	closure	NOUN
ma-234	6	15	of	of	ADP
ma-234	6	16	the	the	DET
ma-234	6	17	lebesgue	lebesgue	NOUN
ma-234	6	18	space	space	NOUN
ma-234	6	19	lα	lα	NOUN
ma-234	6	20	in	in	ADP
ma-234	6	21	the	the	DET
ma-234	6	22	morrey	morrey	NOUN
ma-234	6	23	-	-	PUNCT
ma-234	6	24	type	type	NOUN
ma-234	6	25	space	space	NOUN
ma-234	6	26	f(q	f(q	PROPN
ma-234	6	27	,	,	PUNCT
ma-234	6	28	p	p	X
ma-234	6	29	,	,	PUNCT
ma-234	6	30	α	α	NOUN
ma-234	6	31	)	)	PUNCT
ma-234	6	32	,	,	PUNCT
ma-234	6	33	which	which	PRON
ma-234	6	34	arises	arise	VERB
ma-234	6	35	naturally	naturally	ADV
ma-234	6	36	in	in	ADP
ma-234	6	37	2015	2015	NUM
ma-234	6	38	in	in	ADP
ma-234	6	39	the	the	DET
ma-234	6	40	study	study	NOUN
ma-234	6	41	of	of	ADP
ma-234	6	42	boundednessproperties	boundednesspropertie	NOUN
ma-234	6	43	of	of	ADP
ma-234	6	44	fractional	fractional	ADJ
ma-234	6	45	integral	integral	ADJ
ma-234	6	46	operators	operator	NOUN
ma-234	6	47	.	.	PUNCT
ma-234	7	1	therefore	therefore	ADV
ma-234	7	2	,	,	PUNCT
ma-234	7	3	we	we	PRON
ma-234	7	4	establish	establish	VERB
ma-234	7	5	in	in	ADP
ma-234	7	6	mα	mα	PROPN
ma-234	7	7	q	q	NOUN
ma-234	7	8	,	,	PUNCT
ma-234	7	9	p	p	PRON
ma-234	7	10	some	some	DET
ma-234	7	11	approximationresults	approximationresult	NOUN
ma-234	7	12	by	by	ADP
ma-234	7	13	compactly	compactly	ADV
ma-234	7	14	supported	support	VERB
ma-234	7	15	and/or	and/or	CCONJ
ma-234	7	16	regular	regular	ADJ
ma-234	7	17	functions	function	NOUN
ma-234	7	18	.	.	PUNCT
ma-234	8	1	as	as	ADP
ma-234	8	2	an	an	DET
ma-234	8	3	application	application	NOUN
ma-234	8	4	of	of	ADP
ma-234	8	5	these	these	DET
ma-234	8	6	results	result	NOUN
ma-234	8	7	,	,	PUNCT
ma-234	8	8	weobtain	weobtain	VERB
ma-234	8	9	an	an	DET
ma-234	8	10	explicit	explicit	ADJ
ma-234	8	11	solution	solution	NOUN
ma-234	8	12	in	in	ADP
ma-234	8	13	[	[	PUNCT
ma-234	8	14	lp(rd)]d	lp(rd)]d	NOUN
ma-234	8	15	of	of	ADP
ma-234	8	16	the	the	DET
ma-234	8	17	equation	equation	NOUN
ma-234	8	18	divf	divf	NOUN
ma-234	8	19	=	=	PUNCT
ma-234	9	1	f	f	PROPN
ma-234	9	2	whenever	whenever	SCONJ
ma-234	9	3	f	f	PROPN
ma-234	9	4	is	be	AUX
ma-234	9	5	inmα	inmα	ADV
ma-234	9	6	q	q	ADJ
ma-234	9	7	,	,	PUNCT
ma-234	9	8	p	p	X
ma-234	9	9	,	,	PUNCT
ma-234	9	10	with	with	ADP
ma-234	9	11	d	d	PROPN
ma-234	9	12	≥	≥	NUM
ma-234	9	13	3	3	NUM
ma-234	9	14	,	,	PUNCT
ma-234	9	15	1	1	NUM
ma-234	9	16	≤	≤	NUM
ma-234	9	17	q	q	ADJ
ma-234	9	18	≤	≤	NUM
ma-234	9	19	α	α	NOUN
ma-234	9	20	<	<	X
ma-234	9	21	d	d	PROPN
ma-234	9	22	and	and	CCONJ
ma-234	9	23	1	1	NUM
ma-234	9	24	p	p	NOUN
ma-234	10	1	=	=	NOUN
ma-234	10	2	1	1	NUM
ma-234	10	3	α	α	NOUN
ma-234	10	4	−	−	NOUN
ma-234	10	5	1	1	NUM
ma-234	10	6	d	d	NOUN
ma-234	10	7	.	.	PUNCT
ma-234	11	1	1	1	X
ma-234	11	2	.	.	X
ma-234	11	3	introduction	introduction	NOUN
ma-234	11	4	let	let	VERB
ma-234	11	5	d	d	PRON
ma-234	11	6	be	be	AUX
ma-234	11	7	a	a	DET
ma-234	11	8	fixed	fix	VERB
ma-234	11	9	positive	positive	ADJ
ma-234	11	10	integer	integer	NOUN
ma-234	11	11	.	.	PUNCT
ma-234	12	1	rd	rd	NOUN
ma-234	12	2	is	be	AUX
ma-234	12	3	equipped	equip	VERB
ma-234	12	4	with	with	ADP
ma-234	12	5	its	its	PRON
ma-234	12	6	usual	usual	ADJ
ma-234	12	7	hilbert	hilbert	NOUN
ma-234	12	8	space	space	NOUN
ma-234	12	9	structure	structure	NOUN
ma-234	12	10	and	and	CCONJ
ma-234	12	11	theeuclidean	theeuclidean	ADJ
ma-234	12	12	norm	norm	NOUN
ma-234	12	13	of	of	ADP
ma-234	12	14	any	any	DET
ma-234	12	15	element	element	NOUN
ma-234	12	16	x	x	PUNCT
ma-234	12	17	of	of	ADP
ma-234	12	18	rd	rd	PROPN
ma-234	12	19	is	be	AUX
ma-234	12	20	denoted	denote	VERB
ma-234	12	21	by	by	ADP
ma-234	12	22	|x	|x	PROPN
ma-234	12	23	|.recall	|.recall	PROPN
ma-234	12	24	that	that	SCONJ
ma-234	12	25	the	the	DET
ma-234	12	26	classical	classical	ADJ
ma-234	12	27	lebesgue	lebesgue	NOUN
ma-234	12	28	space	space	NOUN
ma-234	12	29	lq	lq	VERB
ma-234	12	30	:	:	PUNCT
ma-234	12	31	=	=	SYM
ma-234	12	32	lq(rd	lq(rd	PROPN
ma-234	12	33	)	)	PUNCT
ma-234	12	34	,	,	PUNCT
ma-234	12	35	with	with	ADP
ma-234	12	36	q	q	PROPN
ma-234	12	37	∈	∈	PROPN
ma-234	13	1	[	[	X
ma-234	13	2	1,∞	1,∞	NUM
ma-234	13	3	]	]	PUNCT
ma-234	13	4	,	,	PUNCT
ma-234	13	5	is	be	AUX
ma-234	13	6	defined	define	VERB
ma-234	13	7	to	to	PART
ma-234	13	8	be	be	AUX
ma-234	13	9	theset	theset	VERB
ma-234	13	10	of	of	ADP
ma-234	13	11	all	all	DET
ma-234	13	12	measurable	measurable	ADJ
ma-234	13	13	complex	complex	ADJ
ma-234	13	14	functions	function	NOUN
ma-234	13	15	f	f	PROPN
ma-234	13	16	on	on	ADP
ma-234	13	17	rd	rd	NOUN
ma-234	14	1	such	such	ADJ
ma-234	14	2	that	that	SCONJ
ma-234	14	3	‖f	‖f	PRON
ma-234	14	4	‖q	‖q	NOUN
ma-234	14	5	:	:	PUNCT
ma-234	14	6	=	=	PUNCT
ma-234	15	1	[	[	X
ma-234	15	2	∫	∫	PROPN
ma-234	15	3	rd	rd	PROPN
ma-234	15	4	|f	|f	PROPN
ma-234	15	5	(	(	PUNCT
ma-234	15	6	x)|q	x)|q	PROPN
ma-234	15	7	dx	dx	PROPN
ma-234	15	8	]	]	PUNCT
ma-234	15	9	1	1	NUM
ma-234	15	10	q	q	NOUN
ma-234	15	11	<	<	X
ma-234	15	12	∞	∞	NOUN
ma-234	15	13	with	with	ADP
ma-234	15	14	the	the	DET
ma-234	15	15	usual	usual	ADJ
ma-234	15	16	modification	modification	NOUN
ma-234	15	17	made	make	VERB
ma-234	15	18	when	when	SCONJ
ma-234	15	19	q	q	PROPN
ma-234	15	20	=	=	AUX
ma-234	15	21	∞.	∞.	PROPN
ma-234	15	22	in	in	ADP
ma-234	15	23	what	what	PRON
ma-234	15	24	follows	follow	VERB
ma-234	15	25	,	,	PUNCT
ma-234	15	26	|e|	|e|	PRON
ma-234	15	27	and	and	CCONJ
ma-234	15	28	χe	χe	PROPN
ma-234	15	29	denote	denote	VERB
ma-234	15	30	the	the	DET
ma-234	15	31	lebesguemeasure	lebesguemeasure	NOUN
ma-234	15	32	and	and	CCONJ
ma-234	15	33	the	the	DET
ma-234	15	34	characteristic	characteristic	ADJ
ma-234	15	35	function	function	NOUN
ma-234	15	36	of	of	ADP
ma-234	15	37	any	any	DET
ma-234	15	38	measurable	measurable	ADJ
ma-234	15	39	set	set	NOUN
ma-234	15	40	e	e	PROPN
ma-234	15	41	⊂	⊂	PROPN
ma-234	15	42	rd	rd	PROPN
ma-234	15	43	,	,	PUNCT
ma-234	15	44	respectively	respectively	ADV
ma-234	15	45	.	.	PUNCT
ma-234	15	46	lqloc	lqloc	ADJ
ma-234	15	47	denotesthe	denotesthe	PROPN
ma-234	15	48	set	set	NOUN
ma-234	15	49	of	of	ADP
ma-234	15	50	all	all	DET
ma-234	15	51	measurable	measurable	ADJ
ma-234	15	52	complex	complex	ADJ
ma-234	15	53	functions	function	NOUN
ma-234	15	54	f	f	PROPN
ma-234	15	55	on	on	ADP
ma-234	15	56	rd	rd	NOUN
ma-234	15	57	such	such	ADJ
ma-234	15	58	that	that	SCONJ
ma-234	15	59	f	f	PROPN
ma-234	15	60	χk	χk	NOUN
ma-234	15	61	∈	∈	PROPN
ma-234	15	62	lq	lq	VERB
ma-234	15	63	for	for	ADP
ma-234	15	64	any	any	DET
ma-234	15	65	bounded	bounded	ADJ
ma-234	15	66	measurablesubset	measurablesubset	NOUN
ma-234	15	67	k	k	PROPN
ma-234	15	68	of	of	ADP
ma-234	15	69	rd	rd	PROPN
ma-234	15	70	.for	.for	ADP
ma-234	15	71	1	1	NUM
ma-234	15	72	≤	≤	PROPN
ma-234	15	73	q	q	NOUN
ma-234	15	74	,	,	PUNCT
ma-234	15	75	α	α	PROPN
ma-234	15	76	≤	≤	NOUN
ma-234	15	77	∞	∞	PROPN
ma-234	15	78	,	,	PUNCT
ma-234	16	1	the	the	DET
ma-234	16	2	morrey	morrey	PROPN
ma-234	16	3	spacemα	spacemα	PROPN
ma-234	16	4	q	q	X
ma-234	16	5	:	:	PUNCT
ma-234	16	6	=	=	SYM
ma-234	16	7	mα	mα	X
ma-234	16	8	q	q	NOUN
ma-234	16	9	(	(	PUNCT
ma-234	16	10	rd	rd	NOUN
ma-234	16	11	)	)	PUNCT
ma-234	16	12	is	be	AUX
ma-234	16	13	defined	define	VERB
ma-234	16	14	as	as	ADP
ma-234	16	15	the	the	DET
ma-234	16	16	set	set	NOUN
ma-234	16	17	of	of	ADP
ma-234	16	18	all	all	DET
ma-234	16	19	elements	element	NOUN
ma-234	16	20	f	f	PROPN
ma-234	16	21	of	of	ADP
ma-234	16	22	lqloc	lqloc	NOUN
ma-234	16	23	for	for	ADP
ma-234	16	24	which	which	PRON
ma-234	16	25	‖f	‖f	ADP
ma-234	17	1	‖mα	‖mα	NUM
ma-234	17	2	q	q	NOUN
ma-234	17	3	:	:	PUNCT
ma-234	17	4	=	=	NUM
ma-234	17	5	sup	sup	NOUN
ma-234	17	6	x∈rd	x∈rd	PROPN
ma-234	17	7	,	,	PUNCT
ma-234	17	8	r>0	r>0	PROPN
ma-234	17	9	|q(x	|q(x	PROPN
ma-234	17	10	,	,	PUNCT
ma-234	17	11	r)|	r)|	NOUN
ma-234	17	12	1	1	NUM
ma-234	17	13	α	α	NOUN
ma-234	17	14	−	−	PROPN
ma-234	17	15	1	1	NUM
ma-234	17	16	q	q	NOUN
ma-234	17	17	∥∥f	∥∥f	PROPN
ma-234	17	18	χq(x	χq(x	ADP
ma-234	17	19	,	,	PUNCT
ma-234	17	20	r	r	NOUN
ma-234	17	21	)	)	PUNCT
ma-234	17	22	∥∥	∥∥	X
ma-234	17	23	q	q	X
ma-234	17	24	<	<	X
ma-234	17	25	∞	∞	PROPN
ma-234	17	26	,	,	PUNCT
ma-234	17	27	received	receive	VERB
ma-234	17	28	:	:	PUNCT
ma-234	17	29	20	20	NUM
ma-234	17	30	mar	mar	PROPN
ma-234	17	31	2024	2024	NUM
ma-234	17	32	.	.	PUNCT
ma-234	18	1	key	key	ADJ
ma-234	18	2	words	word	NOUN
ma-234	18	3	and	and	CCONJ
ma-234	18	4	phrases	phrase	NOUN
ma-234	18	5	.	.	PUNCT
ma-234	19	1	bourgain	bourgain	NOUN
ma-234	19	2	-	-	PUNCT
ma-234	19	3	morrey	morrey	NOUN
ma-234	19	4	spaces	space	NOUN
ma-234	19	5	;	;	PUNCT
ma-234	19	6	morrey	morrey	NOUN
ma-234	19	7	-	-	PUNCT
ma-234	19	8	type	type	NOUN
ma-234	19	9	space	space	NOUN
ma-234	19	10	;	;	PUNCT
ma-234	19	11	maximal	maximal	ADJ
ma-234	19	12	operator	operator	NOUN
ma-234	19	13	;	;	PUNCT
ma-234	19	14	hardy	hardy	ADJ
ma-234	19	15	-	-	PUNCT
ma-234	19	16	littlewood	littlewood	NOUN
ma-234	19	17	-	-	PUNCT
ma-234	19	18	sobolevtheorem	sobolevtheorem	NOUN
ma-234	19	19	;	;	PUNCT
ma-234	19	20	approximation	approximation	NOUN
ma-234	19	21	;	;	PUNCT
ma-234	19	22	divergence	divergence	NOUN
ma-234	19	23	equation	equation	NOUN
ma-234	19	24	.	.	PUNCT
ma-234	20	1	1	1	NUM
ma-234	21	1	https://adac.ee	https://adac.ee	PROPN
ma-234	21	2	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	21	3	https://orcid.org/0000-0003-1123-506x	https://orcid.org/0000-0003-1123-506x	PROPN
ma-234	21	4	eur	eur	PROPN
ma-234	21	5	.	.	PUNCT
ma-234	22	1	j.	j.	PROPN
ma-234	22	2	math	math	PROPN
ma-234	22	3	.	.	PUNCT
ma-234	23	1	anal	anal	PROPN
ma-234	23	2	.	.	PUNCT
ma-234	24	1	10.28924	10.28924	NUM
ma-234	24	2	/	/	SYM
ma-234	24	3	ada	ada	PROPN
ma-234	24	4	/	/	SYM
ma-234	24	5	ma.4.16	ma.4.16	PROPN
ma-234	24	6	2where	2where	NUM
ma-234	24	7	q(x	q(x	PROPN
ma-234	24	8	,	,	PUNCT
ma-234	24	9	r	r	NOUN
ma-234	24	10	)	)	PUNCT
ma-234	24	11	=	=	PUNCT
ma-234	25	1	d∏	d∏	PROPN
ma-234	25	2	j=1	j=1	PROPN
ma-234	25	3	[	[	PUNCT
ma-234	25	4	xj	xj	PROPN
ma-234	25	5	−	−	PROPN
ma-234	25	6	r	r	NOUN
ma-234	25	7	2	2	NUM
ma-234	25	8	,	,	PUNCT
ma-234	25	9	xj	xj	PROPN
ma-234	25	10	+	+	CCONJ
ma-234	25	11	r	r	NOUN
ma-234	25	12	2	2	NUM
ma-234	25	13	)	)	PUNCT
ma-234	25	14	,	,	PUNCT
ma-234	25	15	x	x	PUNCT
ma-234	25	16	=	=	PRON
ma-234	25	17	(	(	PUNCT
ma-234	25	18	x1	x1	PROPN
ma-234	25	19	,	,	PUNCT
ma-234	25	20	x2	x2	PROPN
ma-234	25	21	,	,	PUNCT
ma-234	25	22	...	...	PUNCT
ma-234	25	23	,	,	PUNCT
ma-234	25	24	xd	xd	INTJ
ma-234	25	25	)	)	PUNCT
ma-234	25	26	∈	∈	PROPN
ma-234	25	27	rd	rd	PROPN
ma-234	25	28	and	and	CCONJ
ma-234	25	29	0	0	NUM
ma-234	26	1	<	<	X
ma-234	26	2	r	r	NOUN
ma-234	26	3	<	<	X
ma-234	26	4	∞.	∞.	PROPN
ma-234	26	5	morrey	morrey	NOUN
ma-234	26	6	spaces	space	NOUN
ma-234	26	7	were	be	AUX
ma-234	26	8	introduced	introduce	VERB
ma-234	26	9	in	in	ADP
ma-234	26	10	1938	1938	NUM
ma-234	26	11	by	by	ADP
ma-234	26	12	c.	c.	PROPN
ma-234	26	13	morrey	morrey	PROPN
ma-234	27	1	[	[	X
ma-234	27	2	11	11	NUM
ma-234	27	3	]	]	PUNCT
ma-234	27	4	in	in	ADP
ma-234	27	5	order	order	NOUN
ma-234	27	6	to	to	PART
ma-234	27	7	study	study	VERB
ma-234	27	8	both	both	CCONJ
ma-234	27	9	the	the	DET
ma-234	27	10	regularityproblem	regularityproblem	NOUN
ma-234	27	11	of	of	ADP
ma-234	27	12	solutions	solution	NOUN
ma-234	27	13	for	for	ADP
ma-234	27	14	quasi	quasi	ADJ
ma-234	27	15	-	-	ADJ
ma-234	27	16	linear	linear	ADJ
ma-234	27	17	elliptic	elliptic	ADJ
ma-234	27	18	partial	partial	ADJ
ma-234	27	19	differential	differential	NOUN
ma-234	27	20	equations	equation	NOUN
ma-234	27	21	and	and	CCONJ
ma-234	27	22	the	the	DET
ma-234	27	23	calculus	calculus	NOUN
ma-234	27	24	of	of	ADP
ma-234	27	25	vari	vari	NOUN
ma-234	27	26	-	-	PUNCT
ma-234	27	27	ations	ation	NOUN
ma-234	27	28	.	.	PUNCT
ma-234	28	1	note	note	VERB
ma-234	28	2	that	that	SCONJ
ma-234	28	3	,	,	PUNCT
ma-234	28	4	for	for	ADP
ma-234	28	5	1	1	NUM
ma-234	28	6	≤	≤	NUM
ma-234	28	7	q	q	PROPN
ma-234	28	8	≤	≤	NUM
ma-234	28	9	α	α	NOUN
ma-234	28	10	≤	≤	NUM
ma-234	28	11	∞	∞	PROPN
ma-234	28	12	,	,	PUNCT
ma-234	28	13	lα	lα	ADJ
ma-234	28	14	is	be	AUX
ma-234	28	15	included	include	VERB
ma-234	28	16	in	in	ADP
ma-234	28	17	mα	mα	PROPN
ma-234	28	18	q	q	NOUN
ma-234	28	19	and	and	CCONJ
ma-234	28	20	the	the	DET
ma-234	28	21	inclusion	inclusion	NOUN
ma-234	28	22	is	be	AUX
ma-234	28	23	proper	proper	ADJ
ma-234	28	24	when	when	SCONJ
ma-234	28	25	q	q	X
ma-234	28	26	<	<	X
ma-234	28	27	α	α	X
ma-234	28	28	<	<	X
ma-234	28	29	∞.	∞.	PROPN
ma-234	28	30	moreover	moreover	ADV
ma-234	28	31	,	,	PUNCT
ma-234	28	32	morrey	morrey	PROPN
ma-234	28	33	spaces	space	NOUN
ma-234	28	34	describe	describe	VERB
ma-234	28	35	local	local	ADJ
ma-234	28	36	regularity	regularity	NOUN
ma-234	28	37	of	of	ADP
ma-234	28	38	functions	function	NOUN
ma-234	28	39	more	more	ADV
ma-234	28	40	precisely	precisely	ADV
ma-234	28	41	thanlebesgue	thanlebesgue	NOUN
ma-234	28	42	spaces	space	NOUN
ma-234	28	43	.	.	PUNCT
ma-234	29	1	however	however	ADV
ma-234	29	2	,	,	PUNCT
ma-234	29	3	some	some	DET
ma-234	29	4	nice	nice	ADJ
ma-234	29	5	and	and	CCONJ
ma-234	29	6	useful	useful	ADJ
ma-234	29	7	properties	property	NOUN
ma-234	29	8	of	of	ADP
ma-234	29	9	lα	lα	NOUN
ma-234	29	10	are	be	AUX
ma-234	29	11	not	not	PART
ma-234	29	12	shared	share	VERB
ma-234	29	13	by	by	ADP
ma-234	29	14	mα	mα	PROPN
ma-234	29	15	q	q	PROPN
ma-234	29	16	when	when	SCONJ
ma-234	29	17	1	1	NUM
ma-234	29	18	≤	≤	NOUN
ma-234	29	19	q	q	NOUN
ma-234	29	20	<	<	X
ma-234	29	21	α	α	X
ma-234	29	22	<	<	X
ma-234	29	23	∞.	∞.	PROPN
ma-234	29	24	for	for	ADP
ma-234	29	25	example	example	NOUN
ma-234	29	26	,	,	PUNCT
ma-234	29	27	in	in	ADP
ma-234	29	28	this	this	DET
ma-234	29	29	case	case	NOUN
ma-234	29	30	,	,	PUNCT
ma-234	29	31	the	the	DET
ma-234	29	32	set	set	NOUN
ma-234	29	33	of	of	ADP
ma-234	29	34	all	all	PRON
ma-234	29	35	compactly	compactly	ADV
ma-234	29	36	supported	support	VERB
ma-234	29	37	and/or	and/or	CCONJ
ma-234	29	38	regularelements	regularelement	NOUN
ma-234	29	39	is	be	AUX
ma-234	29	40	not	not	PART
ma-234	29	41	dense	dense	ADJ
ma-234	29	42	in	in	ADP
ma-234	29	43	mα	mα	PROPN
ma-234	29	44	q	q	NOUN
ma-234	29	45	.	.	PUNCT
ma-234	30	1	because	because	SCONJ
ma-234	30	2	of	of	ADP
ma-234	30	3	this	this	DET
ma-234	30	4	unpleasant	unpleasant	ADJ
ma-234	30	5	issue	issue	NOUN
ma-234	30	6	,	,	PUNCT
ma-234	30	7	several	several	ADJ
ma-234	30	8	distinguished	distinguished	ADJ
ma-234	30	9	linear	linear	PROPN
ma-234	30	10	sub	sub	NOUN
ma-234	30	11	-	-	NOUN
ma-234	30	12	spaces	space	NOUN
ma-234	30	13	of	of	ADP
ma-234	30	14	morrey	morrey	PROPN
ma-234	30	15	spaces	space	NOUN
ma-234	30	16	have	have	AUX
ma-234	30	17	been	be	AUX
ma-234	30	18	considered	consider	VERB
ma-234	30	19	for	for	ADP
ma-234	30	20	their	their	PRON
ma-234	30	21	easy	easy	ADJ
ma-234	30	22	use	use	NOUN
ma-234	30	23	in	in	ADP
ma-234	30	24	harmonic	harmonic	ADJ
ma-234	30	25	analysis	analysis	NOUN
ma-234	30	26	,	,	PUNCT
ma-234	30	27	specificallyin	specificallyin	NOUN
ma-234	30	28	boundedness	boundedness	PROPN
ma-234	30	29	problem	problem	NOUN
ma-234	30	30	of	of	ADP
ma-234	30	31	classical	classical	ADJ
ma-234	30	32	operators.the	operators.the	PRON
ma-234	30	33	present	present	ADJ
ma-234	30	34	paper	paper	NOUN
ma-234	30	35	focuses	focus	VERB
ma-234	30	36	on	on	ADP
ma-234	30	37	bourgain	bourgain	NOUN
ma-234	30	38	-	-	PUNCT
ma-234	30	39	morrey	morrey	PROPN
ma-234	30	40	spacesmα	spacesmα	PROPN
ma-234	30	41	q	q	PROPN
ma-234	30	42	,	,	PUNCT
ma-234	30	43	p	p	NOUN
ma-234	30	44	with	with	ADP
ma-234	30	45	1	1	NUM
ma-234	30	46	≤	≤	NUM
ma-234	30	47	q	q	NOUN
ma-234	30	48	,	,	PUNCT
ma-234	30	49	α	α	X
ma-234	30	50	,	,	PUNCT
ma-234	30	51	p	p	ADJ
ma-234	30	52	≤	≤	PUNCT
ma-234	30	53	∞.	∞.	PROPN
ma-234	30	54	recall	recall	VERB
ma-234	30	55	that	that	SCONJ
ma-234	30	56	,	,	PUNCT
ma-234	30	57	a	a	DET
ma-234	30	58	special	special	ADJ
ma-234	30	59	case	case	NOUN
ma-234	30	60	of	of	ADP
ma-234	30	61	these	these	DET
ma-234	30	62	spaces	space	NOUN
ma-234	30	63	was	be	AUX
ma-234	30	64	first	first	ADV
ma-234	30	65	introduced	introduce	VERB
ma-234	30	66	by	by	ADP
ma-234	30	67	bourgain	bourgain	NOUN
ma-234	31	1	[	[	X
ma-234	31	2	2	2	X
ma-234	31	3	]	]	PUNCT
ma-234	31	4	in	in	ADP
ma-234	31	5	1991	1991	NUM
ma-234	31	6	in	in	ADP
ma-234	31	7	order	order	NOUN
ma-234	31	8	to	to	PART
ma-234	31	9	study	study	VERB
ma-234	31	10	thestein	thestein	ADJ
ma-234	31	11	-	-	PUNCT
ma-234	31	12	tomas	toma	NOUN
ma-234	31	13	estimate	estimate	NOUN
ma-234	31	14	.	.	PUNCT
ma-234	32	1	later	later	ADV
ma-234	32	2	on	on	ADV
ma-234	32	3	,	,	PUNCT
ma-234	32	4	bourgain	bourgain	NOUN
ma-234	32	5	-	-	PUNCT
ma-234	32	6	morrey	morrey	NOUN
ma-234	32	7	spaces	space	NOUN
ma-234	32	8	have	have	AUX
ma-234	32	9	been	be	AUX
ma-234	32	10	used	use	VERB
ma-234	32	11	fruitfully	fruitfully	ADV
ma-234	32	12	in	in	ADP
ma-234	32	13	the	the	DET
ma-234	32	14	studyof	studyof	ADJ
ma-234	32	15	fourier	fourier	NOUN
ma-234	32	16	restriction	restriction	NOUN
ma-234	32	17	,	,	PUNCT
ma-234	32	18	multipliers	multiplier	VERB
ma-234	32	19	problems	problem	NOUN
ma-234	32	20	and	and	CCONJ
ma-234	32	21	partial	partial	ADJ
ma-234	32	22	differential	differential	NOUN
ma-234	32	23	equations	equation	NOUN
ma-234	32	24	,	,	PUNCT
ma-234	32	25	and	and	CCONJ
ma-234	32	26	in	in	ADP
ma-234	32	27	the	the	DET
ma-234	32	28	proof	proof	ADJ
ma-234	32	29	ofrefinements	ofrefinement	NOUN
ma-234	32	30	of	of	ADP
ma-234	32	31	strichartz	strichartz	NOUN
ma-234	32	32	inequality	inequality	NOUN
ma-234	32	33	(	(	PUNCT
ma-234	32	34	see	see	VERB
ma-234	32	35	[	[	X
ma-234	32	36	8–10	8–10	NOUN
ma-234	32	37	]	]	PUNCT
ma-234	32	38	and	and	CCONJ
ma-234	32	39	the	the	DET
ma-234	32	40	references	reference	NOUN
ma-234	32	41	therein	therein	ADV
ma-234	32	42	)	)	PUNCT
ma-234	32	43	.	.	PUNCT
ma-234	33	1	they	they	PRON
ma-234	33	2	are	be	AUX
ma-234	33	3	defined	define	VERB
ma-234	33	4	asfollows	asfollow	NOUN
ma-234	33	5	.	.	PUNCT
ma-234	34	1	definition	definition	NOUN
ma-234	34	2	1.1	1.1	NUM
ma-234	34	3	.	.	PUNCT
ma-234	35	1	let	let	VERB
ma-234	35	2	1	1	NUM
ma-234	35	3	≤	≤	NOUN
ma-234	35	4	q	q	ADJ
ma-234	35	5	,	,	PUNCT
ma-234	35	6	α	α	X
ma-234	35	7	,	,	PUNCT
ma-234	35	8	p	p	NOUN
ma-234	35	9	≤	≤	PUNCT
ma-234	35	10	∞.	∞.	PROPN
ma-234	35	11	the	the	DET
ma-234	35	12	bourgain	bourgain	NOUN
ma-234	35	13	-	-	PUNCT
ma-234	35	14	morrey	morrey	NOUN
ma-234	35	15	space	space	NOUN
ma-234	35	16	mα	mα	PROPN
ma-234	35	17	q	q	NOUN
ma-234	35	18	,	,	PUNCT
ma-234	35	19	p	p	X
ma-234	35	20	:	:	PUNCT
ma-234	35	21	=	=	PUNCT
ma-234	35	22	mα	mα	PROPN
ma-234	35	23	q	q	ADJ
ma-234	35	24	,	,	PUNCT
ma-234	35	25	p(rd	p(rd	ADJ
ma-234	35	26	)	)	PUNCT
ma-234	35	27	is	be	AUX
ma-234	35	28	defined	define	VERB
ma-234	35	29	as	as	ADP
ma-234	35	30	the	the	DET
ma-234	35	31	set	set	NOUN
ma-234	35	32	of	of	ADP
ma-234	35	33	all	all	DET
ma-234	35	34	f	f	PROPN
ma-234	35	35	∈	∈	PROPN
ma-234	35	36	lqloc	lqloc	NOUN
ma-234	35	37	for	for	ADP
ma-234	35	38	which	which	PRON
ma-234	35	39	‖f	‖f	ADP
ma-234	35	40	‖mα	‖mα	NUM
ma-234	35	41	q	q	NOUN
ma-234	35	42	,	,	PUNCT
ma-234	35	43	p	p	X
ma-234	35	44	:	:	PUNCT
ma-234	35	45	=	=	SYM
ma-234	35	46	∥∥∥∥{|qk	∥∥∥∥{|qk	PROPN
ma-234	35	47	,	,	PUNCT
ma-234	35	48	m|	m|	NOUN
ma-234	35	49	1	1	NUM
ma-234	35	50	α	α	NOUN
ma-234	35	51	−	−	NOUN
ma-234	35	52	1	1	NUM
ma-234	35	53	q	q	NOUN
ma-234	35	54	∥∥f	∥∥f	NOUN
ma-234	35	55	χqk	χqk	NOUN
ma-234	35	56	,	,	PUNCT
ma-234	35	57	m∥∥q}(k	m∥∥q}(k	PROPN
ma-234	35	58	,	,	PUNCT
ma-234	35	59	m)∈zd×z	m)∈zd×z	NOUN
ma-234	35	60	∥∥∥∥	∥∥∥∥	PUNCT
ma-234	35	61	`	`	PUNCT
ma-234	35	62	p	p	X
ma-234	35	63	<	<	X
ma-234	35	64	∞	∞	PROPN
ma-234	35	65	,	,	PUNCT
ma-234	35	66	where	where	SCONJ
ma-234	35	67	the	the	DET
ma-234	35	68	sets	set	NOUN
ma-234	35	69	qk	qk	VERB
ma-234	35	70	,	,	PUNCT
ma-234	35	71	m	m	VERB
ma-234	36	1	=	=	PUNCT
ma-234	36	2	d∏	d∏	PROPN
ma-234	36	3	j=1	j=1	PROPN
ma-234	36	4	[	[	PUNCT
ma-234	36	5	kj2	kj2	NOUN
ma-234	36	6	m	m	PRON
ma-234	36	7	,	,	PUNCT
ma-234	36	8	(	(	PUNCT
ma-234	36	9	kj	kj	NOUN
ma-234	36	10	+	+	NOUN
ma-234	36	11	1	1	NUM
ma-234	36	12	)	)	PUNCT
ma-234	36	13	2	2	NUM
ma-234	36	14	m	m	NOUN
ma-234	36	15	)	)	PUNCT
ma-234	36	16	,	,	PUNCT
ma-234	36	17	k	k	X
ma-234	36	18	=	=	PRON
ma-234	36	19	(	(	PUNCT
ma-234	36	20	k1	k1	PROPN
ma-234	36	21	,	,	PUNCT
ma-234	36	22	k2	k2	NOUN
ma-234	36	23	,	,	PUNCT
ma-234	36	24	...	...	PUNCT
ma-234	36	25	,	,	PUNCT
ma-234	36	26	kd	kd	PROPN
ma-234	36	27	)	)	PUNCT
ma-234	36	28	∈	∈	PROPN
ma-234	36	29	zd	zd	PROPN
ma-234	36	30	,	,	PUNCT
ma-234	36	31	m	m	VERB
ma-234	36	32	∈	∈	PROPN
ma-234	36	33	z	z	NOUN
ma-234	36	34	are	be	AUX
ma-234	36	35	the	the	DET
ma-234	36	36	usual	usual	ADJ
ma-234	36	37	dyadic	dyadic	ADJ
ma-234	36	38	cubes	cube	NOUN
ma-234	36	39	of	of	ADP
ma-234	36	40	rd	rd	PROPN
ma-234	36	41	and	and	CCONJ
ma-234	36	42	for	for	ADP
ma-234	36	43	any	any	DET
ma-234	36	44	sequence	sequence	NOUN
ma-234	36	45	{	{	PUNCT
ma-234	36	46	ai}i∈i	ai}i∈i	CCONJ
ma-234	36	47	included	include	VERB
ma-234	36	48	in	in	ADP
ma-234	36	49	c	c	PROPN
ma-234	36	50	,	,	PUNCT
ma-234	36	51	‖{ai}i∈i‖`p	‖{ai}i∈i‖`p	NUM
ma-234	36	52	:	:	PUNCT
ma-234	36	53	=	=	SYM
ma-234	36	54			PROPN
ma-234	36	55	(	(	PUNCT
ma-234	36	56	∑	∑	ADP
ma-234	36	57	i∈i	i∈i	ADJ
ma-234	36	58	|ai	|ai	X
ma-234	36	59	|p	|p	X
ma-234	36	60	)	)	PUNCT
ma-234	36	61	1	1	NUM
ma-234	36	62	p	p	NOUN
ma-234	36	63	if	if	SCONJ
ma-234	36	64	p	p	PROPN
ma-234	36	65	<	<	X
ma-234	36	66	∞	∞	NUM
ma-234	36	67	sup	sup	NOUN
ma-234	36	68	i∈i	i∈i	ADJ
ma-234	36	69	|ai	|ai	NUM
ma-234	36	70	|	|	ADV
ma-234	36	71	if	if	SCONJ
ma-234	36	72	p	p	PROPN
ma-234	36	73	=	=	NOUN
ma-234	36	74	∞.	∞.	PROPN
ma-234	36	75	it	it	PRON
ma-234	36	76	is	be	AUX
ma-234	36	77	well	well	ADV
ma-234	36	78	known	know	VERB
ma-234	36	79	that	that	SCONJ
ma-234	36	80	,	,	PUNCT
ma-234	36	81	when	when	SCONJ
ma-234	36	82	1	1	NUM
ma-234	36	83	≤	≤	NOUN
ma-234	36	84	q	q	NOUN
ma-234	36	85	<	<	X
ma-234	36	86	α	α	X
ma-234	36	87	<	<	X
ma-234	36	88	p	p	X
ma-234	36	89	≤	≤	NUM
ma-234	36	90	∞	∞	PROPN
ma-234	36	91	,	,	PUNCT
ma-234	36	92	lα	lα	ADJ
ma-234	36	93	is	be	AUX
ma-234	36	94	properly	properly	ADV
ma-234	36	95	included	include	VERB
ma-234	36	96	in	in	ADP
ma-234	36	97	mα	mα	PROPN
ma-234	36	98	q	q	NOUN
ma-234	36	99	,	,	PUNCT
ma-234	36	100	p	p	X
ma-234	36	101	,	,	PUNCT
ma-234	36	102	which	which	PRON
ma-234	36	103	is	be	AUX
ma-234	36	104	alinear	alinear	ADJ
ma-234	36	105	subspace	subspace	NOUN
ma-234	36	106	of	of	ADP
ma-234	36	107	mα	mα	PROPN
ma-234	36	108	q	q	PROPN
ma-234	36	109	.	.	PUNCT
ma-234	37	1	actually	actually	ADV
ma-234	37	2	we	we	PRON
ma-234	37	3	have	have	VERB
ma-234	37	4	{	{	PUNCT
ma-234	37	5	lα	lα	PROPN
ma-234	37	6	⊂mα	⊂mα	X
ma-234	38	1	q	q	NOUN
ma-234	38	2	,	,	PUNCT
ma-234	38	3	p	p	NOUN
ma-234	38	4	⊂mα	⊂mα	X
ma-234	38	5	q	q	NOUN
ma-234	38	6	,	,	PUNCT
ma-234	38	7	p1	p1	NOUN
ma-234	38	8	⊂mα	⊂mα	X
ma-234	38	9	q,∞	q,∞	PROPN
ma-234	38	10	=	=	PROPN
ma-234	38	11	mα	mα	PROPN
ma-234	38	12	q	q	NOUN
ma-234	38	13	,	,	PUNCT
ma-234	38	14	1	1	NUM
ma-234	38	15	≤	≤	NOUN
ma-234	39	1	q	q	NOUN
ma-234	39	2	<	<	X
ma-234	39	3	α	α	X
ma-234	39	4	<	<	X
ma-234	39	5	p	p	X
ma-234	39	6	≤	≤	ADJ
ma-234	39	7	p1	p1	NOUN
ma-234	39	8	≤	≤	PUNCT
ma-234	39	9	∞.	∞.	PROPN
ma-234	39	10	mα	mα	PROPN
ma-234	40	1	q	q	NOUN
ma-234	40	2	,	,	PUNCT
ma-234	40	3	p	p	NOUN
ma-234	40	4	⊂mα	⊂mα	ADP
ma-234	40	5	q1,p	q1,p	PROPN
ma-234	40	6	,	,	PUNCT
ma-234	40	7	1	1	NUM
ma-234	40	8	≤	≤	NUM
ma-234	40	9	q1	q1	PROPN
ma-234	40	10	≤	≤	NUM
ma-234	40	11	q	q	PROPN
ma-234	40	12	≤	≤	NUM
ma-234	40	13	α	α	NOUN
ma-234	40	14	≤	≤	NOUN
ma-234	40	15	p	p	PROPN
ma-234	40	16	≤	≤	ADJ
ma-234	40	17	∞.	∞.	PROPN
ma-234	40	18	(	(	PUNCT
ma-234	40	19	1	1	NUM
ma-234	40	20	)	)	PUNCT
ma-234	40	21	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	40	22	eur	eur	PROPN
ma-234	40	23	.	.	PUNCT
ma-234	41	1	j.	j.	PROPN
ma-234	41	2	math	math	PROPN
ma-234	41	3	.	.	PUNCT
ma-234	42	1	anal	anal	PROPN
ma-234	42	2	.	.	PUNCT
ma-234	43	1	10.28924	10.28924	NUM
ma-234	43	2	/	/	SYM
ma-234	43	3	ada	ada	PROPN
ma-234	43	4	/	/	SYM
ma-234	43	5	ma.4.16	ma.4.16	PROPN
ma-234	43	6	3many	3many	NUM
ma-234	43	7	useful	useful	ADJ
ma-234	43	8	results	result	NOUN
ma-234	43	9	,	,	PUNCT
ma-234	43	10	well	well	INTJ
ma-234	43	11	known	know	VERB
ma-234	43	12	for	for	ADP
ma-234	43	13	lebesgue	lebesgue	NOUN
ma-234	43	14	or	or	CCONJ
ma-234	43	15	morrey	morrey	PROPN
ma-234	43	16	spaces	space	NOUN
ma-234	43	17	,	,	PUNCT
ma-234	43	18	have	have	AUX
ma-234	43	19	been	be	AUX
ma-234	43	20	extended	extend	VERB
ma-234	43	21	to	to	ADP
ma-234	43	22	thesetting	thesetting	NOUN
ma-234	43	23	of	of	ADP
ma-234	43	24	bourgain	bourgain	NOUN
ma-234	43	25	-	-	PUNCT
ma-234	43	26	morrey	morrey	NOUN
ma-234	43	27	spaces	space	NOUN
ma-234	43	28	(	(	PUNCT
ma-234	43	29	see	see	VERB
ma-234	43	30	[	[	X
ma-234	43	31	8	8	NUM
ma-234	43	32	,	,	PUNCT
ma-234	43	33	9	9	NUM
ma-234	43	34	]	]	PUNCT
ma-234	43	35	)	)	PUNCT
ma-234	43	36	.	.	PUNCT
ma-234	44	1	for	for	ADP
ma-234	44	2	instance	instance	NOUN
ma-234	44	3	,	,	PUNCT
ma-234	44	4	the	the	DET
ma-234	44	5	boundedness	boundedness	NOUN
ma-234	44	6	of	of	ADP
ma-234	44	7	some	some	DET
ma-234	44	8	classicaloperators	classicaloperator	NOUN
ma-234	44	9	on	on	ADP
ma-234	44	10	these	these	DET
ma-234	44	11	spaces	space	NOUN
ma-234	44	12	has	have	AUX
ma-234	44	13	been	be	AUX
ma-234	44	14	investigated	investigate	VERB
ma-234	44	15	by	by	ADP
ma-234	44	16	hatano	hatano	PROPN
ma-234	44	17	et	et	PROPN
ma-234	44	18	al	al	PROPN
ma-234	44	19	.	.	PUNCT
ma-234	45	1	[	[	X
ma-234	45	2	8].let	8].let	NUM
ma-234	45	3	0	0	NUM
ma-234	45	4	<	<	X
ma-234	45	5	γ	γ	X
ma-234	45	6	<	<	X
ma-234	45	7	1	1	NUM
ma-234	45	8	.	.	PUNCT
ma-234	45	9	define	define	VERB
ma-234	45	10	the	the	DET
ma-234	45	11	fractional	fractional	ADJ
ma-234	45	12	integral	integral	ADJ
ma-234	45	13	operator	operator	NOUN
ma-234	45	14	iγ	iγ	NOUN
ma-234	45	15	as	as	ADP
ma-234	45	16	iγf	iγf	NOUN
ma-234	45	17	(	(	PUNCT
ma-234	45	18	x	x	NOUN
ma-234	45	19	)	)	PUNCT
ma-234	45	20	=	=	SYM
ma-234	45	21	∫	∫	PROPN
ma-234	45	22	rd	rd	PROPN
ma-234	45	23	|x	|x	PROPN
ma-234	45	24	−	−	PROPN
ma-234	45	25	y	y	PROPN
ma-234	45	26	|d(γ−1)f	|d(γ−1)f	PROPN
ma-234	45	27	(	(	PUNCT
ma-234	45	28	y)dy	y)dy	PROPN
ma-234	45	29	when	when	SCONJ
ma-234	45	30	the	the	DET
ma-234	45	31	above	above	ADJ
ma-234	45	32	integral	integral	ADJ
ma-234	45	33	makes	make	VERB
ma-234	45	34	sense.recall	sense.recall	PRON
ma-234	45	35	that	that	SCONJ
ma-234	45	36	the	the	DET
ma-234	45	37	hardy	hardy	ADJ
ma-234	45	38	-	-	PUNCT
ma-234	45	39	littlewood	littlewood	NOUN
ma-234	45	40	-	-	PUNCT
ma-234	45	41	sobolev	sobolev	NOUN
ma-234	45	42	theorem	theorem	NOUN
ma-234	45	43	of	of	ADP
ma-234	45	44	fractional	fractional	ADJ
ma-234	45	45	integration	integration	NOUN
ma-234	45	46	is	be	AUX
ma-234	45	47	one	one	NUM
ma-234	45	48	of	of	ADP
ma-234	45	49	the	the	DET
ma-234	45	50	mostimportant	mostimportant	NOUN
ma-234	45	51	tools	tool	NOUN
ma-234	45	52	in	in	ADP
ma-234	45	53	the	the	DET
ma-234	45	54	study	study	NOUN
ma-234	45	55	of	of	ADP
ma-234	45	56	partial	partial	ADJ
ma-234	45	57	differential	differential	ADJ
ma-234	45	58	equations	equation	NOUN
ma-234	45	59	.	.	PUNCT
ma-234	46	1	it	it	PRON
ma-234	46	2	reads	read	VERB
ma-234	46	3	as	as	SCONJ
ma-234	46	4	follows	follow	VERB
ma-234	46	5	.	.	PUNCT
ma-234	47	1	theorem	theorem	VERB
ma-234	47	2	1.2	1.2	NUM
ma-234	47	3	.	.	PUNCT
ma-234	48	1	[	[	X
ma-234	48	2	14	14	NUM
ma-234	48	3	]	]	PUNCT
ma-234	48	4	let	let	VERB
ma-234	48	5	0	0	PUNCT
ma-234	48	6	<	<	X
ma-234	48	7	γ	γ	X
ma-234	48	8	<	<	X
ma-234	48	9	1	1	NUM
ma-234	48	10	α	α	NOUN
ma-234	48	11	≤	≤	NUM
ma-234	48	12	1	1	NUM
ma-234	48	13	and	and	CCONJ
ma-234	48	14	1	1	NUM
ma-234	48	15	p	p	NOUN
ma-234	48	16	=	=	NOUN
ma-234	48	17	1	1	NUM
ma-234	48	18	α	α	NOUN
ma-234	48	19	−	−	PROPN
ma-234	48	20	γ	γ	PROPN
ma-234	48	21	.	.	PUNCT
ma-234	49	1	then	then	ADV
ma-234	49	2	there	there	PRON
ma-234	49	3	is	be	VERB
ma-234	49	4	a	a	DET
ma-234	49	5	real	real	ADJ
ma-234	49	6	number	number	NOUN
ma-234	49	7	aα	aα	NOUN
ma-234	49	8	,	,	PUNCT
ma-234	49	9	γ	γ	PROPN
ma-234	49	10	such	such	ADJ
ma-234	49	11	that	that	PRON
ma-234	49	12	:	:	PUNCT
ma-234	49	13	(	(	PUNCT
ma-234	49	14	i	i	NOUN
ma-234	49	15	)	)	PUNCT
ma-234	50	1	when	when	SCONJ
ma-234	50	2	1	1	NUM
ma-234	50	3	<	<	X
ma-234	50	4	α	α	PROPN
ma-234	50	5	‖iγf	‖iγf	PROPN
ma-234	50	6	‖p	‖p	PROPN
ma-234	50	7	≤	≤	PROPN
ma-234	50	8	aα	aα	PROPN
ma-234	50	9	,	,	PUNCT
ma-234	50	10	γ‖f	γ‖f	ADJ
ma-234	50	11	‖α	‖α	PROPN
ma-234	50	12	,	,	PUNCT
ma-234	50	13	f	f	PROPN
ma-234	50	14	∈	∈	PROPN
ma-234	50	15	lα	lα	PROPN
ma-234	50	16	(	(	PUNCT
ma-234	50	17	2	2	NUM
ma-234	50	18	)	)	PUNCT
ma-234	50	19	(	(	PUNCT
ma-234	50	20	ii	ii	NOUN
ma-234	50	21	)	)	PUNCT
ma-234	50	22	when	when	SCONJ
ma-234	50	23	α	α	NOUN
ma-234	50	24	=	=	NOUN
ma-234	50	25	1	1	NUM
ma-234	50	26	‖f	‖f	NUM
ma-234	50	27	‖∗	‖∗	PUNCT
ma-234	50	28	1	1	NUM
ma-234	50	29	1−γ	1−γ	NUM
ma-234	50	30	,	,	PUNCT
ma-234	50	31	∞	∞	PROPN
ma-234	50	32	≤	≤	PROPN
ma-234	50	33	a1,γ‖f	a1,γ‖f	PROPN
ma-234	50	34	‖1	‖1	PROPN
ma-234	50	35	,	,	PUNCT
ma-234	50	36	f	f	PROPN
ma-234	50	37	∈	∈	PROPN
ma-234	50	38	l1	l1	PROPN
ma-234	50	39	.	.	PUNCT
ma-234	51	1	(	(	PUNCT
ma-234	51	2	3	3	X
ma-234	51	3	)	)	PUNCT
ma-234	51	4	recall	recall	NOUN
ma-234	51	5	that	that	SCONJ
ma-234	51	6	,	,	PUNCT
ma-234	51	7	for	for	ADP
ma-234	51	8	q	q	PROPN
ma-234	51	9	∈	∈	PROPN
ma-234	51	10	[	[	X
ma-234	51	11	1,∞	1,∞	NUM
ma-234	51	12	)	)	PUNCT
ma-234	51	13	,	,	PUNCT
ma-234	51	14	‖	‖	PROPN
ma-234	51	15	·	·	PUNCT
ma-234	51	16	‖∗q,∞	‖∗q,∞	PROPN
ma-234	51	17	denotes	denote	VERB
ma-234	51	18	the	the	DET
ma-234	51	19	quasi	quasi	NOUN
ma-234	51	20	-	-	NOUN
ma-234	51	21	norm	norm	NOUN
ma-234	51	22	of	of	ADP
ma-234	51	23	the	the	DET
ma-234	51	24	weak	weak	ADJ
ma-234	51	25	-	-	PUNCT
ma-234	51	26	lebesgue	lebesgue	NOUN
ma-234	51	27	wlq	wlq	PROPN
ma-234	51	28	definedby	definedby	ADV
ma-234	51	29	wlq	wlq	PROPN
ma-234	52	1	=	=	PRON
ma-234	52	2	{	{	PUNCT
ma-234	52	3	f	f	PROPN
ma-234	52	4	∈	∈	PROPN
ma-234	52	5	l1	l1	PROPN
ma-234	52	6	loc	loc	PROPN
ma-234	52	7	:	:	PUNCT
ma-234	52	8	‖f	‖f	ADP
ma-234	52	9	‖∗q,∞	‖∗q,∞	ADJ
ma-234	52	10	=	=	PUNCT
ma-234	52	11	sup	sup	NOUN
ma-234	52	12	λ>0	λ>0	NOUN
ma-234	52	13	λ	λ	PROPN
ma-234	52	14	∣∣{x	∣∣{x	PROPN
ma-234	52	15	∈	∈	PROPN
ma-234	52	16	rd	rd	PROPN
ma-234	52	17	:	:	PUNCT
ma-234	52	18	|f	|f	PROPN
ma-234	52	19	(	(	PUNCT
ma-234	52	20	x)|	x)|	PROPN
ma-234	52	21	>	>	X
ma-234	52	22	λ	λ	PROPN
ma-234	52	23	}	}	PUNCT
ma-234	52	24	∣∣	∣∣	X
ma-234	52	25	1	1	NUM
ma-234	52	26	q	q	NOUN
ma-234	52	27	<	<	X
ma-234	52	28	∞	∞	NUM
ma-234	52	29	}	}	PUNCT
ma-234	52	30	.	.	PUNCT
ma-234	53	1	the	the	DET
ma-234	53	2	first	first	ADJ
ma-234	53	3	aim	aim	NOUN
ma-234	53	4	of	of	ADP
ma-234	53	5	the	the	DET
ma-234	53	6	present	present	ADJ
ma-234	53	7	paper	paper	NOUN
ma-234	53	8	is	be	AUX
ma-234	53	9	to	to	PART
ma-234	53	10	establish	establish	VERB
ma-234	53	11	an	an	DET
ma-234	53	12	extension	extension	NOUN
ma-234	53	13	of	of	ADP
ma-234	53	14	the	the	DET
ma-234	53	15	above	above	ADJ
ma-234	53	16	useful	useful	ADJ
ma-234	53	17	theorem	theorem	ADJ
ma-234	53	18	tothe	tothe	NOUN
ma-234	53	19	setting	setting	NOUN
ma-234	53	20	of	of	ADP
ma-234	53	21	bourgain	bourgain	NOUN
ma-234	53	22	-	-	PUNCT
ma-234	53	23	morrey	morrey	NOUN
ma-234	53	24	spaces	space	NOUN
ma-234	53	25	.	.	PUNCT
ma-234	54	1	note	note	VERB
ma-234	54	2	that	that	SCONJ
ma-234	54	3	,	,	PUNCT
ma-234	54	4	our	our	PRON
ma-234	54	5	result	result	NOUN
ma-234	54	6	refines	refine	VERB
ma-234	54	7	that	that	PRON
ma-234	54	8	of	of	ADP
ma-234	54	9	hatano	hatano	PROPN
ma-234	54	10	et	et	PROPN
ma-234	54	11	al	al	PROPN
ma-234	54	12	.	.	PROPN
ma-234	54	13	,	,	PUNCT
ma-234	54	14	whichstates	whichstate	VERB
ma-234	54	15	that	that	PRON
ma-234	54	16	fractional	fractional	ADJ
ma-234	54	17	integral	integral	ADJ
ma-234	54	18	operators	operator	NOUN
ma-234	54	19	map	map	VERB
ma-234	54	20	bourgain	bourgain	NOUN
ma-234	54	21	-	-	PUNCT
ma-234	54	22	morrey	morrey	NOUN
ma-234	54	23	spaces	space	NOUN
ma-234	54	24	into	into	ADP
ma-234	54	25	the	the	DET
ma-234	54	26	same	same	ADJ
ma-234	54	27	type	type	NOUN
ma-234	54	28	spaces(see	spaces(see	NOUN
ma-234	54	29	[	[	X
ma-234	54	30	8	8	NUM
ma-234	54	31	,	,	PUNCT
ma-234	54	32	theorem	theorem	VERB
ma-234	54	33	4.4]).another	4.4]).another	NUM
ma-234	54	34	morrey	morrey	ADJ
ma-234	54	35	-	-	PUNCT
ma-234	54	36	type	type	NOUN
ma-234	54	37	space	space	NOUN
ma-234	54	38	considered	consider	VERB
ma-234	54	39	in	in	ADP
ma-234	54	40	this	this	DET
ma-234	54	41	paper	paper	NOUN
ma-234	54	42	is	be	AUX
ma-234	54	43	the	the	DET
ma-234	54	44	space	space	NOUN
ma-234	54	45	f(q	f(q	PROPN
ma-234	54	46	,	,	PUNCT
ma-234	54	47	p	p	X
ma-234	54	48	,	,	PUNCT
ma-234	54	49	α	α	NOUN
ma-234	54	50	)	)	PUNCT
ma-234	54	51	(	(	PUNCT
ma-234	54	52	1	1	NUM
ma-234	54	53	≤	≤	NUM
ma-234	54	54	q	q	NOUN
ma-234	54	55	,	,	PUNCT
ma-234	54	56	α	α	X
ma-234	54	57	,	,	PUNCT
ma-234	54	58	p	p	NOUN
ma-234	54	59	≤	≤	NUM
ma-234	54	60	∞),which	∞),which	NOUN
ma-234	54	61	arises	arise	VERB
ma-234	54	62	naturally	naturally	ADV
ma-234	54	63	in	in	ADP
ma-234	54	64	the	the	DET
ma-234	54	65	study	study	NOUN
ma-234	54	66	of	of	ADP
ma-234	54	67	boundedness	boundedness	NOUN
ma-234	54	68	properties	property	NOUN
ma-234	54	69	of	of	ADP
ma-234	54	70	fractional	fractional	ADJ
ma-234	54	71	integral	integral	ADJ
ma-234	54	72	operators.it	operators.it	NUM
ma-234	54	73	has	have	AUX
ma-234	54	74	been	be	AUX
ma-234	54	75	introduced	introduce	VERB
ma-234	54	76	in	in	ADP
ma-234	54	77	2015	2015	NUM
ma-234	54	78	by	by	ADP
ma-234	54	79	fofana	fofana	PROPN
ma-234	54	80	et	et	PROPN
ma-234	54	81	al	al	PROPN
ma-234	54	82	.	.	PUNCT
ma-234	55	1	[	[	X
ma-234	55	2	7	7	NUM
ma-234	55	3	]	]	PUNCT
ma-234	55	4	.	.	PUNCT
ma-234	56	1	note	note	VERB
ma-234	56	2	that	that	SCONJ
ma-234	56	3	,	,	PUNCT
ma-234	56	4	recently	recently	ADV
ma-234	56	5	in	in	ADP
ma-234	56	6	2020	2020	NUM
ma-234	56	7	,	,	PUNCT
ma-234	56	8	the	the	DET
ma-234	56	9	space	space	NOUN
ma-234	56	10	f(q	f(q	PROPN
ma-234	56	11	,	,	PUNCT
ma-234	56	12	p	p	X
ma-234	56	13	,	,	PUNCT
ma-234	56	14	α	α	NOUN
ma-234	56	15	)	)	PUNCT
ma-234	56	16	has	have	AUX
ma-234	56	17	been	be	AUX
ma-234	56	18	studied	study	VERB
ma-234	56	19	also	also	ADV
ma-234	56	20	in	in	ADP
ma-234	56	21	[	[	X
ma-234	56	22	15	15	NUM
ma-234	56	23	]	]	PUNCT
ma-234	56	24	,	,	PUNCT
ma-234	56	25	where	where	SCONJ
ma-234	56	26	it	it	PRON
ma-234	56	27	is	be	AUX
ma-234	56	28	called	call	VERB
ma-234	56	29	the	the	DET
ma-234	56	30	riesz	riesz	NOUN
ma-234	56	31	-	-	PUNCT
ma-234	56	32	morrey	morrey	NOUN
ma-234	56	33	space	space	NOUN
ma-234	56	34	and	and	CCONJ
ma-234	56	35	denoted	denote	VERB
ma-234	56	36	by	by	ADP
ma-234	56	37	rmp	rmp	PROPN
ma-234	56	38	,	,	PUNCT
ma-234	56	39	q	q	X
ma-234	56	40	,	,	PUNCT
ma-234	56	41	1	1	NUM
ma-234	56	42	p	p	NOUN
ma-234	56	43	−	−	PROPN
ma-234	56	44	1	1	NUM
ma-234	56	45	α	α	PROPN
ma-234	56	46	(	(	PUNCT
ma-234	56	47	rd	rd	NOUN
ma-234	56	48	)	)	PUNCT
ma-234	56	49	.	.	PUNCT
ma-234	57	1	it	it	PRON
ma-234	57	2	is	be	AUX
ma-234	57	3	defined	define	VERB
ma-234	57	4	as	as	ADP
ma-234	57	5	follows	follow	VERB
ma-234	57	6	.	.	PUNCT
ma-234	58	1	definition	definition	NOUN
ma-234	58	2	1.3	1.3	NUM
ma-234	58	3	.	.	PUNCT
ma-234	59	1	let	let	VERB
ma-234	59	2	1	1	NUM
ma-234	59	3	≤	≤	NOUN
ma-234	59	4	q	q	NOUN
ma-234	60	1	,	,	PUNCT
ma-234	60	2	p	p	X
ma-234	60	3	,	,	PUNCT
ma-234	60	4	α	α	PROPN
ma-234	60	5	≤	≤	PUNCT
ma-234	60	6	∞.	∞.	PROPN
ma-234	60	7	the	the	DET
ma-234	60	8	space	space	NOUN
ma-234	60	9	f(q	f(q	PROPN
ma-234	60	10	,	,	PUNCT
ma-234	60	11	p	p	X
ma-234	60	12	,	,	PUNCT
ma-234	60	13	α	α	NOUN
ma-234	60	14	)	)	PUNCT
ma-234	60	15	:	:	PUNCT
ma-234	60	16	=	=	PUNCT
ma-234	60	17	f(q	f(q	PROPN
ma-234	60	18	,	,	PUNCT
ma-234	60	19	p	p	X
ma-234	60	20	,	,	PUNCT
ma-234	60	21	α)(rd	α)(rd	NUM
ma-234	60	22	)	)	PUNCT
ma-234	60	23	is	be	AUX
ma-234	60	24	defined	define	VERB
ma-234	60	25	as	as	ADP
ma-234	60	26	the	the	DET
ma-234	60	27	set	set	NOUN
ma-234	60	28	of	of	ADP
ma-234	60	29	all	all	DET
ma-234	60	30	f	f	PROPN
ma-234	60	31	∈	∈	PROPN
ma-234	60	32	lqloc	lqloc	NOUN
ma-234	60	33	for	for	ADP
ma-234	60	34	which	which	PRON
ma-234	60	35	‖f	‖f	ADP
ma-234	60	36	‖f(q	‖f(q	NOUN
ma-234	60	37	,	,	PUNCT
ma-234	60	38	p	p	X
ma-234	60	39	,	,	PUNCT
ma-234	60	40	α	α	NOUN
ma-234	60	41	)	)	PUNCT
ma-234	60	42	is	be	AUX
ma-234	60	43	finite	finite	ADJ
ma-234	60	44	,	,	PUNCT
ma-234	60	45	where	where	SCONJ
ma-234	60	46	‖f	‖f	ADP
ma-234	60	47	‖f(q	‖f(q	NOUN
ma-234	60	48	,	,	PUNCT
ma-234	60	49	p	p	X
ma-234	60	50	,	,	PUNCT
ma-234	60	51	α	α	NOUN
ma-234	60	52	)	)	PUNCT
ma-234	60	53	=	=	SYM
ma-234	61	1			PRON
ma-234	61	2	sup	sup	INTJ
ma-234	61	3	{	{	PUNCT
ma-234	61	4	qi}∈p	qi}∈p	PROPN
ma-234	61	5	∥∥∥{|qi	∥∥∥{|qi	PROPN
ma-234	61	6	|	|	ADV
ma-234	62	1	1	1	NUM
ma-234	62	2	α	α	NOUN
ma-234	62	3	−	−	PROPN
ma-234	62	4	1	1	NUM
ma-234	62	5	q	q	NOUN
ma-234	62	6	‖f	‖f	ADJ
ma-234	62	7	χqi‖q	χqi‖q	PROPN
ma-234	62	8	}	}	PUNCT
ma-234	62	9	i∈i	i∈i	ADJ
ma-234	62	10	∥∥∥	∥∥∥	PROPN
ma-234	62	11	`	`	PUNCT
ma-234	62	12	p	p	NOUN
ma-234	62	13	if	if	SCONJ
ma-234	62	14	p	p	PROPN
ma-234	62	15	<	<	NOUN
ma-234	62	16	∞	∞	PROPN
ma-234	62	17	sup	sup	NOUN
ma-234	62	18	q∈q	q∈q	NOUN
ma-234	62	19	|q|	|q|	VERB
ma-234	62	20	1	1	NUM
ma-234	62	21	α	α	NOUN
ma-234	62	22	−	−	PROPN
ma-234	62	23	1	1	NUM
ma-234	62	24	q	q	NOUN
ma-234	62	25	‖f	‖f	PRON
ma-234	62	26	χqi‖q	χqi‖q	PROPN
ma-234	62	27	if	if	SCONJ
ma-234	62	28	p	p	PROPN
ma-234	62	29	=	=	NOUN
ma-234	62	30	∞	∞	PROPN
ma-234	62	31	,	,	PUNCT
ma-234	62	32	with	with	ADP
ma-234	62	33	•	•	NOUN
ma-234	62	34	q	q	NOUN
ma-234	62	35	=	=	PUNCT
ma-234	62	36	{	{	PUNCT
ma-234	62	37	q(x	q(x	PROPN
ma-234	62	38	,	,	PUNCT
ma-234	62	39	r	r	NOUN
ma-234	62	40	)	)	PUNCT
ma-234	62	41	:	:	PUNCT
ma-234	62	42	(	(	PUNCT
ma-234	62	43	r	r	NOUN
ma-234	62	44	,	,	PUNCT
ma-234	62	45	x	x	NOUN
ma-234	62	46	)	)	PUNCT
ma-234	62	47	∈	∈	PROPN
ma-234	62	48	(	(	PUNCT
ma-234	62	49	0,∞)×	0,∞)×	NUM
ma-234	62	50	rd	rd	NOUN
ma-234	62	51	}	}	PUNCT
ma-234	62	52	•	•	NOUN
ma-234	62	53	p	p	NOUN
ma-234	63	1	=	=	X
ma-234	63	2	{	{	PUNCT
ma-234	63	3	{	{	PUNCT
ma-234	63	4	qi}i∈i	qi}i∈i	NOUN
ma-234	63	5	⊂	⊂	PROPN
ma-234	63	6	q	q	X
ma-234	63	7	:	:	PUNCT
ma-234	63	8	i	i	PRON
ma-234	63	9	is	be	AUX
ma-234	63	10	countable	countable	ADJ
ma-234	63	11	and	and	CCONJ
ma-234	63	12	qi	qi	NOUN
ma-234	63	13	∩qj	∩qj	NOUN
ma-234	63	14	=	=	NOUN
ma-234	63	15	∅	∅	NOUN
ma-234	64	1	if	if	SCONJ
ma-234	64	2	i	i	PRON
ma-234	64	3	6=	6=	PROPN
ma-234	64	4	j	j	PROPN
ma-234	64	5	}	}	PUNCT
ma-234	64	6	.	.	PUNCT
ma-234	65	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	65	2	eur	eur	PROPN
ma-234	65	3	.	.	PUNCT
ma-234	66	1	j.	j.	PROPN
ma-234	66	2	math	math	PROPN
ma-234	66	3	.	.	PUNCT
ma-234	67	1	anal	anal	PROPN
ma-234	67	2	.	.	PUNCT
ma-234	68	1	10.28924	10.28924	NUM
ma-234	68	2	/	/	SYM
ma-234	68	3	ada	ada	PROPN
ma-234	68	4	/	/	SYM
ma-234	68	5	ma.4.16	ma.4.16	PROPN
ma-234	68	6	4it	4it	PROPN
ma-234	68	7	is	be	AUX
ma-234	68	8	well	well	ADV
ma-234	68	9	known	know	VERB
ma-234	68	10	that	that	SCONJ
ma-234	68	11	the	the	DET
ma-234	68	12	space	space	NOUN
ma-234	68	13	f(q	f(q	PROPN
ma-234	68	14	,	,	PUNCT
ma-234	68	15	p	p	X
ma-234	68	16	,	,	PUNCT
ma-234	68	17	α	α	NOUN
ma-234	68	18	)	)	PUNCT
ma-234	68	19	is	be	AUX
ma-234	68	20	a	a	DET
ma-234	68	21	linear	linear	ADJ
ma-234	68	22	subspace	subspace	NOUN
ma-234	68	23	of	of	ADP
ma-234	68	24	lqloc	lqloc	NOUN
ma-234	68	25	and	and	CCONJ
ma-234	68	26	a	a	DET
ma-234	68	27	banach	banach	NOUN
ma-234	68	28	space	space	NOUN
ma-234	68	29	,	,	PUNCT
ma-234	68	30	whenendowed	whenendowe	VERB
ma-234	68	31	with	with	ADP
ma-234	68	32	‖	‖	PROPN
ma-234	68	33	·	·	PUNCT
ma-234	68	34	‖f(q	‖f(q	NOUN
ma-234	68	35	,	,	PUNCT
ma-234	68	36	p	p	X
ma-234	68	37	,	,	PUNCT
ma-234	68	38	α	α	NOUN
ma-234	68	39	)	)	PUNCT
ma-234	68	40	.	.	PUNCT
ma-234	69	1	f(q	f(q	PROPN
ma-234	69	2	,	,	PUNCT
ma-234	69	3	p	p	X
ma-234	69	4	,	,	PUNCT
ma-234	69	5	α	α	NOUN
ma-234	69	6	)	)	PUNCT
ma-234	69	7	is	be	AUX
ma-234	69	8	nontrivial	nontrivial	ADJ
ma-234	69	9	if	if	SCONJ
ma-234	69	10	and	and	CCONJ
ma-234	69	11	only	only	ADV
ma-234	69	12	if	if	SCONJ
ma-234	69	13	q	q	PROPN
ma-234	69	14	≤	≤	X
ma-234	69	15	α	α	NOUN
ma-234	69	16	≤	≤	ADJ
ma-234	69	17	p	p	NOUN
ma-234	69	18	(	(	PUNCT
ma-234	69	19	see	see	VERB
ma-234	69	20	[	[	X
ma-234	69	21	7	7	NUM
ma-234	69	22	]	]	NUM
ma-234	69	23	)	)	PUNCT
ma-234	69	24	.	.	PUNCT
ma-234	70	1	moreover	moreover	ADV
ma-234	70	2	,	,	PUNCT
ma-234	70	3	when	when	SCONJ
ma-234	70	4	1	1	NUM
ma-234	70	5	≤	≤	NUM
ma-234	70	6	q1	q1	PROPN
ma-234	70	7	≤	≤	NUM
ma-234	70	8	q	q	PROPN
ma-234	70	9	≤	≤	NUM
ma-234	70	10	α	α	NOUN
ma-234	70	11	≤	≤	NOUN
ma-234	70	12	p	p	PROPN
ma-234	70	13	≤	≤	PROPN
ma-234	70	14	p1	p1	NOUN
ma-234	70	15	,	,	PUNCT
ma-234	70	16	the	the	DET
ma-234	70	17	following	follow	VERB
ma-234	70	18	inclusion	inclusion	NOUN
ma-234	70	19	and	and	CCONJ
ma-234	70	20	equality	equality	NOUN
ma-234	70	21	relations	relation	NOUN
ma-234	70	22	hold	hold	VERB
ma-234	70	23	:	:	PUNCT
ma-234	70	24	{	{	PUNCT
ma-234	70	25	lα	lα	PROPN
ma-234	70	26	=	=	PROPN
ma-234	70	27	f(q	f(q	PROPN
ma-234	70	28	,	,	PUNCT
ma-234	70	29	α	α	NOUN
ma-234	70	30	,	,	PUNCT
ma-234	70	31	α	α	NOUN
ma-234	70	32	)	)	PUNCT
ma-234	71	1	⊂	⊂	PROPN
ma-234	71	2	f(q	f(q	PROPN
ma-234	71	3	,	,	PUNCT
ma-234	71	4	p	p	X
ma-234	71	5	,	,	PUNCT
ma-234	71	6	α	α	NOUN
ma-234	71	7	)	)	PUNCT
ma-234	71	8	⊂	⊂	PROPN
ma-234	71	9	f(q	f(q	PROPN
ma-234	71	10	,	,	PUNCT
ma-234	71	11	p1	p1	PROPN
ma-234	71	12	,	,	PUNCT
ma-234	71	13	α	α	X
ma-234	71	14	)	)	PUNCT
ma-234	71	15	⊂	⊂	PROPN
ma-234	71	16	f(q,∞	f(q,∞	PROPN
ma-234	71	17	,	,	PUNCT
ma-234	71	18	α	α	NOUN
ma-234	71	19	)	)	PUNCT
ma-234	71	20	=	=	NOUN
ma-234	71	21	mα	mα	PROPN
ma-234	71	22	q	q	PROPN
ma-234	71	23	f(q	f(q	PROPN
ma-234	71	24	,	,	PUNCT
ma-234	71	25	p	p	X
ma-234	71	26	,	,	PUNCT
ma-234	71	27	α	α	NOUN
ma-234	71	28	)	)	PUNCT
ma-234	71	29	⊂	⊂	PROPN
ma-234	71	30	f(q1	f(q1	PROPN
ma-234	71	31	,	,	PUNCT
ma-234	71	32	p	p	X
ma-234	71	33	,	,	PUNCT
ma-234	71	34	α	α	NOUN
ma-234	71	35	)	)	PUNCT
ma-234	71	36	.	.	PUNCT
ma-234	72	1	(	(	PUNCT
ma-234	72	2	4	4	X
ma-234	72	3	)	)	PUNCT
ma-234	72	4	note	note	NOUN
ma-234	72	5	that	that	SCONJ
ma-234	72	6	(	(	PUNCT
ma-234	72	7	4	4	X
ma-234	72	8	)	)	PUNCT
ma-234	72	9	shows	show	VERB
ma-234	72	10	that	that	SCONJ
ma-234	72	11	the	the	DET
ma-234	72	12	spaces	space	NOUN
ma-234	72	13	f(q	f(q	PROPN
ma-234	72	14	,	,	PUNCT
ma-234	72	15	p	p	X
ma-234	72	16	,	,	PUNCT
ma-234	72	17	α	α	NOUN
ma-234	72	18	)	)	PUNCT
ma-234	72	19	provide	provide	VERB
ma-234	72	20	a	a	DET
ma-234	72	21	bridge	bridge	NOUN
ma-234	72	22	connecting	connect	VERB
ma-234	72	23	both	both	CCONJ
ma-234	72	24	lebesgue	lebesgue	PROPN
ma-234	72	25	spacesand	spacesand	PROPN
ma-234	72	26	morrey	morrey	PROPN
ma-234	72	27	spaces	space	VERB
ma-234	72	28	.	.	PUNCT
ma-234	73	1	many	many	ADJ
ma-234	73	2	results	result	NOUN
ma-234	73	3	,	,	PUNCT
ma-234	73	4	well	well	INTJ
ma-234	73	5	known	know	VERB
ma-234	73	6	for	for	ADP
ma-234	73	7	lebesgue	lebesgue	NOUN
ma-234	73	8	or	or	CCONJ
ma-234	73	9	morrey	morrey	PROPN
ma-234	73	10	spaces	space	NOUN
ma-234	73	11	,	,	PUNCT
ma-234	73	12	have	have	AUX
ma-234	73	13	been	be	AUX
ma-234	73	14	extendedin	extendedin	VERB
ma-234	73	15	the	the	DET
ma-234	73	16	framework	framework	NOUN
ma-234	73	17	of	of	ADP
ma-234	73	18	these	these	DET
ma-234	73	19	spaces	space	NOUN
ma-234	73	20	(	(	PUNCT
ma-234	73	21	see	see	VERB
ma-234	73	22	[	[	X
ma-234	73	23	5	5	NUM
ma-234	73	24	,	,	PUNCT
ma-234	73	25	7	7	NUM
ma-234	73	26	]	]	NUM
ma-234	73	27	)	)	PUNCT
ma-234	73	28	.	.	PUNCT
ma-234	74	1	furthermore	furthermore	ADV
ma-234	74	2	,	,	PUNCT
ma-234	74	3	the	the	DET
ma-234	74	4	relations	relation	NOUN
ma-234	74	5	(	(	PUNCT
ma-234	74	6	1	1	NUM
ma-234	74	7	)	)	PUNCT
ma-234	74	8	and	and	CCONJ
ma-234	74	9	(	(	PUNCT
ma-234	74	10	4	4	X
ma-234	74	11	)	)	PUNCT
ma-234	74	12	point	point	NOUN
ma-234	74	13	out	out	ADP
ma-234	74	14	thatthe	thatthe	NOUN
ma-234	74	15	spaces	space	NOUN
ma-234	74	16	mα	mα	PROPN
ma-234	74	17	q	q	ADJ
ma-234	74	18	,	,	PUNCT
ma-234	74	19	p	p	NOUN
ma-234	74	20	and	and	CCONJ
ma-234	74	21	f(q	f(q	PROPN
ma-234	74	22	,	,	PUNCT
ma-234	74	23	p	p	X
ma-234	74	24	,	,	PUNCT
ma-234	74	25	α	α	NOUN
ma-234	74	26	)	)	PUNCT
ma-234	74	27	satisfy	satisfy	VERB
ma-234	74	28	almost	almost	ADV
ma-234	74	29	the	the	DET
ma-234	74	30	same	same	ADJ
ma-234	74	31	inclusion	inclusion	NOUN
ma-234	74	32	relations	relation	NOUN
ma-234	74	33	.	.	PUNCT
ma-234	75	1	we	we	PRON
ma-234	75	2	also	also	ADV
ma-234	75	3	observe	observe	VERB
ma-234	75	4	thatthe	thatthe	NOUN
ma-234	75	5	norm	norm	NOUN
ma-234	75	6	structures	structure	NOUN
ma-234	75	7	of	of	ADP
ma-234	75	8	these	these	DET
ma-234	75	9	two	two	NUM
ma-234	75	10	spaces	space	NOUN
ma-234	75	11	are	be	AUX
ma-234	75	12	very	very	ADV
ma-234	75	13	similar	similar	ADJ
ma-234	75	14	.	.	PUNCT
ma-234	76	1	thus	thus	ADV
ma-234	76	2	,	,	PUNCT
ma-234	76	3	a	a	DET
ma-234	76	4	natural	natural	ADJ
ma-234	76	5	question	question	NOUN
ma-234	76	6	is	be	AUX
ma-234	76	7	that	that	SCONJ
ma-234	76	8	,	,	PUNCT
ma-234	76	9	what	what	PRON
ma-234	76	10	isthe	isthe	ADJ
ma-234	76	11	link	link	NOUN
ma-234	76	12	between	between	ADP
ma-234	76	13	the	the	DET
ma-234	76	14	spaces	space	NOUN
ma-234	76	15	mα	mα	X
ma-234	76	16	q	q	ADJ
ma-234	76	17	,	,	PUNCT
ma-234	76	18	p	p	NOUN
ma-234	76	19	and	and	CCONJ
ma-234	76	20	f(q	f(q	PROPN
ma-234	76	21	,	,	PUNCT
ma-234	76	22	p	p	X
ma-234	76	23	,	,	PUNCT
ma-234	76	24	α	α	NOUN
ma-234	76	25	)	)	PUNCT
ma-234	76	26	?	?	PUNCT
ma-234	77	1	the	the	DET
ma-234	77	2	second	second	ADJ
ma-234	77	3	aim	aim	NOUN
ma-234	77	4	of	of	ADP
ma-234	77	5	this	this	DET
ma-234	77	6	paper	paper	NOUN
ma-234	77	7	is	be	AUX
ma-234	77	8	to	to	PART
ma-234	77	9	study	study	VERB
ma-234	77	10	the	the	DET
ma-234	77	11	above	above	ADV
ma-234	77	12	mentioned	mention	VERB
ma-234	77	13	question	question	NOUN
ma-234	77	14	.	.	PUNCT
ma-234	78	1	we	we	PRON
ma-234	78	2	succeeded	succeed	VERB
ma-234	78	3	in	in	ADP
ma-234	78	4	provingthatmα	provingthatmα	PROPN
ma-234	78	5	q	q	NOUN
ma-234	78	6	,	,	PUNCT
ma-234	78	7	p	p	PRON
ma-234	78	8	is	be	AUX
ma-234	78	9	continuously	continuously	ADV
ma-234	78	10	included	include	VERB
ma-234	78	11	in	in	ADP
ma-234	78	12	f(q	f(q	PROPN
ma-234	78	13	,	,	PUNCT
ma-234	78	14	p	p	X
ma-234	78	15	,	,	PUNCT
ma-234	78	16	α	α	NOUN
ma-234	78	17	)	)	PUNCT
ma-234	78	18	and	and	CCONJ
ma-234	78	19	,	,	PUNCT
ma-234	78	20	when	when	SCONJ
ma-234	78	21	p	p	NOUN
ma-234	78	22	<	<	X
ma-234	78	23	∞,mα	∞,mα	NOUN
ma-234	78	24	q	q	NOUN
ma-234	78	25	,	,	PUNCT
ma-234	78	26	p	p	PRON
ma-234	78	27	is	be	AUX
ma-234	78	28	included	include	VERB
ma-234	78	29	in	in	ADP
ma-234	78	30	the	the	DET
ma-234	78	31	closureof	closureof	NOUN
ma-234	78	32	lα	lα	NOUN
ma-234	78	33	in	in	ADP
ma-234	78	34	f(q	f(q	PROPN
ma-234	78	35	,	,	PUNCT
ma-234	78	36	p	p	X
ma-234	78	37	,	,	PUNCT
ma-234	78	38	α	α	NOUN
ma-234	78	39	)	)	PUNCT
ma-234	78	40	.	.	PUNCT
ma-234	79	1	therefore	therefore	ADV
ma-234	79	2	,	,	PUNCT
ma-234	79	3	we	we	PRON
ma-234	79	4	also	also	ADV
ma-234	79	5	establish	establish	VERB
ma-234	79	6	inmα	inmα	ADJ
ma-234	79	7	q	q	NOUN
ma-234	79	8	,	,	PUNCT
ma-234	79	9	p	p	NOUN
ma-234	79	10	some	some	DET
ma-234	79	11	approximation	approximation	NOUN
ma-234	79	12	results	result	NOUN
ma-234	79	13	by	by	ADP
ma-234	79	14	compactlysupported	compactlysupporte	VERB
ma-234	79	15	and/or	and/or	CCONJ
ma-234	79	16	regular	regular	ADJ
ma-234	79	17	functions.as	functions.as	X
ma-234	79	18	an	an	DET
ma-234	79	19	application	application	NOUN
ma-234	79	20	of	of	ADP
ma-234	79	21	the	the	DET
ma-234	79	22	above	above	ADJ
ma-234	79	23	mentioned	mention	VERB
ma-234	79	24	results	result	NOUN
ma-234	79	25	,	,	PUNCT
ma-234	79	26	we	we	PRON
ma-234	79	27	obtain	obtain	VERB
ma-234	79	28	an	an	DET
ma-234	79	29	explicit	explicit	ADJ
ma-234	79	30	solution	solution	NOUN
ma-234	79	31	in	in	ADP
ma-234	79	32	(	(	PUNCT
ma-234	79	33	lp)d	lp)d	PROPN
ma-234	79	34	of	of	ADP
ma-234	79	35	theequation	theequation	NOUN
ma-234	79	36	divf	divf	NOUN
ma-234	79	37	=	=	PROPN
ma-234	80	1	f	f	PROPN
ma-234	80	2	whenever	whenever	SCONJ
ma-234	80	3	f	f	PROPN
ma-234	80	4	is	be	AUX
ma-234	80	5	in	in	ADP
ma-234	80	6	mα	mα	PROPN
ma-234	80	7	q	q	NOUN
ma-234	80	8	,	,	PUNCT
ma-234	80	9	p	p	X
ma-234	80	10	,	,	PUNCT
ma-234	80	11	with	with	ADP
ma-234	80	12	d	d	PROPN
ma-234	80	13	≥	≥	NUM
ma-234	80	14	3	3	NUM
ma-234	80	15	,	,	PUNCT
ma-234	80	16	1	1	NUM
ma-234	80	17	≤	≤	NUM
ma-234	80	18	q	q	ADJ
ma-234	80	19	≤	≤	NUM
ma-234	80	20	α	α	NOUN
ma-234	80	21	<	<	X
ma-234	80	22	d	d	PROPN
ma-234	80	23	and	and	CCONJ
ma-234	80	24	1	1	NUM
ma-234	80	25	p	p	NOUN
ma-234	80	26	=	=	NOUN
ma-234	81	1	1	1	NUM
ma-234	81	2	α	α	NOUN
ma-234	81	3	−	−	NOUN
ma-234	81	4	1	1	NUM
ma-234	81	5	d	d	PROPN
ma-234	81	6	.the	.the	DET
ma-234	81	7	remainder	remainder	NOUN
ma-234	81	8	of	of	ADP
ma-234	81	9	the	the	DET
ma-234	81	10	paper	paper	NOUN
ma-234	81	11	is	be	AUX
ma-234	81	12	organized	organize	VERB
ma-234	81	13	as	as	SCONJ
ma-234	81	14	follows	follow	VERB
ma-234	81	15	.	.	PUNCT
ma-234	82	1	section	section	NOUN
ma-234	82	2	2	2	NUM
ma-234	82	3	contains	contain	VERB
ma-234	82	4	a	a	DET
ma-234	82	5	more	more	ADV
ma-234	82	6	detailed	detailed	ADJ
ma-234	82	7	presen	presen	NOUN
ma-234	82	8	-	-	PUNCT
ma-234	82	9	tation	tation	NOUN
ma-234	82	10	of	of	ADP
ma-234	82	11	our	our	PRON
ma-234	82	12	main	main	ADJ
ma-234	82	13	results	result	NOUN
ma-234	82	14	.	.	PUNCT
ma-234	83	1	section	section	NOUN
ma-234	83	2	3	3	NUM
ma-234	83	3	deals	deal	NOUN
ma-234	83	4	with	with	ADP
ma-234	83	5	some	some	DET
ma-234	83	6	preliminary	preliminary	ADJ
ma-234	83	7	results	result	NOUN
ma-234	83	8	on	on	ADP
ma-234	83	9	mα	mα	PROPN
ma-234	83	10	q	q	NOUN
ma-234	83	11	,	,	PUNCT
ma-234	83	12	p	p	X
ma-234	83	13	.	.	PUNCT
ma-234	84	1	in	in	ADP
ma-234	84	2	section	section	NOUN
ma-234	84	3	4we	4we	NOUN
ma-234	84	4	prove	prove	VERB
ma-234	84	5	the	the	DET
ma-234	84	6	inclusion	inclusion	NOUN
ma-234	84	7	of	of	ADP
ma-234	84	8	mα	mα	PROPN
ma-234	84	9	q	q	NOUN
ma-234	84	10	,	,	PUNCT
ma-234	84	11	p	p	NOUN
ma-234	84	12	in	in	ADP
ma-234	84	13	f(q	f(q	PROPN
ma-234	84	14	,	,	PUNCT
ma-234	84	15	p	p	X
ma-234	84	16	,	,	PUNCT
ma-234	84	17	α	α	NOUN
ma-234	84	18	)	)	PUNCT
ma-234	84	19	and	and	CCONJ
ma-234	84	20	also	also	ADV
ma-234	84	21	approximation	approximation	NOUN
ma-234	84	22	results	result	NOUN
ma-234	84	23	.	.	PUNCT
ma-234	85	1	section	section	NOUN
ma-234	85	2	5	5	NUM
ma-234	85	3	is	be	AUX
ma-234	85	4	devotedto	devotedto	ADJ
ma-234	85	5	prove	prove	VERB
ma-234	85	6	our	our	PRON
ma-234	85	7	main	main	ADJ
ma-234	85	8	theorem	theorem	NOUN
ma-234	85	9	showing	show	VERB
ma-234	85	10	the	the	DET
ma-234	85	11	action	action	NOUN
ma-234	85	12	of	of	ADP
ma-234	85	13	fractional	fractional	ADJ
ma-234	85	14	integral	integral	ADJ
ma-234	85	15	operators	operator	NOUN
ma-234	85	16	on	on	ADP
ma-234	85	17	mα	mα	PROPN
ma-234	85	18	q	q	NOUN
ma-234	85	19	,	,	PUNCT
ma-234	85	20	p	p	NOUN
ma-234	85	21	.	.	PUNCT
ma-234	86	1	section	section	NOUN
ma-234	86	2	6contains	6contains	NUM
ma-234	86	3	an	an	DET
ma-234	86	4	application	application	NOUN
ma-234	86	5	to	to	ADP
ma-234	86	6	the	the	DET
ma-234	86	7	divergence	divergence	NOUN
ma-234	86	8	equation	equation	NOUN
ma-234	86	9	div	div	X
ma-234	86	10	f	f	PROPN
ma-234	86	11	=	=	SYM
ma-234	86	12	f	f	PROPN
ma-234	86	13	.finally	.finally	ADV
ma-234	86	14	,	,	PUNCT
ma-234	86	15	let	let	VERB
ma-234	86	16	us	we	PRON
ma-234	86	17	make	make	VERB
ma-234	86	18	some	some	DET
ma-234	86	19	conventions	convention	NOUN
ma-234	86	20	on	on	ADP
ma-234	86	21	notations	notation	NOUN
ma-234	86	22	used	use	VERB
ma-234	86	23	in	in	ADP
ma-234	86	24	this	this	DET
ma-234	86	25	paper	paper	NOUN
ma-234	86	26	.	.	PUNCT
ma-234	87	1	•	•	NUM
ma-234	88	1	c∞	c∞	PROPN
ma-234	88	2	denotes	denote	VERB
ma-234	88	3	the	the	DET
ma-234	88	4	set	set	NOUN
ma-234	88	5	of	of	ADP
ma-234	88	6	all	all	DET
ma-234	88	7	infinitely	infinitely	ADV
ma-234	88	8	differentiable	differentiable	ADJ
ma-234	88	9	functions	function	NOUN
ma-234	88	10	on	on	ADP
ma-234	88	11	rd	rd	NOUN
ma-234	88	12	and	and	CCONJ
ma-234	88	13	c∞c	c∞c	PROPN
ma-234	88	14	stands	stand	VERB
ma-234	88	15	for	for	ADP
ma-234	88	16	the	the	DET
ma-234	88	17	set	set	VERB
ma-234	88	18	ofall	ofall	ADJ
ma-234	88	19	elements	element	NOUN
ma-234	88	20	of	of	ADP
ma-234	88	21	c∞	c∞	PROPN
ma-234	88	22	with	with	ADP
ma-234	88	23	compact	compact	ADJ
ma-234	88	24	support	support	NOUN
ma-234	88	25	in	in	ADP
ma-234	88	26	rd	rd	PROPN
ma-234	88	27	.	.	PUNCT
ma-234	89	1	•	•	INTJ
ma-234	89	2	let	let	VERB
ma-234	89	3	φ	φ	PROPN
ma-234	89	4	be	be	AUX
ma-234	89	5	a	a	DET
ma-234	89	6	fixed	fix	VERB
ma-234	89	7	nonnegative	nonnegative	ADJ
ma-234	89	8	element	element	NOUN
ma-234	89	9	of	of	ADP
ma-234	89	10	c∞	c∞	PROPN
ma-234	89	11	such	such	ADJ
ma-234	89	12	that	that	SCONJ
ma-234	89	13	its	its	PRON
ma-234	89	14	support	support	NOUN
ma-234	89	15	is	be	AUX
ma-234	89	16	included	include	VERB
ma-234	89	17	in	in	ADP
ma-234	89	18	the	the	DET
ma-234	89	19	unit	unit	NOUN
ma-234	89	20	cube	cube	NOUN
ma-234	89	21	[	[	X
ma-234	89	22	0	0	NUM
ma-234	89	23	,	,	PUNCT
ma-234	89	24	1]d	1]d	NUM
ma-234	89	25	and	and	CCONJ
ma-234	89	26	satisfying	satisfy	VERB
ma-234	89	27	∫	∫	PROPN
ma-234	89	28	rd	rd	PROPN
ma-234	89	29	φ(x)dx	φ(x)dx	NOUN
ma-234	89	30	=	=	NOUN
ma-234	89	31	1	1	X
ma-234	89	32	.	.	X
ma-234	90	1	for	for	ADP
ma-234	90	2	any	any	DET
ma-234	90	3	integer	integer	NOUN
ma-234	90	4	n	n	PRON
ma-234	90	5	≥	≥	NOUN
ma-234	90	6	1	1	NUM
ma-234	90	7	,	,	PUNCT
ma-234	90	8	we	we	PRON
ma-234	90	9	denote	denote	VERB
ma-234	90	10	by	by	ADP
ma-234	90	11	φn	φn	ADP
ma-234	90	12	the	the	DET
ma-234	90	13	dilation	dilation	NOUN
ma-234	90	14	definedby	definedby	ADV
ma-234	90	15	φn(x	φn(x	NOUN
ma-234	90	16	)	)	PUNCT
ma-234	90	17	=	=	SYM
ma-234	90	18	ndφ(nx	ndφ(nx	NOUN
ma-234	90	19	)	)	PUNCT
ma-234	90	20	,	,	PUNCT
ma-234	90	21	x	x	PUNCT
ma-234	90	22	∈	∈	PROPN
ma-234	90	23	rd	rd	PROPN
ma-234	90	24	.	.	PUNCT
ma-234	91	1	•	•	INTJ
ma-234	91	2	let	let	VERB
ma-234	91	3	ω	ω	X
ma-234	91	4	be	be	AUX
ma-234	91	5	a	a	DET
ma-234	91	6	fixed	fix	VERB
ma-234	91	7	element	element	NOUN
ma-234	91	8	of	of	ADP
ma-234	91	9	c∞	c∞	PROPN
ma-234	91	10	satisfying	satisfying	NOUN
ma-234	91	11	χq(0,1	χq(0,1	NOUN
ma-234	91	12	)	)	PUNCT
ma-234	91	13	≤	≤	NUM
ma-234	91	14	ω	ω	NUM
ma-234	91	15	≤	≤	PROPN
ma-234	91	16	χq(0,2	χq(0,2	PROPN
ma-234	91	17	)	)	PUNCT
ma-234	91	18	.	.	PUNCT
ma-234	92	1	for	for	ADP
ma-234	92	2	any	any	DET
ma-234	92	3	integer	integer	NOUN
ma-234	92	4	n	n	PRON
ma-234	92	5	≥	≥	NOUN
ma-234	92	6	1	1	NUM
ma-234	92	7	,	,	PUNCT
ma-234	92	8	ωn	ωn	PRON
ma-234	92	9	isdefined	isdefine	VERB
ma-234	92	10	by	by	ADP
ma-234	92	11	ωn(x	ωn(x	NUM
ma-234	92	12	)	)	PUNCT
ma-234	93	1	=	=	SYM
ma-234	93	2	ω	ω	NOUN
ma-234	93	3	(	(	PUNCT
ma-234	93	4	x	x	NOUN
ma-234	93	5	n	n	PROPN
ma-234	93	6	)	)	PUNCT
ma-234	93	7	,	,	PUNCT
ma-234	93	8	x	x	PUNCT
ma-234	93	9	∈	∈	PROPN
ma-234	93	10	rd	rd	PROPN
ma-234	93	11	.	.	PUNCT
ma-234	94	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	94	2	eur	eur	PROPN
ma-234	94	3	.	.	PUNCT
ma-234	95	1	j.	j.	PROPN
ma-234	95	2	math	math	PROPN
ma-234	95	3	.	.	PUNCT
ma-234	96	1	anal	anal	PROPN
ma-234	96	2	.	.	PUNCT
ma-234	97	1	10.28924	10.28924	NUM
ma-234	97	2	/	/	SYM
ma-234	97	3	ada	ada	PROPN
ma-234	97	4	/	/	SYM
ma-234	97	5	ma.4.16	ma.4.16	PROPN
ma-234	97	6	52	52	NUM
ma-234	97	7	.	.	PUNCT
ma-234	98	1	statement	statement	NOUN
ma-234	98	2	of	of	ADP
ma-234	98	3	the	the	DET
ma-234	98	4	main	main	ADJ
ma-234	98	5	results	result	NOUN
ma-234	98	6	for	for	ADP
ma-234	98	7	0	0	NUM
ma-234	98	8	<	<	X
ma-234	98	9	γ	γ	X
ma-234	98	10	<	<	X
ma-234	98	11	1	1	NUM
ma-234	98	12	,	,	PUNCT
ma-234	98	13	the	the	DET
ma-234	98	14	fractional	fractional	ADJ
ma-234	98	15	integral	integral	ADJ
ma-234	98	16	operator	operator	NOUN
ma-234	98	17	iγ	iγ	NOUN
ma-234	98	18	is	be	AUX
ma-234	98	19	known	know	VERB
ma-234	98	20	to	to	PART
ma-234	98	21	be	be	AUX
ma-234	98	22	closely	closely	ADV
ma-234	98	23	related	relate	VERB
ma-234	98	24	to	to	ADP
ma-234	98	25	the	the	DET
ma-234	98	26	fractionalmaximal	fractionalmaximal	NOUN
ma-234	98	27	operator	operator	NOUN
ma-234	98	28	mγ	mγ	PRON
ma-234	98	29	defined	define	VERB
ma-234	98	30	by	by	ADP
ma-234	98	31	mγf	mγf	X
ma-234	98	32	(	(	PUNCT
ma-234	98	33	x	x	NOUN
ma-234	98	34	)	)	PUNCT
ma-234	98	35	=	=	SYM
ma-234	98	36	sup	sup	NOUN
ma-234	98	37	q3x	q3x	NOUN
ma-234	98	38	|q|γ−1	|q|γ−1	X
ma-234	98	39	∫	∫	PROPN
ma-234	98	40	q	q	PROPN
ma-234	98	41	|f	|f	PROPN
ma-234	98	42	(	(	PUNCT
ma-234	98	43	y)|dy	y)|dy	PROPN
ma-234	98	44	,	,	PUNCT
ma-234	98	45	f	f	PROPN
ma-234	98	46	∈	∈	PROPN
ma-234	98	47	l1	l1	PROPN
ma-234	98	48	loc	loc	PROPN
ma-234	98	49	,	,	PUNCT
ma-234	98	50	x	x	PROPN
ma-234	98	51	∈	∈	PROPN
ma-234	98	52	rd	rd	PROPN
ma-234	98	53	,	,	PUNCT
ma-234	98	54	where	where	SCONJ
ma-234	98	55	the	the	DET
ma-234	98	56	supremum	supremum	NOUN
ma-234	98	57	is	be	AUX
ma-234	98	58	taken	take	VERB
ma-234	98	59	over	over	ADP
ma-234	98	60	all	all	DET
ma-234	98	61	cubes	cube	NOUN
ma-234	98	62	q	q	NOUN
ma-234	98	63	in	in	ADP
ma-234	98	64	rd	rd	NOUN
ma-234	98	65	containing	contain	VERB
ma-234	98	66	x	x	PUNCT
ma-234	98	67	.our	.our	PRON
ma-234	98	68	first	first	ADJ
ma-234	98	69	result	result	NOUN
ma-234	98	70	reads	read	VERB
ma-234	98	71	as	as	SCONJ
ma-234	98	72	follows	follow	VERB
ma-234	98	73	.	.	PUNCT
ma-234	99	1	theorem	theorem	VERB
ma-234	99	2	2.1	2.1	NUM
ma-234	99	3	.	.	PUNCT
ma-234	100	1	let	let	VERB
ma-234	100	2	us	we	PRON
ma-234	100	3	assume	assume	VERB
ma-234	100	4	that	that	SCONJ
ma-234	100	5	0	0	NUM
ma-234	100	6	<	<	X
ma-234	100	7	γ	γ	X
ma-234	100	8	<	<	X
ma-234	100	9	1	1	NUM
ma-234	100	10	α	α	NOUN
ma-234	100	11	≤	≤	NUM
ma-234	100	12	1	1	NUM
ma-234	100	13	and	and	CCONJ
ma-234	100	14	1	1	NUM
ma-234	100	15	p	p	NOUN
ma-234	100	16	=	=	NOUN
ma-234	101	1	1	1	NUM
ma-234	101	2	α	α	NOUN
ma-234	101	3	−	−	PROPN
ma-234	101	4	γ	γ	PROPN
ma-234	101	5	.	.	PROPN
ma-234	101	6	then	then	ADV
ma-234	101	7	,	,	PUNCT
ma-234	101	8	for	for	ADP
ma-234	101	9	any	any	DET
ma-234	101	10	element	element	NOUN
ma-234	101	11	f	f	PROPN
ma-234	101	12	of	of	ADP
ma-234	101	13	mα	mα	PROPN
ma-234	101	14	1,p	1,p	PROPN
ma-234	101	15	,	,	PUNCT
ma-234	101	16	we	we	PRON
ma-234	101	17	have	have	VERB
ma-234	101	18	‖mγf	‖mγf	NOUN
ma-234	101	19	‖p	‖p	VERB
ma-234	101	20	≤	≤	NOUN
ma-234	101	21	2	2	NUM
ma-234	101	22	d	d	NOUN
ma-234	101	23	(	(	PUNCT
ma-234	101	24	2−	2−	NUM
ma-234	101	25	1	1	NUM
ma-234	101	26	p	p	NOUN
ma-234	101	27	)	)	PUNCT
ma-234	101	28	3d(2−γ	3d(2−γ	NUM
ma-234	101	29	)	)	PUNCT
ma-234	101	30	‖f	‖f	PUNCT
ma-234	102	1	‖mα	‖mα	NUM
ma-234	102	2	1,p	1,p	NOUN
ma-234	102	3	.	.	PUNCT
ma-234	103	1	(	(	PUNCT
ma-234	103	2	5	5	X
ma-234	103	3	)	)	PUNCT
ma-234	103	4	note	note	NOUN
ma-234	103	5	that	that	PRON
ma-234	103	6	theorem	theorem	VERB
ma-234	103	7	2.1	2.1	NUM
ma-234	103	8	refines	refine	NOUN
ma-234	103	9	[	[	X
ma-234	103	10	8	8	NUM
ma-234	103	11	,	,	PUNCT
ma-234	103	12	corollary	corollary	ADJ
ma-234	103	13	4.5	4.5	NUM
ma-234	103	14	]	]	PUNCT
ma-234	103	15	.	.	PUNCT
ma-234	104	1	as	as	ADP
ma-234	104	2	an	an	DET
ma-234	104	3	immediate	immediate	ADJ
ma-234	104	4	consequence	consequence	NOUN
ma-234	104	5	of	of	ADP
ma-234	104	6	this	this	DET
ma-234	104	7	theorem	theorem	NOUN
ma-234	104	8	,	,	PUNCT
ma-234	104	9	we	we	PRON
ma-234	104	10	obtain	obtain	VERB
ma-234	104	11	the	the	DET
ma-234	104	12	following	following	ADJ
ma-234	104	13	result	result	NOUN
ma-234	104	14	which	which	PRON
ma-234	104	15	refines	refine	VERB
ma-234	104	16	[	[	X
ma-234	104	17	8	8	NUM
ma-234	104	18	,	,	PUNCT
ma-234	104	19	theorem	theorem	VERB
ma-234	104	20	4.4	4.4	NUM
ma-234	104	21	]	]	PUNCT
ma-234	104	22	and	and	CCONJ
ma-234	104	23	is	be	AUX
ma-234	104	24	our	our	PRON
ma-234	104	25	most	most	ADV
ma-234	104	26	significant	significant	ADJ
ma-234	104	27	result	result	NOUN
ma-234	104	28	.	.	PUNCT
ma-234	105	1	theorem	theorem	VERB
ma-234	105	2	2.2	2.2	NUM
ma-234	105	3	.	.	PUNCT
ma-234	106	1	let	let	VERB
ma-234	106	2	us	we	PRON
ma-234	106	3	assume	assume	VERB
ma-234	106	4	that	that	SCONJ
ma-234	106	5	0	0	NUM
ma-234	106	6	<	<	X
ma-234	106	7	γ	γ	X
ma-234	106	8	<	<	X
ma-234	106	9	1	1	NUM
ma-234	106	10	α	α	NOUN
ma-234	106	11	≤	≤	NUM
ma-234	106	12	1	1	NUM
ma-234	106	13	and	and	CCONJ
ma-234	106	14	1	1	NUM
ma-234	106	15	p	p	NOUN
ma-234	106	16	=	=	NOUN
ma-234	106	17	1	1	NUM
ma-234	106	18	α−γ	α−γ	NOUN
ma-234	106	19	.	.	PUNCT
ma-234	107	1	then	then	ADV
ma-234	107	2	there	there	PRON
ma-234	107	3	exists	exist	VERB
ma-234	107	4	a	a	DET
ma-234	107	5	real	real	ADJ
ma-234	107	6	constant	constant	ADJ
ma-234	107	7	c	c	NOUN
ma-234	107	8	>	>	X
ma-234	107	9	0	0	NUM
ma-234	107	10	such	such	ADJ
ma-234	107	11	that	that	SCONJ
ma-234	107	12	,	,	PUNCT
ma-234	107	13	for	for	ADP
ma-234	107	14	any	any	DET
ma-234	107	15	element	element	NOUN
ma-234	107	16	f	f	PROPN
ma-234	107	17	of	of	ADP
ma-234	107	18	mα	mα	PROPN
ma-234	107	19	1,p	1,p	PROPN
ma-234	107	20	,	,	PUNCT
ma-234	107	21	we	we	PRON
ma-234	107	22	have	have	VERB
ma-234	107	23	‖iγf	‖iγf	PROPN
ma-234	107	24	‖p	‖p	PROPN
ma-234	107	25	≤	≤	PUNCT
ma-234	108	1	c	c	NOUN
ma-234	108	2	‖f	‖f	PUNCT
ma-234	108	3	‖mα	‖mα	NUM
ma-234	108	4	1,p	1,p	NOUN
ma-234	108	5	.	.	PUNCT
ma-234	109	1	(	(	PUNCT
ma-234	109	2	6	6	NUM
ma-234	109	3	)	)	PUNCT
ma-234	109	4	since	since	SCONJ
ma-234	109	5	lα	lα	PROPN
ma-234	109	6	⊂	⊂	PROPN
ma-234	109	7	mα	mα	PROPN
ma-234	109	8	q	q	ADJ
ma-234	109	9	,	,	PUNCT
ma-234	109	10	p	p	PROPN
ma-234	109	11	⊂	⊂	X
ma-234	109	12	mα	mα	PROPN
ma-234	109	13	1,p	1,p	PROPN
ma-234	109	14	when	when	SCONJ
ma-234	109	15	1	1	NUM
ma-234	109	16	≤	≤	NOUN
ma-234	109	17	q	q	NOUN
ma-234	109	18	<	<	X
ma-234	109	19	α	α	X
ma-234	109	20	<	<	X
ma-234	109	21	p	p	X
ma-234	109	22	,	,	PUNCT
ma-234	109	23	theorem	theorem	VERB
ma-234	109	24	2.2	2.2	NUM
ma-234	109	25	provides	provide	VERB
ma-234	109	26	an	an	DET
ma-234	109	27	extension	extension	NOUN
ma-234	109	28	of	of	ADP
ma-234	109	29	thehardy	thehardy	ADJ
ma-234	109	30	-	-	PUNCT
ma-234	109	31	littlewood	littlewood	NOUN
ma-234	109	32	-	-	PUNCT
ma-234	109	33	sobolev	sobolev	NOUN
ma-234	109	34	theorem	theorem	NOUN
ma-234	109	35	(	(	PUNCT
ma-234	109	36	theorem	theorem	NOUN
ma-234	109	37	1.2	1.2	NUM
ma-234	109	38	)	)	PUNCT
ma-234	109	39	to	to	ADP
ma-234	109	40	the	the	DET
ma-234	109	41	setting	setting	NOUN
ma-234	109	42	of	of	ADP
ma-234	109	43	bourgain	bourgain	NOUN
ma-234	109	44	-	-	PUNCT
ma-234	109	45	morrey	morrey	NOUN
ma-234	110	1	spaces.from	spaces.from	X
ma-234	110	2	(	(	PUNCT
ma-234	110	3	1	1	NUM
ma-234	110	4	)	)	PUNCT
ma-234	110	5	and	and	CCONJ
ma-234	110	6	(	(	PUNCT
ma-234	110	7	6	6	NUM
ma-234	110	8	)	)	PUNCT
ma-234	110	9	,	,	PUNCT
ma-234	110	10	we	we	PRON
ma-234	110	11	have	have	VERB
ma-234	110	12	lα	lα	ADP
ma-234	110	13	⊂mα	⊂mα	ADP
ma-234	110	14	1,p	1,p	PROPN
ma-234	110	15	⊂	⊂	PUNCT
ma-234	110	16	b(γ	b(γ	PROPN
ma-234	110	17	,	,	PUNCT
ma-234	110	18	p	p	NOUN
ma-234	110	19	)	)	PUNCT
ma-234	110	20	,	,	PUNCT
ma-234	110	21	0	0	NUM
ma-234	110	22	<	<	X
ma-234	110	23	γ	γ	X
ma-234	110	24	<	<	X
ma-234	110	25	1	1	NUM
ma-234	110	26	α	α	NOUN
ma-234	110	27	<	<	X
ma-234	110	28	1	1	NUM
ma-234	110	29	and	and	CCONJ
ma-234	110	30	1	1	NUM
ma-234	110	31	p	p	NOUN
ma-234	110	32	=	=	NOUN
ma-234	111	1	1	1	NUM
ma-234	111	2	α	α	NOUN
ma-234	111	3	−	−	PROPN
ma-234	111	4	γ	γ	X
ma-234	111	5	,	,	PUNCT
ma-234	111	6	(	(	PUNCT
ma-234	111	7	7	7	NUM
ma-234	111	8	)	)	PUNCT
ma-234	111	9	where	where	SCONJ
ma-234	111	10	b(γ	b(γ	PROPN
ma-234	111	11	,	,	PUNCT
ma-234	111	12	p	p	X
ma-234	111	13	)	)	PUNCT
ma-234	111	14	=	=	NOUN
ma-234	111	15	{	{	PUNCT
ma-234	111	16	f	f	PROPN
ma-234	111	17	∈	∈	PROPN
ma-234	111	18	l1	l1	PROPN
ma-234	111	19	loc	loc	PROPN
ma-234	111	20	:	:	PUNCT
ma-234	111	21	iγ(|f	iγ(|f	NOUN
ma-234	111	22	|	|	NOUN
ma-234	111	23	)	)	PUNCT
ma-234	111	24	∈	∈	NOUN
ma-234	111	25	lp	lp	NOUN
ma-234	111	26	}	}	PUNCT
ma-234	111	27	,	,	PUNCT
ma-234	111	28	0	0	PUNCT
ma-234	111	29	<	<	X
ma-234	111	30	γ	γ	X
ma-234	111	31	<	<	X
ma-234	111	32	1	1	NUM
ma-234	111	33	≤	≤	NOUN
ma-234	111	34	p	p	NOUN
ma-234	111	35	≤	≤	NUM
ma-234	111	36	∞.	∞.	PROPN
ma-234	111	37	we	we	PRON
ma-234	111	38	recall	recall	VERB
ma-234	111	39	that	that	SCONJ
ma-234	111	40	,	,	PUNCT
ma-234	111	41	for	for	ADP
ma-234	111	42	1	1	NUM
ma-234	111	43	≤	≤	NUM
ma-234	111	44	q	q	NOUN
ma-234	111	45	,	,	PUNCT
ma-234	111	46	p	p	X
ma-234	111	47	,	,	PUNCT
ma-234	111	48	α	α	PROPN
ma-234	111	49	≤	≤	NOUN
ma-234	111	50	∞	∞	PROPN
ma-234	111	51	,	,	PUNCT
ma-234	111	52	the	the	DET
ma-234	111	53	space	space	NOUN
ma-234	111	54	f(q	f(q	PROPN
ma-234	111	55	,	,	PUNCT
ma-234	111	56	p	p	X
ma-234	111	57	,	,	PUNCT
ma-234	111	58	α	α	NOUN
ma-234	111	59	)	)	PUNCT
ma-234	111	60	arises	arise	VERB
ma-234	111	61	naturally	naturally	ADV
ma-234	111	62	in	in	ADP
ma-234	111	63	the	the	DET
ma-234	111	64	search	search	NOUN
ma-234	111	65	of	of	ADP
ma-234	111	66	acharacterization	acharacterization	NOUN
ma-234	111	67	of	of	ADP
ma-234	111	68	the	the	DET
ma-234	111	69	set	set	NOUN
ma-234	111	70	b(γ	b(γ	PROPN
ma-234	111	71	,	,	PUNCT
ma-234	111	72	p	p	NOUN
ma-234	111	73	)	)	PUNCT
ma-234	111	74	in	in	ADP
ma-234	111	75	[	[	X
ma-234	111	76	7	7	NUM
ma-234	111	77	]	]	PUNCT
ma-234	111	78	,	,	PUNCT
ma-234	111	79	where	where	SCONJ
ma-234	111	80	it	it	PRON
ma-234	111	81	is	be	AUX
ma-234	111	82	established	establish	VERB
ma-234	111	83	that	that	SCONJ
ma-234	111	84	b(γ	b(γ	PROPN
ma-234	111	85	,	,	PUNCT
ma-234	111	86	p	p	X
ma-234	111	87	)	)	PUNCT
ma-234	111	88	⊂	⊂	PROPN
ma-234	111	89	f(1	f(1	PROPN
ma-234	111	90	,	,	PUNCT
ma-234	111	91	p	p	X
ma-234	111	92	,	,	PUNCT
ma-234	111	93	α)c	α)c	PUNCT
ma-234	111	94	⊂	⊂	PROPN
ma-234	112	1	f(1	f(1	PROPN
ma-234	112	2	,	,	PUNCT
ma-234	112	3	p	p	X
ma-234	112	4	,	,	PUNCT
ma-234	112	5	α	α	NOUN
ma-234	112	6	)	)	PUNCT
ma-234	112	7	⊂	⊂	PROPN
ma-234	112	8	wb(γ	wb(γ	X
ma-234	112	9	,	,	PUNCT
ma-234	112	10	p	p	NOUN
ma-234	112	11	)	)	PUNCT
ma-234	112	12	,	,	PUNCT
ma-234	112	13	0	0	NUM
ma-234	112	14	<	<	X
ma-234	112	15	γ	γ	X
ma-234	112	16	<	<	X
ma-234	112	17	1	1	NUM
ma-234	112	18	α	α	NOUN
ma-234	112	19	≤	≤	NUM
ma-234	112	20	1	1	NUM
ma-234	112	21	and	and	CCONJ
ma-234	112	22	1	1	NUM
ma-234	112	23	p	p	NOUN
ma-234	112	24	=	=	NOUN
ma-234	112	25	1	1	NUM
ma-234	112	26	α	α	NOUN
ma-234	112	27	−	−	PROPN
ma-234	112	28	γ	γ	X
ma-234	112	29	,	,	PUNCT
ma-234	112	30	(	(	PUNCT
ma-234	112	31	8)	8)	NUM
ma-234	112	32	with	with	ADP
ma-234	112	33	f(q	f(q	PROPN
ma-234	112	34	,	,	PUNCT
ma-234	112	35	p	p	X
ma-234	112	36	,	,	PUNCT
ma-234	112	37	α)c	α)c	PUNCT
ma-234	112	38	=	=	SYM
ma-234	112	39	{	{	PUNCT
ma-234	112	40	f	f	PROPN
ma-234	112	41	∈	∈	PROPN
ma-234	112	42	f(q	f(q	PROPN
ma-234	112	43	,	,	PUNCT
ma-234	112	44	p	p	X
ma-234	112	45	,	,	PUNCT
ma-234	112	46	α	α	NOUN
ma-234	112	47	)	)	PUNCT
ma-234	112	48	:	:	PUNCT
ma-234	112	49	lim	lim	PROPN
ma-234	112	50	y→0	y→0	PROPN
ma-234	112	51	‖f	‖f	PRON
ma-234	112	52	−	−	PROPN
ma-234	112	53	f	f	X
ma-234	112	54	(	(	PUNCT
ma-234	112	55	·	·	PUNCT
ma-234	112	56	−	−	PROPN
ma-234	112	57	y)‖f(q	y)‖f(q	PROPN
ma-234	112	58	,	,	PUNCT
ma-234	112	59	p	p	X
ma-234	112	60	,	,	PUNCT
ma-234	112	61	α	α	NOUN
ma-234	112	62	)	)	PUNCT
ma-234	112	63	=	=	SYM
ma-234	112	64	0	0	PUNCT
ma-234	112	65	}	}	PUNCT
ma-234	112	66	and	and	CCONJ
ma-234	112	67	wb(γ	wb(γ	NOUN
ma-234	112	68	,	,	PUNCT
ma-234	112	69	p	p	X
ma-234	112	70	)	)	PUNCT
ma-234	112	71	=	=	SYM
ma-234	112	72	{	{	PUNCT
ma-234	112	73	f	f	PROPN
ma-234	112	74	∈	∈	PROPN
ma-234	112	75	l1	l1	PROPN
ma-234	112	76	loc	loc	PROPN
ma-234	112	77	:	:	PUNCT
ma-234	112	78	iγ(|f	iγ(|f	NOUN
ma-234	112	79	|	|	NOUN
ma-234	112	80	)	)	PUNCT
ma-234	112	81	∈	∈	PROPN
ma-234	112	82	wlp	wlp	PROPN
ma-234	112	83	}	}	PUNCT
ma-234	112	84	,	,	PUNCT
ma-234	112	85	0	0	PUNCT
ma-234	112	86	<	<	X
ma-234	112	87	γ	γ	X
ma-234	112	88	<	<	X
ma-234	112	89	1	1	NUM
ma-234	112	90	≤	≤	NOUN
ma-234	112	91	p	p	NOUN
ma-234	112	92	≤	≤	NUM
ma-234	112	93	∞.	∞.	PROPN
ma-234	112	94	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	112	95	eur	eur	PROPN
ma-234	112	96	.	.	PUNCT
ma-234	113	1	j.	j.	PROPN
ma-234	113	2	math	math	PROPN
ma-234	113	3	.	.	PUNCT
ma-234	114	1	anal	anal	PROPN
ma-234	114	2	.	.	PUNCT
ma-234	115	1	10.28924	10.28924	NUM
ma-234	115	2	/	/	SYM
ma-234	115	3	ada	ada	PROPN
ma-234	115	4	/	/	SYM
ma-234	115	5	ma.4.16	ma.4.16	PROPN
ma-234	116	1	6note	6note	NUM
ma-234	116	2	that	that	SCONJ
ma-234	116	3	,	,	PUNCT
ma-234	116	4	it	it	PRON
ma-234	116	5	is	be	AUX
ma-234	116	6	proved	prove	VERB
ma-234	116	7	in	in	ADP
ma-234	116	8	[	[	X
ma-234	116	9	7	7	X
ma-234	116	10	]	]	PUNCT
ma-234	116	11	that	that	PRON
ma-234	116	12	f(q	f(q	PROPN
ma-234	116	13	,	,	PUNCT
ma-234	116	14	p	p	X
ma-234	116	15	,	,	PUNCT
ma-234	116	16	α)c	α)c	PUNCT
ma-234	116	17	is	be	AUX
ma-234	116	18	the	the	DET
ma-234	116	19	closure	closure	NOUN
ma-234	116	20	of	of	ADP
ma-234	116	21	lα	lα	NOUN
ma-234	116	22	in	in	ADP
ma-234	116	23	f(q	f(q	PROPN
ma-234	116	24	,	,	PUNCT
ma-234	116	25	p	p	X
ma-234	116	26	,	,	PUNCT
ma-234	116	27	α	α	NOUN
ma-234	116	28	)	)	PUNCT
ma-234	116	29	if	if	SCONJ
ma-234	116	30	p	p	PROPN
ma-234	116	31	<	<	X
ma-234	116	32	∞.	∞.	PROPN
ma-234	116	33	it	it	PRON
ma-234	116	34	is	be	AUX
ma-234	116	35	clearthat	clearthat	ADP
ma-234	116	36	the	the	DET
ma-234	116	37	inclusion	inclusion	NOUN
ma-234	116	38	relations	relation	NOUN
ma-234	116	39	(	(	PUNCT
ma-234	116	40	7	7	NUM
ma-234	116	41	)	)	PUNCT
ma-234	116	42	and	and	CCONJ
ma-234	116	43	(	(	PUNCT
ma-234	116	44	8)	8)	NUM
ma-234	116	45	yield	yield	NOUN
ma-234	116	46	what	what	PRON
ma-234	116	47	follows	follow	VERB
ma-234	116	48	mα	mα	PROPN
ma-234	116	49	1,p	1,p	PROPN
ma-234	116	50	⊂	⊂	PROPN
ma-234	116	51	f(1	f(1	PROPN
ma-234	116	52	,	,	PUNCT
ma-234	116	53	p	p	X
ma-234	116	54	,	,	PUNCT
ma-234	116	55	α)c	α)c	PUNCT
ma-234	116	56	,	,	PUNCT
ma-234	117	1	0	0	PUNCT
ma-234	117	2	<	<	X
ma-234	117	3	γ	γ	X
ma-234	117	4	<	<	X
ma-234	117	5	1	1	NUM
ma-234	117	6	α	α	NOUN
ma-234	117	7	<	<	X
ma-234	117	8	1	1	NUM
ma-234	117	9	and	and	CCONJ
ma-234	117	10	1	1	NUM
ma-234	117	11	p	p	NOUN
ma-234	117	12	=	=	NOUN
ma-234	117	13	1	1	NUM
ma-234	117	14	α	α	NOUN
ma-234	117	15	−	−	PROPN
ma-234	117	16	γ	γ	X
ma-234	117	17	.	.	PROPN
ma-234	117	18	(	(	PUNCT
ma-234	117	19	9	9	NUM
ma-234	117	20	)	)	PUNCT
ma-234	117	21	in	in	ADP
ma-234	117	22	the	the	DET
ma-234	117	23	present	present	ADJ
ma-234	117	24	paper	paper	NOUN
ma-234	117	25	we	we	PRON
ma-234	117	26	prove	prove	VERB
ma-234	117	27	,	,	PUNCT
ma-234	117	28	without	without	ADP
ma-234	117	29	the	the	DET
ma-234	117	30	use	use	NOUN
ma-234	117	31	of	of	ADP
ma-234	117	32	fractional	fractional	ADJ
ma-234	117	33	integral	integral	ADJ
ma-234	117	34	operators	operator	NOUN
ma-234	117	35	,	,	PUNCT
ma-234	117	36	the	the	DET
ma-234	117	37	followingextension	followingextension	NOUN
ma-234	117	38	of	of	ADP
ma-234	117	39	the	the	DET
ma-234	117	40	relation	relation	NOUN
ma-234	117	41	(	(	PUNCT
ma-234	117	42	9	9	NUM
ma-234	117	43	)	)	PUNCT
ma-234	117	44	.	.	PUNCT
ma-234	118	1	theorem	theorem	VERB
ma-234	118	2	2.3	2.3	NUM
ma-234	118	3	.	.	PUNCT
ma-234	119	1	let	let	VERB
ma-234	119	2	us	we	PRON
ma-234	119	3	assume	assume	VERB
ma-234	119	4	that	that	SCONJ
ma-234	119	5	1	1	NUM
ma-234	119	6	≤	≤	NUM
ma-234	119	7	q	q	ADJ
ma-234	119	8	≤	≤	NUM
ma-234	119	9	α	α	NOUN
ma-234	119	10	≤	≤	NOUN
ma-234	119	11	p	p	NOUN
ma-234	119	12	≤	≤	NOUN
ma-234	119	13	∞.	∞.	PROPN
ma-234	119	14	then	then	ADV
ma-234	119	15	‖f	‖f	ADP
ma-234	119	16	‖f(q	‖f(q	NOUN
ma-234	119	17	,	,	PUNCT
ma-234	119	18	p	p	X
ma-234	119	19	,	,	PUNCT
ma-234	119	20	α	α	NOUN
ma-234	119	21	)	)	PUNCT
ma-234	119	22	≤	≤	NOUN
ma-234	120	1	3	3	NUM
ma-234	120	2	d	d	NOUN
ma-234	120	3	(	(	PUNCT
ma-234	120	4	1	1	NUM
ma-234	120	5	+	+	NUM
ma-234	120	6	1	1	NUM
ma-234	120	7	α	α	NOUN
ma-234	120	8	−	−	NOUN
ma-234	120	9	1	1	NUM
ma-234	120	10	q	q	NOUN
ma-234	120	11	+	+	NUM
ma-234	120	12	1	1	NUM
ma-234	120	13	p	p	NOUN
ma-234	120	14	)	)	PUNCT
ma-234	120	15	2	2	NUM
ma-234	120	16	d	d	NOUN
ma-234	120	17	(	(	PUNCT
ma-234	120	18	1	1	NUM
ma-234	120	19	q	q	NOUN
ma-234	120	20	−	−	PROPN
ma-234	120	21	1	1	NUM
ma-234	120	22	p	p	NOUN
ma-234	120	23	)	)	PUNCT
ma-234	120	24	‖f	‖f	PUNCT
ma-234	121	1	‖mα	‖mα	NUM
ma-234	121	2	q	q	NOUN
ma-234	121	3	,	,	PUNCT
ma-234	121	4	p	p	NOUN
ma-234	121	5	,	,	PUNCT
ma-234	121	6	f	f	PROPN
ma-234	121	7	∈	∈	PROPN
ma-234	121	8	l1	l1	PROPN
ma-234	121	9	loc	loc	PROPN
ma-234	121	10	(	(	PUNCT
ma-234	121	11	10	10	NUM
ma-234	121	12	)	)	PUNCT
ma-234	121	13	and	and	CCONJ
ma-234	121	14	therefore	therefore	ADV
ma-234	121	15	mα	mα	PROPN
ma-234	121	16	q	q	AUX
ma-234	121	17	,	,	PUNCT
ma-234	121	18	p	p	PRON
ma-234	121	19	is	be	AUX
ma-234	121	20	continuously	continuously	ADV
ma-234	121	21	included	include	VERB
ma-234	121	22	in	in	ADP
ma-234	121	23	f(q	f(q	PROPN
ma-234	121	24	,	,	PUNCT
ma-234	121	25	p	p	X
ma-234	121	26	,	,	PUNCT
ma-234	121	27	α	α	NOUN
ma-234	121	28	)	)	PUNCT
ma-234	121	29	.	.	PUNCT
ma-234	122	1	moreover	moreover	ADV
ma-234	122	2	,	,	PUNCT
ma-234	122	3	if	if	SCONJ
ma-234	122	4	p	p	X
ma-234	122	5	<	<	X
ma-234	122	6	∞	∞	PROPN
ma-234	122	7	then	then	ADV
ma-234	122	8	mα	mα	PROPN
ma-234	122	9	q	q	NOUN
ma-234	122	10	,	,	PUNCT
ma-234	122	11	p	p	NOUN
ma-234	122	12	is	be	AUX
ma-234	122	13	included	include	VERB
ma-234	122	14	in	in	ADP
ma-234	122	15	f(q	f(q	PROPN
ma-234	122	16	,	,	PUNCT
ma-234	122	17	p	p	X
ma-234	122	18	,	,	PUNCT
ma-234	122	19	α)c	α)c	X
ma-234	122	20	.	.	PUNCT
ma-234	123	1	as	as	SCONJ
ma-234	123	2	done	do	VERB
ma-234	123	3	in	in	ADP
ma-234	123	4	[	[	X
ma-234	123	5	4	4	NUM
ma-234	123	6	,	,	PUNCT
ma-234	123	7	6	6	NUM
ma-234	123	8	]	]	PUNCT
ma-234	123	9	for	for	ADP
ma-234	123	10	some	some	DET
ma-234	123	11	special	special	ADJ
ma-234	123	12	subspaces	subspace	NOUN
ma-234	123	13	of	of	ADP
ma-234	123	14	the	the	DET
ma-234	123	15	morrey	morrey	NOUN
ma-234	123	16	-	-	PUNCT
ma-234	123	17	type	type	NOUN
ma-234	123	18	space	space	NOUN
ma-234	123	19	(	(	PUNCT
ma-234	123	20	lq	lq	PROPN
ma-234	123	21	,	,	PUNCT
ma-234	123	22	lp)α	lp)α	PROPN
ma-234	123	23	,	,	PUNCT
ma-234	123	24	usually	usually	ADV
ma-234	123	25	calledthe	calledthe	ADJ
ma-234	123	26	fofana	fofana	ADJ
ma-234	123	27	space	space	NOUN
ma-234	123	28	and	and	CCONJ
ma-234	123	29	closely	closely	ADV
ma-234	123	30	related	relate	VERB
ma-234	123	31	to	to	ADP
ma-234	123	32	f(q	f(q	PROPN
ma-234	123	33	,	,	PUNCT
ma-234	123	34	p	p	X
ma-234	123	35	,	,	PUNCT
ma-234	123	36	α	α	NOUN
ma-234	123	37	)	)	PUNCT
ma-234	123	38	,	,	PUNCT
ma-234	123	39	we	we	PRON
ma-234	123	40	investigate	investigate	VERB
ma-234	123	41	in	in	ADP
ma-234	123	42	bourgain	bourgain	NOUN
ma-234	123	43	-	-	PUNCT
ma-234	123	44	morrey	morrey	NOUN
ma-234	123	45	spacesapproximation	spacesapproximation	NOUN
ma-234	123	46	by	by	ADP
ma-234	123	47	smooth	smooth	ADJ
ma-234	123	48	functions	function	NOUN
ma-234	123	49	.	.	PUNCT
ma-234	124	1	we	we	PRON
ma-234	124	2	shall	shall	AUX
ma-234	124	3	prove	prove	VERB
ma-234	124	4	what	what	PRON
ma-234	124	5	follows	follow	VERB
ma-234	124	6	.	.	PUNCT
ma-234	125	1	theorem	theorem	ADJ
ma-234	125	2	2.4	2.4	NUM
ma-234	125	3	.	.	PUNCT
ma-234	126	1	let	let	VERB
ma-234	126	2	1	1	NUM
ma-234	126	3	≤	≤	NOUN
ma-234	126	4	q	q	ADJ
ma-234	126	5	≤	≤	NUM
ma-234	126	6	α	α	NOUN
ma-234	126	7	≤	≤	NOUN
ma-234	127	1	p	p	DET
ma-234	127	2	<	<	X
ma-234	127	3	∞	∞	PROPN
ma-234	127	4	and	and	CCONJ
ma-234	127	5	f	f	PROPN
ma-234	127	6	be	be	AUX
ma-234	127	7	an	an	DET
ma-234	127	8	element	element	NOUN
ma-234	127	9	of	of	ADP
ma-234	127	10	l1	l1	PROPN
ma-234	127	11	loc	loc	PROPN
ma-234	127	12	.	.	PUNCT
ma-234	128	1	then	then	ADV
ma-234	128	2	the	the	DET
ma-234	128	3	following	follow	VERB
ma-234	128	4	assertions	assertion	NOUN
ma-234	128	5	are	be	AUX
ma-234	128	6	equivalent	equivalent	ADJ
ma-234	128	7	:	:	PUNCT
ma-234	128	8	(	(	PUNCT
ma-234	128	9	i	i	NOUN
ma-234	128	10	)	)	PUNCT
ma-234	128	11	f	f	PROPN
ma-234	128	12	belongs	belong	VERB
ma-234	128	13	to	to	ADP
ma-234	128	14	mα	mα	PROPN
ma-234	128	15	q	q	NOUN
ma-234	128	16	,	,	PUNCT
ma-234	128	17	p	p	X
ma-234	128	18	,	,	PUNCT
ma-234	128	19	(	(	PUNCT
ma-234	128	20	ii	ii	NOUN
ma-234	128	21	)	)	PUNCT
ma-234	128	22	lim	lim	PROPN
ma-234	128	23	n→∞	n→∞	NUM
ma-234	129	1	‖f	‖f	PUNCT
ma-234	129	2	−	−	PROPN
ma-234	129	3	f	f	PROPN
ma-234	129	4	∗	∗	X
ma-234	129	5	φn‖mα	φn‖mα	PUNCT
ma-234	129	6	q	q	NOUN
ma-234	129	7	,	,	PUNCT
ma-234	129	8	p	p	X
ma-234	129	9	=	=	NOUN
ma-234	129	10	0	0	NUM
ma-234	129	11	,	,	PUNCT
ma-234	129	12	where	where	SCONJ
ma-234	129	13	f	f	PROPN
ma-234	129	14	∗	∗	VERB
ma-234	129	15	φn	φn	VERB
ma-234	129	16	is	be	AUX
ma-234	129	17	the	the	DET
ma-234	129	18	convolution	convolution	NOUN
ma-234	129	19	product	product	NOUN
ma-234	129	20	of	of	ADP
ma-234	129	21	f	f	PROPN
ma-234	129	22	and	and	CCONJ
ma-234	129	23	φn	φn	ADP
ma-234	129	24	,	,	PUNCT
ma-234	129	25	(	(	PUNCT
ma-234	129	26	iii	iii	X
ma-234	129	27	)	)	PUNCT
ma-234	129	28	f	f	PROPN
ma-234	129	29	belongs	belong	VERB
ma-234	129	30	to	to	ADP
ma-234	129	31	the	the	DET
ma-234	129	32	closure	closure	NOUN
ma-234	129	33	in	in	ADP
ma-234	129	34	mα	mα	PROPN
ma-234	129	35	q	q	NOUN
ma-234	129	36	,	,	PUNCT
ma-234	129	37	p	p	NOUN
ma-234	129	38	of	of	ADP
ma-234	129	39	the	the	DET
ma-234	129	40	set	set	NOUN
ma-234	129	41	c∞mα	c∞mα	NOUN
ma-234	129	42	q	q	NOUN
ma-234	129	43	,	,	PUNCT
ma-234	129	44	p	p	NOUN
ma-234	129	45	=	=	PUNCT
ma-234	129	46	{	{	PUNCT
ma-234	129	47	g	g	PROPN
ma-234	129	48	∈	∈	PROPN
ma-234	129	49	c∞	c∞	PROPN
ma-234	129	50	:	:	PUNCT
ma-234	129	51	∂βg	∂βg	VERB
ma-234	129	52	∈mα	∈mα	NOUN
ma-234	129	53	q	q	PROPN
ma-234	129	54	,	,	PUNCT
ma-234	129	55	p	p	NOUN
ma-234	129	56	for	for	ADP
ma-234	129	57	any	any	DET
ma-234	129	58	β	β	NOUN
ma-234	129	59	in	in	ADP
ma-234	129	60	nd	nd	ADP
ma-234	129	61	}	}	PUNCT
ma-234	129	62	,	,	PUNCT
ma-234	129	63	where	where	SCONJ
ma-234	129	64	∂βg	∂βg	NOUN
ma-234	129	65	stands	stand	VERB
ma-234	129	66	for	for	ADP
ma-234	129	67	the	the	DET
ma-234	129	68	derivative	derivative	NOUN
ma-234	129	69	of	of	ADP
ma-234	129	70	order	order	NOUN
ma-234	129	71	β	β	X
ma-234	129	72	of	of	ADP
ma-234	129	73	g.	g.	PROPN
ma-234	129	74	note	note	PROPN
ma-234	129	75	that	that	SCONJ
ma-234	129	76	theorem	theorem	VERB
ma-234	129	77	2.4	2.4	NUM
ma-234	129	78	implies	imply	VERB
ma-234	129	79	that	that	SCONJ
ma-234	129	80	both	both	CCONJ
ma-234	129	81	c∞	c∞	PROPN
ma-234	129	82	∩mα	∩mα	NOUN
ma-234	129	83	q	q	NOUN
ma-234	129	84	,	,	PUNCT
ma-234	129	85	p	p	NOUN
ma-234	129	86	and	and	CCONJ
ma-234	129	87	c∞mα	c∞mα	ADJ
ma-234	129	88	q	q	NOUN
ma-234	129	89	,	,	PUNCT
ma-234	129	90	p	p	NOUN
ma-234	129	91	are	be	AUX
ma-234	129	92	dense	dense	ADJ
ma-234	129	93	inmα	inmα	ADJ
ma-234	129	94	q	q	NOUN
ma-234	129	95	,	,	PUNCT
ma-234	129	96	p	p	NOUN
ma-234	129	97	if	if	SCONJ
ma-234	129	98	p	p	PROPN
ma-234	129	99	<	<	X
ma-234	129	100	∞.	∞.	PROPN
ma-234	129	101	asa	asa	PROPN
ma-234	129	102	consequence	consequence	NOUN
ma-234	129	103	of	of	ADP
ma-234	129	104	this	this	DET
ma-234	129	105	theorem	theorem	NOUN
ma-234	129	106	,	,	PUNCT
ma-234	129	107	we	we	PRON
ma-234	129	108	obtain	obtain	VERB
ma-234	129	109	the	the	DET
ma-234	129	110	following	follow	VERB
ma-234	129	111	approximation	approximation	NOUN
ma-234	129	112	result	result	NOUN
ma-234	129	113	.	.	PUNCT
ma-234	130	1	theorem	theorem	VERB
ma-234	130	2	2.5	2.5	NUM
ma-234	130	3	.	.	PUNCT
ma-234	131	1	let	let	VERB
ma-234	131	2	1	1	NUM
ma-234	131	3	≤	≤	NOUN
ma-234	131	4	q	q	ADJ
ma-234	131	5	≤	≤	NUM
ma-234	131	6	α	α	NOUN
ma-234	131	7	≤	≤	NOUN
ma-234	132	1	p	p	DET
ma-234	132	2	<	<	X
ma-234	132	3	∞	∞	PROPN
ma-234	132	4	and	and	CCONJ
ma-234	132	5	f	f	PROPN
ma-234	132	6	be	be	AUX
ma-234	132	7	any	any	DET
ma-234	132	8	element	element	NOUN
ma-234	132	9	of	of	ADP
ma-234	132	10	mα	mα	PROPN
ma-234	132	11	q	q	NOUN
ma-234	132	12	,	,	PUNCT
ma-234	132	13	p	p	NOUN
ma-234	132	14	.	.	PUNCT
ma-234	133	1	then	then	ADV
ma-234	133	2	lim	lim	PROPN
ma-234	133	3	n→∞	n→∞	NUM
ma-234	133	4	‖f	‖f	PUNCT
ma-234	133	5	−	−	PROPN
ma-234	134	1	(	(	PUNCT
ma-234	134	2	f	f	NOUN
ma-234	134	3	ωn	ωn	PROPN
ma-234	134	4	)	)	PUNCT
ma-234	134	5	∗	∗	NOUN
ma-234	134	6	φn‖mα	φn‖mα	PUNCT
ma-234	134	7	q	q	NOUN
ma-234	134	8	,	,	PUNCT
ma-234	134	9	p	p	X
ma-234	134	10	=	=	NOUN
ma-234	134	11	0	0	NUM
ma-234	134	12	.	.	PUNCT
ma-234	135	1	it	it	PRON
ma-234	135	2	is	be	AUX
ma-234	135	3	easy	easy	ADJ
ma-234	135	4	to	to	PART
ma-234	135	5	see	see	VERB
ma-234	135	6	that	that	PRON
ma-234	135	7	,	,	PUNCT
ma-234	135	8	for	for	ADP
ma-234	135	9	any	any	DET
ma-234	135	10	integer	integer	NOUN
ma-234	135	11	n	n	PRON
ma-234	135	12	≥	≥	NOUN
ma-234	135	13	1	1	NUM
ma-234	135	14	,	,	PUNCT
ma-234	135	15	(	(	PUNCT
ma-234	135	16	f	f	NOUN
ma-234	135	17	ωn	ωn	PROPN
ma-234	135	18	)	)	PUNCT
ma-234	135	19	∗	∗	NOUN
ma-234	135	20	φn	φn	PROPN
ma-234	135	21	belongs	belong	VERB
ma-234	135	22	to	to	PART
ma-234	135	23	c∞c	c∞c	VERB
ma-234	135	24	.	.	PUNCT
ma-234	136	1	consequently	consequently	ADV
ma-234	136	2	,	,	PUNCT
ma-234	136	3	theorem2.5	theorem2.5	PROPN
ma-234	136	4	implies	imply	VERB
ma-234	136	5	that	that	SCONJ
ma-234	136	6	c∞c	c∞c	ADJ
ma-234	136	7	∩mα	∩mα	NOUN
ma-234	136	8	q	q	NOUN
ma-234	136	9	,	,	PUNCT
ma-234	136	10	p	p	PRON
ma-234	136	11	is	be	AUX
ma-234	136	12	dense	dense	ADJ
ma-234	136	13	in	in	ADP
ma-234	136	14	mα	mα	PROPN
ma-234	136	15	q	q	NOUN
ma-234	136	16	,	,	PUNCT
ma-234	136	17	p	p	X
ma-234	136	18	if	if	SCONJ
ma-234	136	19	p	p	PROPN
ma-234	136	20	<	<	AUX
ma-234	136	21	∞.let	∞.let	NOUN
ma-234	136	22	us	we	PRON
ma-234	136	23	consider	consider	VERB
ma-234	136	24	the	the	DET
ma-234	136	25	divergence	divergence	NOUN
ma-234	136	26	equation	equation	NOUN
ma-234	136	27	divf	divf	NOUN
ma-234	136	28	=	=	PROPN
ma-234	136	29	f	f	PROPN
ma-234	136	30	,	,	PUNCT
ma-234	136	31	f	f	PROPN
ma-234	136	32	∈	∈	PROPN
ma-234	136	33	l1	l1	PROPN
ma-234	136	34	loc	loc	PROPN
ma-234	136	35	.	.	PUNCT
ma-234	137	1	(	(	PUNCT
ma-234	137	2	11	11	NUM
ma-234	137	3	)	)	PUNCT
ma-234	137	4	to	to	ADP
ma-234	137	5	our	our	PRON
ma-234	137	6	knowledge	knowledge	NOUN
ma-234	137	7	,	,	PUNCT
ma-234	137	8	for	for	ADP
ma-234	137	9	a	a	DET
ma-234	137	10	given	give	VERB
ma-234	137	11	p	p	NOUN
ma-234	137	12	in	in	ADP
ma-234	137	13	[	[	NOUN
ma-234	137	14	1,∞	1,∞	NUM
ma-234	137	15	)	)	PUNCT
ma-234	137	16	,	,	PUNCT
ma-234	137	17	the	the	DET
ma-234	137	18	characterization	characterization	NOUN
ma-234	137	19	of	of	ADP
ma-234	137	20	the	the	DET
ma-234	137	21	class	class	NOUN
ma-234	137	22	of	of	ADP
ma-234	137	23	functions	function	NOUN
ma-234	137	24	f	f	X
ma-234	137	25	forwhich	forwhich	VERB
ma-234	137	26	the	the	DET
ma-234	137	27	equation	equation	NOUN
ma-234	137	28	(	(	PUNCT
ma-234	137	29	11	11	NUM
ma-234	137	30	)	)	PUNCT
ma-234	137	31	has	have	VERB
ma-234	137	32	a	a	DET
ma-234	137	33	solution	solution	NOUN
ma-234	138	1	f	f	NOUN
ma-234	138	2	=	=	PRON
ma-234	138	3	(	(	PUNCT
ma-234	138	4	fj)1≤j≤d	fj)1≤j≤d	PROPN
ma-234	138	5	in	in	ADP
ma-234	138	6	(	(	PUNCT
ma-234	138	7	lp)d	lp)d	PROPN
ma-234	138	8	is	be	AUX
ma-234	138	9	still	still	ADV
ma-234	138	10	an	an	DET
ma-234	138	11	open	open	ADJ
ma-234	138	12	problem	problem	NOUN
ma-234	138	13	.	.	PUNCT
ma-234	139	1	however	however	ADV
ma-234	139	2	,	,	PUNCT
ma-234	139	3	phuc	phuc	PROPN
ma-234	139	4	and	and	CCONJ
ma-234	139	5	torres	torre	NOUN
ma-234	139	6	proved	prove	VERB
ma-234	139	7	that	that	SCONJ
ma-234	139	8	(	(	PUNCT
ma-234	139	9	see	see	VERB
ma-234	139	10	[	[	X
ma-234	139	11	13	13	NUM
ma-234	139	12	,	,	PUNCT
ma-234	139	13	theorem	theorem	VERB
ma-234	139	14	3.2	3.2	NUM
ma-234	139	15	]	]	PUNCT
ma-234	139	16	)	)	PUNCT
ma-234	139	17	,	,	PUNCT
ma-234	139	18	for	for	ADP
ma-234	139	19	d	d	PROPN
ma-234	139	20	d−1	d−1	PROPN
ma-234	139	21	<	<	X
ma-234	139	22	p	p	X
ma-234	139	23	<	<	X
ma-234	139	24	∞	∞	PROPN
ma-234	139	25	,	,	PUNCT
ma-234	139	26	the	the	DET
ma-234	139	27	equation	equation	NOUN
ma-234	139	28	(	(	PUNCT
ma-234	139	29	11	11	NUM
ma-234	139	30	)	)	PUNCT
ma-234	139	31	has	have	VERB
ma-234	139	32	a	a	DET
ma-234	139	33	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	139	34	eur	eur	PROPN
ma-234	139	35	.	.	PUNCT
ma-234	140	1	j.	j.	PROPN
ma-234	140	2	math	math	PROPN
ma-234	140	3	.	.	PUNCT
ma-234	141	1	anal	anal	PROPN
ma-234	141	2	.	.	PUNCT
ma-234	142	1	10.28924	10.28924	NUM
ma-234	142	2	/	/	SYM
ma-234	142	3	ada	ada	PROPN
ma-234	142	4	/	/	SYM
ma-234	142	5	ma.4.16	ma.4.16	PROPN
ma-234	143	1	7solution	7solution	NUM
ma-234	143	2	in	in	ADP
ma-234	143	3	(	(	PUNCT
ma-234	143	4	lp)d	lp)d	PROPN
ma-234	143	5	if	if	SCONJ
ma-234	144	1	and	and	CCONJ
ma-234	144	2	only	only	ADV
ma-234	144	3	if	if	SCONJ
ma-234	144	4	f	f	PROPN
ma-234	144	5	belongs	belong	VERB
ma-234	144	6	to	to	ADP
ma-234	144	7	the	the	DET
ma-234	144	8	set	set	PROPN
ma-234	144	9	b	b	PROPN
ma-234	144	10	(	(	PUNCT
ma-234	144	11	1	1	NUM
ma-234	144	12	d	d	NOUN
ma-234	144	13	,	,	PUNCT
ma-234	144	14	p	p	NOUN
ma-234	144	15	)	)	PUNCT
ma-234	144	16	.	.	PUNCT
ma-234	145	1	this	this	DET
ma-234	145	2	result	result	NOUN
ma-234	145	3	combined	combine	VERB
ma-234	145	4	with	with	ADP
ma-234	145	5	(	(	PUNCT
ma-234	145	6	7	7	X
ma-234	145	7	)	)	PUNCT
ma-234	145	8	showsthat	showsthat	NOUN
ma-234	145	9	,	,	PUNCT
ma-234	145	10	for	for	ADP
ma-234	145	11	1	1	NUM
ma-234	145	12	<	<	X
ma-234	145	13	α	α	X
ma-234	145	14	<	<	X
ma-234	145	15	d	d	NOUN
ma-234	145	16	and	and	CCONJ
ma-234	145	17	1	1	NUM
ma-234	145	18	p	p	NOUN
ma-234	145	19	=	=	NOUN
ma-234	145	20	1	1	NUM
ma-234	145	21	α−	α−	ADP
ma-234	145	22	1	1	NUM
ma-234	145	23	d	d	NOUN
ma-234	145	24	,	,	PUNCT
ma-234	145	25	a	a	DET
ma-234	145	26	sufficient	sufficient	ADJ
ma-234	145	27	condition	condition	NOUN
ma-234	145	28	for	for	ADP
ma-234	145	29	the	the	DET
ma-234	145	30	solvability	solvability	NOUN
ma-234	145	31	in	in	ADP
ma-234	145	32	(	(	PUNCT
ma-234	145	33	lp)d	lp)d	PROPN
ma-234	145	34	of	of	ADP
ma-234	145	35	the	the	DET
ma-234	145	36	equation(11	equation(11	NOUN
ma-234	145	37	)	)	PUNCT
ma-234	145	38	is	be	AUX
ma-234	145	39	that	that	SCONJ
ma-234	145	40	f	f	PROPN
ma-234	145	41	belongs	belong	VERB
ma-234	145	42	to	to	ADP
ma-234	145	43	mα	mα	PROPN
ma-234	145	44	1,p	1,p	PROPN
ma-234	145	45	.	.	PUNCT
ma-234	146	1	moreover	moreover	ADV
ma-234	146	2	,	,	PUNCT
ma-234	146	3	an	an	DET
ma-234	146	4	application	application	NOUN
ma-234	146	5	of	of	ADP
ma-234	146	6	theorem	theorem	ADJ
ma-234	146	7	2.2	2.2	NUM
ma-234	146	8	and	and	CCONJ
ma-234	146	9	theorem	theorem	VERB
ma-234	146	10	2.5	2.5	NUM
ma-234	146	11	allowsus	allowsus	NOUN
ma-234	146	12	to	to	PART
ma-234	146	13	obtain	obtain	VERB
ma-234	146	14	an	an	DET
ma-234	146	15	explicit	explicit	ADJ
ma-234	146	16	solution	solution	NOUN
ma-234	146	17	of	of	ADP
ma-234	146	18	the	the	DET
ma-234	146	19	equation	equation	NOUN
ma-234	146	20	(	(	PUNCT
ma-234	146	21	11	11	NUM
ma-234	146	22	)	)	PUNCT
ma-234	146	23	in	in	ADP
ma-234	146	24	(	(	PUNCT
ma-234	146	25	lp)d	lp)d	PROPN
ma-234	146	26	,	,	PUNCT
ma-234	146	27	as	as	SCONJ
ma-234	146	28	shown	show	VERB
ma-234	146	29	below	below	ADV
ma-234	146	30	.	.	PUNCT
ma-234	147	1	theorem	theorem	VERB
ma-234	147	2	2.6	2.6	NUM
ma-234	147	3	.	.	PUNCT
ma-234	148	1	let	let	VERB
ma-234	148	2	us	we	PRON
ma-234	148	3	assume	assume	VERB
ma-234	148	4	that	that	SCONJ
ma-234	148	5	d	d	PROPN
ma-234	148	6	≥	≥	NUM
ma-234	148	7	3	3	NUM
ma-234	148	8	,	,	PUNCT
ma-234	148	9	1	1	NUM
ma-234	148	10	≤	≤	NUM
ma-234	148	11	q	q	ADJ
ma-234	148	12	≤	≤	NUM
ma-234	148	13	α	α	NOUN
ma-234	148	14	<	<	X
ma-234	148	15	d	d	X
ma-234	148	16	,	,	PUNCT
ma-234	148	17	1	1	NUM
ma-234	148	18	p	p	NOUN
ma-234	148	19	=	=	NOUN
ma-234	149	1	1	1	NUM
ma-234	149	2	α	α	NOUN
ma-234	149	3	−	−	NOUN
ma-234	149	4	1	1	NUM
ma-234	149	5	d	d	NOUN
ma-234	149	6	and	and	CCONJ
ma-234	149	7	f	f	PROPN
ma-234	149	8	is	be	AUX
ma-234	149	9	an	an	DET
ma-234	149	10	element	element	NOUN
ma-234	149	11	ofmα	ofmα	ADJ
ma-234	149	12	q	q	NOUN
ma-234	149	13	,	,	PUNCT
ma-234	149	14	p	p	NOUN
ma-234	149	15	.	.	PUNCT
ma-234	150	1	then	then	ADV
ma-234	150	2	there	there	PRON
ma-234	150	3	exists	exist	VERB
ma-234	150	4	a	a	DET
ma-234	150	5	real	real	ADJ
ma-234	150	6	constant	constant	ADJ
ma-234	150	7	cd	cd	NOUN
ma-234	150	8	such	such	ADJ
ma-234	150	9	that	that	SCONJ
ma-234	150	10	f	f	PROPN
ma-234	150	11	=	=	PRON
ma-234	151	1	(	(	PUNCT
ma-234	151	2	cd	cd	PROPN
ma-234	151	3	rj	rj	PROPN
ma-234	151	4	(	(	PUNCT
ma-234	151	5	i	i	NOUN
ma-234	151	6	1	1	NUM
ma-234	151	7	d	d	PROPN
ma-234	151	8	f	f	PROPN
ma-234	151	9	)	)	PUNCT
ma-234	151	10	)	)	PUNCT
ma-234	152	1	1≤j≤d	1≤j≤d	NUM
ma-234	152	2	is	be	AUX
ma-234	152	3	a	a	DET
ma-234	152	4	solution	solution	NOUN
ma-234	152	5	in	in	ADP
ma-234	152	6	(	(	PUNCT
ma-234	152	7	lp)d	lp)d	PROPN
ma-234	152	8	of	of	ADP
ma-234	152	9	the	the	DET
ma-234	152	10	equation	equation	NOUN
ma-234	152	11	(	(	PUNCT
ma-234	152	12	11	11	NUM
ma-234	152	13	)	)	PUNCT
ma-234	152	14	,	,	PUNCT
ma-234	152	15	where	where	SCONJ
ma-234	152	16	rj	rj	PROPN
ma-234	152	17	(	(	PUNCT
ma-234	152	18	1	1	NUM
ma-234	152	19	≤	≤	NUM
ma-234	152	20	j	j	PROPN
ma-234	152	21	≤	≤	PROPN
ma-234	152	22	d	d	X
ma-234	152	23	)	)	PUNCT
ma-234	152	24	stands	stand	VERB
ma-234	152	25	for	for	ADP
ma-234	152	26	the	the	DET
ma-234	152	27	riesz	riesz	PROPN
ma-234	152	28	transform	transform	NOUN
ma-234	152	29	defined	define	VERB
ma-234	152	30	by	by	ADP
ma-234	152	31	rjϕ(x	rjϕ(x	PROPN
ma-234	152	32	)	)	PUNCT
ma-234	153	1	=	=	SYM
ma-234	153	2	γ	γ	X
ma-234	153	3	(	(	PUNCT
ma-234	153	4	d+1	d+1	PROPN
ma-234	153	5	2	2	NUM
ma-234	153	6	)	)	PUNCT
ma-234	153	7	π	π	PROPN
ma-234	153	8	d+1	d+1	PROPN
ma-234	153	9	2	2	NUM
ma-234	153	10	lim	lim	NOUN
ma-234	153	11	ε→0	ε→0	NOUN
ma-234	153	12	+	+	CCONJ
ma-234	153	13	∫	∫	PROPN
ma-234	153	14	|x−y	|x−y	PROPN
ma-234	153	15	|≥ε	|≥ε	PROPN
ma-234	153	16	xj	xj	PROPN
ma-234	153	17	−	−	PROPN
ma-234	153	18	yj	yj	PROPN
ma-234	153	19	|x	|x	PROPN
ma-234	153	20	−	−	PROPN
ma-234	153	21	y	y	PROPN
ma-234	153	22	|d+1	|d+1	NOUN
ma-234	153	23	ϕ(y)dy	ϕ(y)dy	X
ma-234	153	24	,	,	PUNCT
ma-234	153	25	x	x	PROPN
ma-234	153	26	∈	∈	PROPN
ma-234	153	27	rd	rd	PROPN
ma-234	153	28	,	,	PUNCT
ma-234	153	29	ϕ	ϕ	PROPN
ma-234	153	30	∈	∈	PROPN
ma-234	153	31	lp	lp	NOUN
ma-234	153	32	.	.	PROPN
ma-234	154	1	3	3	X
ma-234	154	2	.	.	X
ma-234	154	3	preliminaries	preliminary	NOUN
ma-234	154	4	this	this	DET
ma-234	154	5	section	section	NOUN
ma-234	154	6	is	be	AUX
ma-234	154	7	devoted	devote	VERB
ma-234	154	8	to	to	PART
ma-234	154	9	prove	prove	VERB
ma-234	154	10	some	some	DET
ma-234	154	11	preliminary	preliminary	ADJ
ma-234	154	12	results	result	NOUN
ma-234	154	13	.	.	PUNCT
ma-234	155	1	3.1	3.1	NUM
ma-234	155	2	.	.	PUNCT
ma-234	155	3	equivalent	equivalent	ADJ
ma-234	155	4	norms	norm	NOUN
ma-234	155	5	on	on	ADP
ma-234	155	6	mα	mα	PROPN
ma-234	155	7	q	q	NOUN
ma-234	155	8	,	,	PUNCT
ma-234	155	9	p.	p.	NOUN
ma-234	155	10	we	we	PRON
ma-234	155	11	begin	begin	VERB
ma-234	155	12	this	this	DET
ma-234	155	13	subsection	subsection	NOUN
ma-234	155	14	by	by	ADP
ma-234	155	15	recalling	recall	VERB
ma-234	155	16	the	the	DET
ma-234	155	17	definition	definition	NOUN
ma-234	155	18	of	of	ADP
ma-234	155	19	classicaldyadic	classicaldyadic	ADJ
ma-234	155	20	grids	grid	NOUN
ma-234	155	21	.	.	PUNCT
ma-234	156	1	definition	definition	NOUN
ma-234	156	2	3.1	3.1	NUM
ma-234	156	3	.	.	PUNCT
ma-234	157	1	a	a	DET
ma-234	157	2	dyadic	dyadic	ADJ
ma-234	157	3	grid	grid	NOUN
ma-234	157	4	is	be	AUX
ma-234	157	5	a	a	DET
ma-234	157	6	countable	countable	ADJ
ma-234	157	7	collection	collection	NOUN
ma-234	157	8	d	d	NOUN
ma-234	157	9	of	of	ADP
ma-234	157	10	cubes	cube	NOUN
ma-234	157	11	of	of	ADP
ma-234	157	12	rd	rd	NOUN
ma-234	157	13	which	which	PRON
ma-234	157	14	are	be	AUX
ma-234	157	15	dyadic	dyadic	ADJ
ma-234	157	16	translates	translate	NOUN
ma-234	157	17	and	and	CCONJ
ma-234	157	18	dilations	dilation	NOUN
ma-234	157	19	of	of	ADP
ma-234	157	20	the	the	DET
ma-234	157	21	unit	unit	NOUN
ma-234	157	22	cube	cube	NOUN
ma-234	158	1	[	[	X
ma-234	158	2	0	0	NUM
ma-234	158	3	,	,	PUNCT
ma-234	158	4	1)d	1)d	NUM
ma-234	158	5	.	.	PUNCT
ma-234	159	1	more	more	ADV
ma-234	159	2	precisely	precisely	ADV
ma-234	159	3	,	,	PUNCT
ma-234	159	4	d	d	NOUN
ma-234	159	5	may	may	AUX
ma-234	159	6	be	be	AUX
ma-234	159	7	characterized	characterize	VERB
ma-234	159	8	as	as	SCONJ
ma-234	159	9	follows	follow	VERB
ma-234	159	10	:	:	PUNCT
ma-234	159	11	(	(	PUNCT
ma-234	159	12	i	i	NOUN
ma-234	159	13	)	)	PUNCT
ma-234	160	1	if	if	SCONJ
ma-234	160	2	q	q	X
ma-234	160	3	∈	∈	PROPN
ma-234	161	1	d	d	X
ma-234	161	2	then	then	ADV
ma-234	161	3	its	its	PRON
ma-234	161	4	side	side	ADJ
ma-234	161	5	-	-	PUNCT
ma-234	161	6	length	length	NOUN
ma-234	161	7	`	`	PUNCT
ma-234	161	8	(	(	PUNCT
ma-234	161	9	q	q	X
ma-234	161	10	)	)	PUNCT
ma-234	161	11	=	=	SYM
ma-234	161	12	2	2	NUM
ma-234	161	13	m	m	NOUN
ma-234	161	14	for	for	ADP
ma-234	161	15	some	some	DET
ma-234	161	16	m	m	NOUN
ma-234	161	17	∈	∈	PROPN
ma-234	161	18	z	z	X
ma-234	161	19	(	(	PUNCT
ma-234	161	20	ii	ii	NOUN
ma-234	161	21	)	)	PUNCT
ma-234	161	22	if	if	SCONJ
ma-234	161	23	q	q	ADJ
ma-234	161	24	,	,	PUNCT
ma-234	161	25	p	p	NOUN
ma-234	161	26	∈	∈	PROPN
ma-234	162	1	d	d	X
ma-234	162	2	then	then	ADV
ma-234	162	3	q	q	PROPN
ma-234	162	4	∩	∩	PROPN
ma-234	162	5	p	p	PROPN
ma-234	162	6	∈	∈	PROPN
ma-234	162	7	{	{	PUNCT
ma-234	162	8	∅	∅	NOUN
ma-234	162	9	,	,	PUNCT
ma-234	162	10	q	q	X
ma-234	162	11	,	,	PUNCT
ma-234	162	12	p	p	NOUN
ma-234	162	13	}	}	PUNCT
ma-234	162	14	(	(	PUNCT
ma-234	162	15	iii	iii	NOUN
ma-234	162	16	)	)	PUNCT
ma-234	162	17	for	for	ADP
ma-234	162	18	each	each	DET
ma-234	162	19	m	m	PROPN
ma-234	162	20	∈	∈	PROPN
ma-234	162	21	z	z	NOUN
ma-234	162	22	,	,	PUNCT
ma-234	162	23	the	the	DET
ma-234	162	24	family	family	NOUN
ma-234	162	25	dm	dm	PROPN
ma-234	162	26	=	=	PUNCT
ma-234	162	27	{	{	PUNCT
ma-234	162	28	q	q	NOUN
ma-234	162	29	∈	∈	PROPN
ma-234	162	30	d	d	PROPN
ma-234	162	31	/	/	SYM
ma-234	162	32	`	`	PUNCT
ma-234	162	33	(	(	PUNCT
ma-234	162	34	q	q	X
ma-234	162	35	)	)	PUNCT
ma-234	162	36	=	=	SYM
ma-234	162	37	2	2	NUM
ma-234	162	38	m	m	NOUN
ma-234	162	39	}	}	PUNCT
ma-234	162	40	form	form	VERB
ma-234	162	41	a	a	DET
ma-234	162	42	partition	partition	NOUN
ma-234	162	43	of	of	ADP
ma-234	162	44	rd	rd	PROPN
ma-234	162	45	.	.	PUNCT
ma-234	162	46	example	example	NOUN
ma-234	163	1	3.2	3.2	NUM
ma-234	163	2	.	.	PUNCT
ma-234	164	1	•	•	NUM
ma-234	164	2	the	the	DET
ma-234	164	3	standard	standard	ADJ
ma-234	164	4	dyadic	dyadic	ADJ
ma-234	164	5	grid	grid	NOUN
ma-234	164	6	d0	d0	NOUN
ma-234	164	7	is	be	AUX
ma-234	164	8	defined	define	VERB
ma-234	164	9	by	by	ADP
ma-234	164	10	d0	d0	NOUN
ma-234	164	11	=	=	SYM
ma-234	164	12	{	{	PUNCT
ma-234	164	13	2	2	NUM
ma-234	164	14	m	m	VERB
ma-234	164	15	(	(	PUNCT
ma-234	164	16	[	[	X
ma-234	164	17	0	0	NUM
ma-234	164	18	,	,	PUNCT
ma-234	164	19	1)d	1)d	NUM
ma-234	164	20	+	+	X
ma-234	165	1	k	k	X
ma-234	165	2	)	)	PUNCT
ma-234	165	3	/	/	PUNCT
ma-234	166	1	m	m	VERB
ma-234	166	2	∈	∈	PROPN
ma-234	166	3	z	z	NOUN
ma-234	166	4	,	,	PUNCT
ma-234	166	5	k	k	PROPN
ma-234	166	6	∈	∈	PROPN
ma-234	166	7	zd	zd	PROPN
ma-234	166	8	}	}	PUNCT
ma-234	166	9	.	.	PUNCT
ma-234	167	1	•	•	NOUN
ma-234	167	2	each	each	PRON
ma-234	167	3	of	of	ADP
ma-234	167	4	the	the	DET
ma-234	167	5	following	follow	VERB
ma-234	167	6	3d	3d	NUM
ma-234	167	7	collections	collection	NOUN
ma-234	167	8	of	of	ADP
ma-234	167	9	cubes	cube	NOUN
ma-234	167	10	in	in	ADP
ma-234	167	11	rd	rd	NOUN
ma-234	167	12	dt	dt	NOUN
ma-234	168	1	=	=	PUNCT
ma-234	168	2	{	{	PUNCT
ma-234	168	3	2	2	NUM
ma-234	168	4	m	m	VERB
ma-234	168	5	(	(	PUNCT
ma-234	168	6	[	[	X
ma-234	168	7	0	0	NUM
ma-234	168	8	,	,	PUNCT
ma-234	168	9	1)d	1)d	NUM
ma-234	168	10	+	+	CCONJ
ma-234	168	11	k	k	PROPN
ma-234	169	1	+	+	NUM
ma-234	169	2	t	t	PROPN
ma-234	169	3	)	)	PUNCT
ma-234	169	4	/	/	PUNCT
ma-234	170	1	m	m	VERB
ma-234	170	2	∈	∈	PROPN
ma-234	170	3	z	z	NOUN
ma-234	170	4	,	,	PUNCT
ma-234	170	5	k	k	PROPN
ma-234	170	6	∈	∈	PROPN
ma-234	170	7	zd	zd	PROPN
ma-234	170	8	}	}	PUNCT
ma-234	170	9	,	,	PUNCT
ma-234	170	10	t	t	PROPN
ma-234	170	11	∈	∈	PROPN
ma-234	170	12	{	{	PUNCT
ma-234	170	13	−1/3	−1/3	ADJ
ma-234	170	14	,	,	PUNCT
ma-234	170	15	0	0	NUM
ma-234	170	16	,	,	PUNCT
ma-234	170	17	1/3}d	1/3}d	NUM
ma-234	170	18	is	be	AUX
ma-234	170	19	a	a	DET
ma-234	170	20	dyadic	dyadic	ADJ
ma-234	170	21	grid	grid	NOUN
ma-234	170	22	.	.	PUNCT
ma-234	171	1	the	the	DET
ma-234	171	2	following	follow	VERB
ma-234	171	3	property	property	NOUN
ma-234	171	4	holds	hold	NOUN
ma-234	171	5	(	(	PUNCT
ma-234	171	6	see	see	VERB
ma-234	171	7	[	[	X
ma-234	171	8	3	3	NUM
ma-234	171	9	,	,	PUNCT
ma-234	171	10	theorem	theorem	VERB
ma-234	171	11	3.1	3.1	NUM
ma-234	171	12	]	]	PUNCT
ma-234	171	13	and	and	CCONJ
ma-234	171	14	its	its	PRON
ma-234	171	15	proof	proof	NOUN
ma-234	171	16	)	)	PUNCT
ma-234	171	17	.	.	PUNCT
ma-234	172	1	proposition	proposition	NOUN
ma-234	172	2	3.3	3.3	NUM
ma-234	172	3	.	.	PUNCT
ma-234	173	1	for	for	ADP
ma-234	173	2	every	every	DET
ma-234	173	3	cube	cube	NOUN
ma-234	173	4	q	q	PROPN
ma-234	173	5	of	of	ADP
ma-234	173	6	rd	rd	PROPN
ma-234	173	7	,	,	PUNCT
ma-234	173	8	there	there	PRON
ma-234	173	9	exists	exist	VERB
ma-234	173	10	an	an	DET
ma-234	173	11	element	element	NOUN
ma-234	173	12	t	t	PROPN
ma-234	173	13	of	of	ADP
ma-234	173	14	{	{	PUNCT
ma-234	173	15	−1/3	−1/3	ADJ
ma-234	173	16	,	,	PUNCT
ma-234	173	17	0	0	NUM
ma-234	173	18	,	,	PUNCT
ma-234	173	19	1/3}d	1/3}d	NOUN
ma-234	173	20	and	and	CCONJ
ma-234	173	21	a	a	DET
ma-234	173	22	cube	cube	NOUN
ma-234	173	23	qt	qt	NOUN
ma-234	173	24	of	of	ADP
ma-234	173	25	dt	dt	PROPN
ma-234	173	26	such	such	ADJ
ma-234	173	27	that	that	DET
ma-234	173	28	q	q	NOUN
ma-234	173	29	is	be	AUX
ma-234	173	30	included	include	VERB
ma-234	173	31	in	in	ADP
ma-234	173	32	qt	qt	NOUN
ma-234	173	33	and	and	CCONJ
ma-234	173	34	`	`	PUNCT
ma-234	173	35	(	(	PUNCT
ma-234	173	36	qt	qt	NOUN
ma-234	173	37	)	)	PUNCT
ma-234	173	38	≤	≤	NOUN
ma-234	173	39	3	3	NUM
ma-234	173	40	`	`	PUNCT
ma-234	173	41	(	(	PUNCT
ma-234	173	42	q	q	NOUN
ma-234	173	43	)	)	PUNCT
ma-234	173	44	.	.	PUNCT
ma-234	174	1	let	let	VERB
ma-234	174	2	us	we	PRON
ma-234	174	3	introduce	introduce	VERB
ma-234	174	4	the	the	DET
ma-234	174	5	following	following	ADJ
ma-234	174	6	definition	definition	NOUN
ma-234	174	7	.	.	PUNCT
ma-234	175	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	175	2	eur	eur	PROPN
ma-234	175	3	.	.	PUNCT
ma-234	176	1	j.	j.	PROPN
ma-234	176	2	math	math	PROPN
ma-234	176	3	.	.	PUNCT
ma-234	177	1	anal	anal	PROPN
ma-234	177	2	.	.	PUNCT
ma-234	178	1	10.28924	10.28924	NUM
ma-234	178	2	/	/	SYM
ma-234	178	3	ada	ada	PROPN
ma-234	178	4	/	/	SYM
ma-234	178	5	ma.4.16	ma.4.16	PROPN
ma-234	178	6	8	8	NUM
ma-234	178	7	definition	definition	NOUN
ma-234	178	8	3.4	3.4	NUM
ma-234	178	9	.	.	PUNCT
ma-234	179	1	let	let	VERB
ma-234	179	2	1	1	NUM
ma-234	179	3	≤	≤	NOUN
ma-234	179	4	q	q	ADJ
ma-234	179	5	≤	≤	NUM
ma-234	179	6	α	α	NOUN
ma-234	179	7	≤	≤	NOUN
ma-234	179	8	p	p	NOUN
ma-234	179	9	≤	≤	ADJ
ma-234	179	10	∞.	∞.	PROPN
ma-234	179	11	for	for	ADP
ma-234	179	12	any	any	DET
ma-234	179	13	dyadic	dyadic	ADJ
ma-234	179	14	grid	grid	NOUN
ma-234	179	15	d	d	NOUN
ma-234	179	16	and	and	CCONJ
ma-234	179	17	any	any	DET
ma-234	179	18	element	element	NOUN
ma-234	179	19	f	f	PROPN
ma-234	179	20	of	of	ADP
ma-234	179	21	lqloc	lqloc	NOUN
ma-234	179	22	,	,	PUNCT
ma-234	179	23	we	we	PRON
ma-234	179	24	define	define	VERB
ma-234	179	25	‖f	‖f	PUNCT
ma-234	179	26	‖mα	‖mα	NUM
ma-234	179	27	q	q	ADJ
ma-234	179	28	,	,	PUNCT
ma-234	179	29	p(d	p(d	NOUN
ma-234	179	30	)	)	PUNCT
ma-234	179	31	=	=	SYM
ma-234	180	1	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-234	180	2	{	{	PUNCT
ma-234	180	3	|q|	|q|	NUM
ma-234	180	4	1	1	NUM
ma-234	180	5	α	α	NOUN
ma-234	180	6	−	−	NOUN
ma-234	180	7	1	1	NUM
ma-234	180	8	q	q	NOUN
ma-234	180	9	(	(	PUNCT
ma-234	180	10	∫	∫	PROPN
ma-234	180	11	q	q	PROPN
ma-234	180	12	|f	|f	PROPN
ma-234	180	13	(	(	PUNCT
ma-234	180	14	x)|qdx	x)|qdx	PROPN
ma-234	180	15	)	)	PUNCT
ma-234	180	16	1	1	NUM
ma-234	180	17	q	q	NOUN
ma-234	180	18	}	}	PUNCT
ma-234	180	19	q∈d	q∈d	NOUN
ma-234	180	20	∥∥∥∥∥	∥∥∥∥∥	PUNCT
ma-234	180	21	`	`	PUNCT
ma-234	180	22	p	p	X
ma-234	180	23	.	.	PUNCT
ma-234	181	1	we	we	PRON
ma-234	181	2	shall	shall	AUX
ma-234	181	3	prove	prove	VERB
ma-234	181	4	that	that	SCONJ
ma-234	181	5	,	,	PUNCT
ma-234	181	6	for	for	ADP
ma-234	181	7	any	any	DET
ma-234	181	8	t	t	NOUN
ma-234	181	9	∈	∈	PROPN
ma-234	181	10	{	{	PUNCT
ma-234	181	11	−1/3	−1/3	ADJ
ma-234	181	12	,	,	PUNCT
ma-234	181	13	0	0	NUM
ma-234	181	14	,	,	PUNCT
ma-234	181	15	1/3}d	1/3}d	NUM
ma-234	181	16	,	,	PUNCT
ma-234	181	17	the	the	DET
ma-234	181	18	norms	norm	NOUN
ma-234	181	19	‖	‖	PROPN
ma-234	181	20	·	·	PUNCT
ma-234	181	21	‖mα	‖mα	NUM
ma-234	181	22	q	q	NOUN
ma-234	181	23	,	,	PUNCT
ma-234	181	24	p(dt	p(dt	PROPN
ma-234	181	25	)	)	PUNCT
ma-234	181	26	and	and	CCONJ
ma-234	181	27	‖	‖	PROPN
ma-234	181	28	·	·	PUNCT
ma-234	181	29	‖mα	‖mα	NUM
ma-234	181	30	q	q	NOUN
ma-234	181	31	,	,	PUNCT
ma-234	181	32	p	p	NOUN
ma-234	181	33	areequivalent	areequivalent	NOUN
ma-234	181	34	.	.	PUNCT
ma-234	182	1	in	in	ADP
ma-234	182	2	order	order	NOUN
ma-234	182	3	to	to	PART
ma-234	182	4	do	do	AUX
ma-234	182	5	this	this	PRON
ma-234	182	6	,	,	PUNCT
ma-234	182	7	we	we	PRON
ma-234	182	8	establish	establish	VERB
ma-234	182	9	the	the	DET
ma-234	182	10	following	follow	VERB
ma-234	182	11	preparatory	preparatory	ADJ
ma-234	182	12	lemma	lemma	PROPN
ma-234	182	13	.	.	PUNCT
ma-234	183	1	lemma	lemma	PROPN
ma-234	183	2	3.5	3.5	NUM
ma-234	183	3	.	.	PUNCT
ma-234	184	1	let	let	VERB
ma-234	184	2	1	1	NUM
ma-234	184	3	≤	≤	NOUN
ma-234	184	4	q	q	ADJ
ma-234	184	5	≤	≤	NUM
ma-234	184	6	α	α	NOUN
ma-234	184	7	≤	≤	NOUN
ma-234	184	8	p	p	NOUN
ma-234	184	9	≤	≤	NUM
ma-234	184	10	∞	∞	PROPN
ma-234	184	11	,	,	PUNCT
ma-234	185	1	d	d	NOUN
ma-234	185	2	and	and	CCONJ
ma-234	185	3	d′	d′	PRON
ma-234	185	4	are	be	AUX
ma-234	185	5	two	two	NUM
ma-234	185	6	dyadic	dyadic	ADJ
ma-234	185	7	grids	grid	NOUN
ma-234	185	8	.	.	PUNCT
ma-234	186	1	then	then	ADV
ma-234	186	2	for	for	ADP
ma-234	186	3	any	any	DET
ma-234	186	4	element	element	NOUN
ma-234	186	5	f	f	PROPN
ma-234	186	6	of	of	ADP
ma-234	186	7	lqloc	lqloc	NOUN
ma-234	186	8	,	,	PUNCT
ma-234	186	9	we	we	PRON
ma-234	186	10	have	have	VERB
ma-234	186	11			PUNCT
ma-234	186	12	‖f	‖f	ADP
ma-234	186	13	‖mα	‖mα	NUM
ma-234	186	14	q	q	ADJ
ma-234	186	15	,	,	PUNCT
ma-234	186	16	p(d	p(d	NOUN
ma-234	186	17	)	)	PUNCT
ma-234	186	18	≤	≤	ADV
ma-234	186	19	2	2	NUM
ma-234	186	20	d	d	NOUN
ma-234	186	21	(	(	PUNCT
ma-234	186	22	1	1	NUM
ma-234	186	23	q	q	NOUN
ma-234	186	24	−	−	PROPN
ma-234	186	25	1	1	NUM
ma-234	186	26	p	p	NOUN
ma-234	186	27	)	)	PUNCT
ma-234	186	28	‖f	‖f	PUNCT
ma-234	187	1	‖mα	‖mα	NUM
ma-234	187	2	q	q	NOUN
ma-234	187	3	,	,	PUNCT
ma-234	187	4	p(d′	p(d′	NUM
ma-234	187	5	)	)	PUNCT
ma-234	187	6	if	if	SCONJ
ma-234	187	7	p	p	NOUN
ma-234	187	8	<	<	X
ma-234	187	9	∞	∞	NUM
ma-234	187	10	‖f	‖f	PUNCT
ma-234	187	11	‖mα	‖mα	NUM
ma-234	187	12	q,∞(d	q,∞(d	NOUN
ma-234	187	13	)	)	PUNCT
ma-234	187	14	≤	≤	NOUN
ma-234	188	1	2d‖f	2d‖f	NUM
ma-234	188	2	‖mα	‖mα	NUM
ma-234	188	3	q,∞(d′	q,∞(d′	NOUN
ma-234	188	4	)	)	PUNCT
ma-234	188	5	.	.	PUNCT
ma-234	189	1	proof	proof	NOUN
ma-234	189	2	.	.	PUNCT
ma-234	190	1	let	let	VERB
ma-234	190	2	f	f	PRON
ma-234	190	3	be	be	AUX
ma-234	190	4	any	any	DET
ma-234	190	5	element	element	NOUN
ma-234	190	6	of	of	ADP
ma-234	190	7	lqloc	lqloc	NOUN
ma-234	190	8	and	and	CCONJ
ma-234	190	9	fix	fix	VERB
ma-234	190	10	m	m	PROPN
ma-234	190	11	∈	∈	NOUN
ma-234	190	12	z.we	z.we	NOUN
ma-234	190	13	recall	recall	VERB
ma-234	190	14	that	that	SCONJ
ma-234	190	15	both	both	CCONJ
ma-234	190	16	the	the	DET
ma-234	190	17	families	family	NOUN
ma-234	191	1	dm	dm	NOUN
ma-234	191	2	=	=	SYM
ma-234	191	3	{	{	PUNCT
ma-234	191	4	q	q	NOUN
ma-234	191	5	∈	∈	PROPN
ma-234	191	6	d	d	PROPN
ma-234	191	7	/	/	SYM
ma-234	191	8	`	`	PUNCT
ma-234	191	9	(	(	PUNCT
ma-234	191	10	q	q	X
ma-234	191	11	)	)	PUNCT
ma-234	191	12	=	=	SYM
ma-234	191	13	2	2	NUM
ma-234	191	14	m	m	NOUN
ma-234	191	15	}	}	PUNCT
ma-234	191	16	and	and	CCONJ
ma-234	191	17	d′m	d′m	VERB
ma-234	191	18	=	=	PUNCT
ma-234	191	19	{	{	PUNCT
ma-234	191	20	q′	q′	NOUN
ma-234	191	21	∈	∈	NOUN
ma-234	191	22	d′	d′	NUM
ma-234	191	23	/	/	SYM
ma-234	191	24	`	`	PUNCT
ma-234	191	25	(	(	PUNCT
ma-234	191	26	q′	q′	NOUN
ma-234	191	27	)	)	PUNCT
ma-234	192	1	=	=	SYM
ma-234	192	2	2m}form	2m}form	NUM
ma-234	192	3	partitions	partition	NOUN
ma-234	192	4	of	of	ADP
ma-234	192	5	rd	rd	PROPN
ma-234	192	6	.	.	PUNCT
ma-234	193	1	moreover	moreover	ADV
ma-234	193	2	,	,	PUNCT
ma-234	193	3	it	it	PRON
ma-234	193	4	is	be	AUX
ma-234	193	5	easy	easy	ADJ
ma-234	193	6	to	to	PART
ma-234	193	7	see	see	VERB
ma-234	193	8	that	that	PRON
ma-234	193	9	,	,	PUNCT
ma-234	193	10	for	for	ADP
ma-234	193	11	any	any	DET
ma-234	193	12	element	element	NOUN
ma-234	193	13	q	q	PROPN
ma-234	193	14	of	of	ADP
ma-234	193	15	dm	dm	PROPN
ma-234	193	16	,	,	PUNCT
ma-234	193	17	the	the	DET
ma-234	193	18	subset	subset	NOUN
ma-234	193	19	{	{	PUNCT
ma-234	193	20	q′	q′	NOUN
ma-234	193	21	∈	∈	PROPN
ma-234	193	22	d′	d′	NUM
ma-234	193	23	/	/	SYM
ma-234	193	24	q	q	PROPN
ma-234	193	25	∩q′	∩q′	PROPN
ma-234	193	26	6=	6=	ADP
ma-234	193	27	∅	∅	NOUN
ma-234	193	28	}	}	PUNCT
ma-234	193	29	of	of	ADP
ma-234	193	30	d′	d′	PRON
ma-234	193	31	has	have	VERB
ma-234	193	32	at	at	ADP
ma-234	193	33	most	most	ADJ
ma-234	193	34	2d	2d	NUM
ma-234	193	35	elements.a	elements.a	NUM
ma-234	193	36	)	)	PUNCT
ma-234	193	37	suppose	suppose	VERB
ma-234	193	38	that	that	SCONJ
ma-234	193	39	p	p	PROPN
ma-234	193	40	<	<	X
ma-234	193	41	∞.	∞.	PROPN
ma-234	193	42	we	we	PRON
ma-234	193	43	have(∫	have(∫	VERB
ma-234	193	44	q	q	X
ma-234	193	45	|f	|f	PROPN
ma-234	193	46	(	(	PUNCT
ma-234	193	47	x)|qdx	x)|qdx	PROPN
ma-234	193	48	)	)	PUNCT
ma-234	194	1	p	p	X
ma-234	194	2	q	q	NOUN
ma-234	195	1	=	=	PUNCT
ma-234	195	2			PROPN
ma-234	195	3	∑	∑	PUNCT
ma-234	195	4	q′∈d′m	q′∈d′m	PROPN
ma-234	196	1	∫	∫	PROPN
ma-234	196	2	q∩q′	q∩q′	X
ma-234	196	3	|f	|f	PROPN
ma-234	197	1	(	(	PUNCT
ma-234	197	2	x)|qdx	x)|qdx	PROPN
ma-234	197	3			PROPN
ma-234	198	1	p	p	NOUN
ma-234	198	2	q	q	NOUN
ma-234	198	3	≤	≤	NUM
ma-234	198	4	2	2	NUM
ma-234	198	5	d	d	NOUN
ma-234	198	6	(	(	PUNCT
ma-234	198	7	1−	1−	NUM
ma-234	198	8	q	q	NOUN
ma-234	198	9	p	p	NOUN
ma-234	198	10	)	)	PUNCT
ma-234	198	11	p	p	X
ma-234	198	12	q	q	PUNCT
ma-234	198	13	∑	∑	PUNCT
ma-234	198	14	q′∈d′m	q′∈d′m	PROPN
ma-234	198	15	(	(	PUNCT
ma-234	198	16	∫	∫	PROPN
ma-234	198	17	q∩q′	q∩q′	PROPN
ma-234	198	18	|f	|f	PROPN
ma-234	198	19	(	(	PUNCT
ma-234	198	20	x)|qdx	x)|qdx	PROPN
ma-234	198	21	)	)	PUNCT
ma-234	199	1	p	p	X
ma-234	199	2	q	q	X
ma-234	199	3	.	.	PUNCT
ma-234	200	1	consequently∑	consequently∑	PROPN
ma-234	200	2	q∈dm	q∈dm	NOUN
ma-234	200	3	[	[	PUNCT
ma-234	200	4	|q|	|q|	NUM
ma-234	200	5	1	1	NUM
ma-234	200	6	α	α	NOUN
ma-234	200	7	−	−	NOUN
ma-234	200	8	1	1	NUM
ma-234	200	9	q	q	NOUN
ma-234	200	10	(	(	PUNCT
ma-234	200	11	∫	∫	PROPN
ma-234	200	12	q	q	PROPN
ma-234	200	13	|f	|f	PROPN
ma-234	200	14	(	(	PUNCT
ma-234	200	15	x)|qdx	x)|qdx	PROPN
ma-234	200	16	)	)	PUNCT
ma-234	200	17	1	1	NUM
ma-234	200	18	q	q	NOUN
ma-234	201	1	]	]	X
ma-234	201	2	p	p	X
ma-234	201	3	=	=	SYM
ma-234	201	4	2	2	NUM
ma-234	201	5	d	d	NOUN
ma-234	201	6	m	m	PROPN
ma-234	201	7	(	(	PUNCT
ma-234	201	8	1	1	NUM
ma-234	201	9	α	α	NOUN
ma-234	201	10	−	−	PROPN
ma-234	201	11	1	1	NUM
ma-234	201	12	q	q	NOUN
ma-234	201	13	)	)	PUNCT
ma-234	201	14	p	p	NOUN
ma-234	201	15	∑	∑	PUNCT
ma-234	201	16	q∈dm	q∈dm	NOUN
ma-234	201	17	(	(	PUNCT
ma-234	201	18	∫	∫	PROPN
ma-234	201	19	q	q	PROPN
ma-234	201	20	|f	|f	PROPN
ma-234	201	21	(	(	PUNCT
ma-234	201	22	x)|qdx	x)|qdx	PROPN
ma-234	201	23	)	)	PUNCT
ma-234	202	1	p	p	X
ma-234	202	2	q	q	PUNCT
ma-234	202	3	≤	≤	NUM
ma-234	202	4	2	2	NUM
ma-234	202	5	d	d	NOUN
ma-234	202	6	(	(	PUNCT
ma-234	202	7	p	p	X
ma-234	202	8	q	q	X
ma-234	202	9	−1	−1	NOUN
ma-234	202	10	)	)	PUNCT
ma-234	202	11	2	2	NUM
ma-234	203	1	d	d	NOUN
ma-234	203	2	m	m	PROPN
ma-234	203	3	(	(	PUNCT
ma-234	203	4	1	1	NUM
ma-234	203	5	α	α	NOUN
ma-234	203	6	−	−	PROPN
ma-234	203	7	1	1	NUM
ma-234	203	8	q	q	NOUN
ma-234	203	9	)	)	PUNCT
ma-234	203	10	p	p	NOUN
ma-234	203	11	∑	∑	PUNCT
ma-234	203	12	q∈dm	q∈dm	PROPN
ma-234	203	13	∑	∑	PUNCT
ma-234	203	14	q′∈d′m	q′∈d′m	PROPN
ma-234	203	15	,	,	PUNCT
ma-234	203	16	q∩q′	q∩q′	NOUN
ma-234	203	17	6=∅	6=∅	NUM
ma-234	204	1	(	(	PUNCT
ma-234	204	2	∫	∫	PROPN
ma-234	204	3	q∩q′	q∩q′	PROPN
ma-234	204	4	|f	|f	PROPN
ma-234	204	5	(	(	PUNCT
ma-234	204	6	x)|qdx	x)|qdx	PROPN
ma-234	204	7	)	)	PUNCT
ma-234	205	1	p	p	X
ma-234	205	2	q	q	NOUN
ma-234	205	3	=	=	SYM
ma-234	205	4	2	2	NUM
ma-234	205	5	d	d	NOUN
ma-234	205	6	(	(	PUNCT
ma-234	205	7	p	p	X
ma-234	205	8	q	q	X
ma-234	205	9	−1	−1	NOUN
ma-234	205	10	)	)	PUNCT
ma-234	205	11	2	2	NUM
ma-234	205	12	d	d	NOUN
ma-234	205	13	m	m	PROPN
ma-234	205	14	(	(	PUNCT
ma-234	205	15	1	1	NUM
ma-234	205	16	α	α	NOUN
ma-234	205	17	−	−	PROPN
ma-234	205	18	1	1	NUM
ma-234	205	19	q	q	NOUN
ma-234	205	20	)	)	PUNCT
ma-234	205	21	p	p	NOUN
ma-234	205	22	∑	∑	PUNCT
ma-234	205	23	q′∈d′m	q′∈d′m	NUM
ma-234	205	24	∑	∑	PUNCT
ma-234	205	25	q∈dm	q∈dm	NOUN
ma-234	205	26	(	(	PUNCT
ma-234	205	27	∫	∫	PROPN
ma-234	205	28	q∩q′	q∩q′	PROPN
ma-234	205	29	|f	|f	PROPN
ma-234	205	30	(	(	PUNCT
ma-234	205	31	x)|qdx	x)|qdx	PROPN
ma-234	205	32	)	)	PUNCT
ma-234	206	1	p	p	X
ma-234	206	2	q	q	PUNCT
ma-234	206	3	≤	≤	NUM
ma-234	206	4	2	2	NUM
ma-234	206	5	d	d	NOUN
ma-234	206	6	(	(	PUNCT
ma-234	206	7	p	p	X
ma-234	206	8	q	q	X
ma-234	206	9	−1	−1	NOUN
ma-234	206	10	)	)	PUNCT
ma-234	206	11	2	2	NUM
ma-234	207	1	d	d	NOUN
ma-234	207	2	m	m	PROPN
ma-234	207	3	(	(	PUNCT
ma-234	207	4	1	1	NUM
ma-234	207	5	α	α	NOUN
ma-234	207	6	−	−	PROPN
ma-234	207	7	1	1	NUM
ma-234	207	8	q	q	NOUN
ma-234	207	9	)	)	PUNCT
ma-234	207	10	p	p	NOUN
ma-234	207	11	∑	∑	PROPN
ma-234	207	12	q′∈d′m	q′∈d′m	PROPN
ma-234	207	13	(	(	PUNCT
ma-234	207	14	∫	∫	PROPN
ma-234	207	15	q′	q′	NOUN
ma-234	207	16	|f	|f	PROPN
ma-234	207	17	(	(	PUNCT
ma-234	207	18	x)|qdx	x)|qdx	PROPN
ma-234	207	19	)	)	PUNCT
ma-234	208	1	p	p	X
ma-234	208	2	q	q	NOUN
ma-234	208	3	=	=	SYM
ma-234	208	4	2	2	NUM
ma-234	208	5	d	d	NOUN
ma-234	208	6	(	(	PUNCT
ma-234	208	7	p	p	X
ma-234	208	8	q	q	X
ma-234	208	9	−1	−1	NOUN
ma-234	208	10	)	)	PUNCT
ma-234	208	11	∑	∑	PUNCT
ma-234	208	12	q′∈d′m	q′∈d′m	PROPN
ma-234	209	1	[	[	X
ma-234	209	2	∣∣q′∣∣	∣∣q′∣∣	NOUN
ma-234	209	3	1	1	NUM
ma-234	209	4	α	α	NOUN
ma-234	209	5	−	−	PROPN
ma-234	209	6	1	1	NUM
ma-234	209	7	q	q	NOUN
ma-234	209	8	(	(	PUNCT
ma-234	209	9	∫	∫	PROPN
ma-234	209	10	q′	q′	NOUN
ma-234	209	11	|f	|f	PROPN
ma-234	210	1	(	(	PUNCT
ma-234	210	2	x)|qdx	x)|qdx	PROPN
ma-234	210	3	)	)	PUNCT
ma-234	210	4	1	1	NUM
ma-234	210	5	q	q	NOUN
ma-234	210	6	]	]	X
ma-234	210	7	p	p	NOUN
ma-234	210	8	and	and	CCONJ
ma-234	210	9	so	so	ADV
ma-234	210	10	‖f	‖f	ADP
ma-234	210	11	‖mα	‖mα	NUM
ma-234	210	12	q	q	ADJ
ma-234	210	13	,	,	PUNCT
ma-234	210	14	p(d	p(d	NOUN
ma-234	210	15	)	)	PUNCT
ma-234	210	16	≤	≤	ADV
ma-234	210	17	2	2	NUM
ma-234	210	18	d	d	NOUN
ma-234	210	19	(	(	PUNCT
ma-234	210	20	1	1	NUM
ma-234	210	21	q	q	NOUN
ma-234	210	22	−	−	PROPN
ma-234	210	23	1	1	NUM
ma-234	210	24	p	p	NOUN
ma-234	210	25	)	)	PUNCT
ma-234	210	26	‖f	‖f	PUNCT
ma-234	211	1	‖mα	‖mα	NUM
ma-234	211	2	q	q	NOUN
ma-234	211	3	,	,	PUNCT
ma-234	211	4	p(d′	p(d′	NUM
ma-234	211	5	)	)	PUNCT
ma-234	211	6	.	.	PUNCT
ma-234	212	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	212	2	eur	eur	PROPN
ma-234	212	3	.	.	PUNCT
ma-234	213	1	j.	j.	PROPN
ma-234	213	2	math	math	PROPN
ma-234	213	3	.	.	PUNCT
ma-234	214	1	anal	anal	PROPN
ma-234	214	2	.	.	PUNCT
ma-234	215	1	10.28924	10.28924	NUM
ma-234	215	2	/	/	SYM
ma-234	215	3	ada	ada	PROPN
ma-234	215	4	/	/	SYM
ma-234	215	5	ma.4.16	ma.4.16	PROPN
ma-234	215	6	9b	9b	PROPN
ma-234	215	7	)	)	PUNCT
ma-234	215	8	suppose	suppose	VERB
ma-234	215	9	that	that	SCONJ
ma-234	215	10	p	p	PROPN
ma-234	215	11	=	=	PRON
ma-234	215	12	∞.	∞.	PROPN
ma-234	215	13	then	then	ADV
ma-234	215	14	,	,	PUNCT
ma-234	215	15	for	for	ADP
ma-234	215	16	any	any	DET
ma-234	215	17	q	q	PROPN
ma-234	215	18	∈	∈	PROPN
ma-234	215	19	dm	dm	NOUN
ma-234	215	20	,	,	PUNCT
ma-234	215	21	we	we	PRON
ma-234	215	22	have	have	AUX
ma-234	215	23	|q|	|q|	VERB
ma-234	215	24	1	1	NUM
ma-234	215	25	α	α	NOUN
ma-234	215	26	−	−	NOUN
ma-234	215	27	1	1	NUM
ma-234	215	28	q	q	NOUN
ma-234	215	29	(	(	PUNCT
ma-234	215	30	∫	∫	PROPN
ma-234	215	31	q	q	PROPN
ma-234	215	32	|f	|f	PROPN
ma-234	215	33	(	(	PUNCT
ma-234	215	34	x)|qdx	x)|qdx	PROPN
ma-234	215	35	)	)	PUNCT
ma-234	215	36	1	1	NUM
ma-234	215	37	q	q	NOUN
ma-234	215	38	=	=	PUNCT
ma-234	215	39	|q|	|q|	VERB
ma-234	215	40	1	1	NUM
ma-234	215	41	α	α	NOUN
ma-234	215	42	−	−	NOUN
ma-234	215	43	1	1	NUM
ma-234	215	44	q	q	NOUN
ma-234	215	45			PROPN
ma-234	215	46	∑	∑	PUNCT
ma-234	215	47	q′∈d′m	q′∈d′m	PROPN
ma-234	216	1	∫	∫	PROPN
ma-234	216	2	q∩q′	q∩q′	X
ma-234	216	3	|f	|f	PROPN
ma-234	217	1	(	(	PUNCT
ma-234	217	2	x)|qdx	x)|qdx	PROPN
ma-234	217	3			PROPN
ma-234	217	4	1	1	NUM
ma-234	217	5	q	q	NOUN
ma-234	217	6	≤	≤	NUM
ma-234	217	7	|q|	|q|	VERB
ma-234	217	8	1	1	NUM
ma-234	217	9	α	α	NOUN
ma-234	217	10	−	−	PROPN
ma-234	217	11	1	1	NUM
ma-234	217	12	q	q	NOUN
ma-234	217	13	∑	∑	PROPN
ma-234	217	14	q′∈d′m	q′∈d′m	PROPN
ma-234	217	15	(	(	PUNCT
ma-234	217	16	∫	∫	PROPN
ma-234	217	17	q∩q′	q∩q′	PROPN
ma-234	217	18	|f	|f	PROPN
ma-234	218	1	(	(	PUNCT
ma-234	218	2	x)|qdx	x)|qdx	PROPN
ma-234	218	3	)	)	PUNCT
ma-234	218	4	1	1	NUM
ma-234	218	5	q	q	NOUN
ma-234	218	6	=	=	PUNCT
ma-234	218	7	∑	∑	PUNCT
ma-234	218	8	q′∈d′m	q′∈d′m	PROPN
ma-234	218	9	∣∣q′∣∣	∣∣q′∣∣	NOUN
ma-234	218	10	1	1	NUM
ma-234	218	11	α	α	NOUN
ma-234	218	12	−	−	PROPN
ma-234	218	13	1	1	NUM
ma-234	218	14	q	q	NOUN
ma-234	218	15	(	(	PUNCT
ma-234	218	16	∫	∫	PROPN
ma-234	218	17	q∩q′	q∩q′	PROPN
ma-234	218	18	|f	|f	PROPN
ma-234	219	1	(	(	PUNCT
ma-234	219	2	x)|qdx	x)|qdx	PROPN
ma-234	219	3	)	)	PUNCT
ma-234	219	4	1	1	NUM
ma-234	219	5	q	q	NOUN
ma-234	219	6	≤	≤	PROPN
ma-234	219	7	∑	∑	PUNCT
ma-234	219	8	q′∈d′m	q′∈d′m	PROPN
ma-234	219	9	,	,	PUNCT
ma-234	219	10	q∩q′	q∩q′	NOUN
ma-234	219	11	6=∅	6=∅	NUM
ma-234	220	1	∣∣q′∣∣	∣∣q′∣∣	NOUN
ma-234	220	2	1	1	NUM
ma-234	220	3	α	α	NOUN
ma-234	220	4	−	−	PROPN
ma-234	220	5	1	1	NUM
ma-234	220	6	q	q	NOUN
ma-234	220	7	(	(	PUNCT
ma-234	220	8	∫	∫	PROPN
ma-234	220	9	q′	q′	NOUN
ma-234	220	10	|f	|f	PROPN
ma-234	220	11	(	(	PUNCT
ma-234	220	12	x)|qdx	x)|qdx	PROPN
ma-234	220	13	)	)	PUNCT
ma-234	220	14	1	1	NUM
ma-234	220	15	q	q	NOUN
ma-234	220	16	≤	≤	NUM
ma-234	220	17	2d‖f	2d‖f	NOUN
ma-234	220	18	‖mα	‖mα	NUM
ma-234	220	19	q,∞(d′	q,∞(d′	NOUN
ma-234	220	20	)	)	PUNCT
ma-234	220	21	and	and	CCONJ
ma-234	220	22	so	so	ADV
ma-234	220	23	‖f	‖f	PRON
ma-234	220	24	‖mα	‖mα	NUM
ma-234	220	25	q,∞(d	q,∞(d	NOUN
ma-234	220	26	)	)	PUNCT
ma-234	220	27	≤	≤	NOUN
ma-234	221	1	2d‖f	2d‖f	NUM
ma-234	221	2	‖mα	‖mα	NUM
ma-234	221	3	q,∞(d′).the	q,∞(d′).the	DET
ma-234	221	4	proof	proof	NOUN
ma-234	221	5	is	be	AUX
ma-234	221	6	complete	complete	ADJ
ma-234	221	7	.	.	PUNCT
ma-234	222	1	�	�	PROPN
ma-234	222	2	the	the	DET
ma-234	222	3	above	above	ADJ
ma-234	222	4	lemma	lemma	PROPN
ma-234	222	5	leads	lead	VERB
ma-234	222	6	to	to	ADP
ma-234	222	7	the	the	DET
ma-234	222	8	following	follow	VERB
ma-234	222	9	corollary	corollary	NOUN
ma-234	222	10	.	.	PUNCT
ma-234	223	1	corollary	corollary	ADJ
ma-234	223	2	3.6	3.6	NUM
ma-234	223	3	.	.	PUNCT
ma-234	224	1	let	let	VERB
ma-234	224	2	1	1	NUM
ma-234	224	3	≤	≤	NOUN
ma-234	224	4	q	q	ADJ
ma-234	224	5	≤	≤	NUM
ma-234	224	6	α	α	NOUN
ma-234	224	7	≤	≤	NOUN
ma-234	224	8	p	p	NOUN
ma-234	224	9	≤	≤	NOUN
ma-234	224	10	∞	∞	PROPN
ma-234	224	11	and	and	CCONJ
ma-234	224	12	t	t	PROPN
ma-234	224	13	be	be	AUX
ma-234	224	14	in	in	ADP
ma-234	224	15	{	{	PUNCT
ma-234	224	16	−1/3	−1/3	ADJ
ma-234	224	17	,	,	PUNCT
ma-234	224	18	0	0	NUM
ma-234	224	19	,	,	PUNCT
ma-234	224	20	1/3}d	1/3}d	NOUN
ma-234	224	21	.	.	PUNCT
ma-234	225	1	then	then	ADV
ma-234	225	2	for	for	ADP
ma-234	225	3	any	any	DET
ma-234	225	4	element	element	NOUN
ma-234	225	5	f	f	PROPN
ma-234	225	6	of	of	ADP
ma-234	225	7	lqloc	lqloc	NOUN
ma-234	225	8	,	,	PUNCT
ma-234	225	9	we	we	PRON
ma-234	225	10	have	have	NUM
ma-234	225	11	2	2	NUM
ma-234	225	12	d	d	NOUN
ma-234	225	13	(	(	PUNCT
ma-234	225	14	1	1	NUM
ma-234	225	15	p	p	NOUN
ma-234	225	16	−	−	PROPN
ma-234	225	17	1	1	NUM
ma-234	225	18	q	q	NOUN
ma-234	225	19	)	)	PUNCT
ma-234	225	20	‖f	‖f	PUNCT
ma-234	225	21	‖mα	‖mα	NUM
ma-234	225	22	q	q	NOUN
ma-234	225	23	,	,	PUNCT
ma-234	225	24	p	p	NOUN
ma-234	225	25	≤	≤	NOUN
ma-234	225	26	‖f	‖f	PUNCT
ma-234	226	1	‖mα	‖mα	NUM
ma-234	226	2	q	q	ADJ
ma-234	226	3	,	,	PUNCT
ma-234	226	4	p(dt	p(dt	PROPN
ma-234	226	5	)	)	PUNCT
ma-234	226	6	≤	≤	NUM
ma-234	226	7	2	2	NUM
ma-234	226	8	d	d	NOUN
ma-234	226	9	(	(	PUNCT
ma-234	226	10	1	1	NUM
ma-234	226	11	q	q	NOUN
ma-234	226	12	−	−	PROPN
ma-234	226	13	1	1	NUM
ma-234	226	14	p	p	NOUN
ma-234	226	15	)	)	PUNCT
ma-234	226	16	‖f	‖f	PUNCT
ma-234	227	1	‖mα	‖mα	NUM
ma-234	227	2	q	q	NOUN
ma-234	227	3	,	,	PUNCT
ma-234	227	4	p	p	NOUN
ma-234	227	5	if	if	SCONJ
ma-234	227	6	p	p	PROPN
ma-234	227	7	<	<	X
ma-234	227	8	∞	∞	NOUN
ma-234	227	9	2−d‖f	2−d‖f	NOUN
ma-234	227	10	‖mα	‖mα	NUM
ma-234	227	11	q,∞	q,∞	PROPN
ma-234	227	12	≤	≤	NOUN
ma-234	227	13	‖f	‖f	PUNCT
ma-234	227	14	‖mα	‖mα	NUM
ma-234	227	15	q,∞(dt	q,∞(dt	NOUN
ma-234	227	16	)	)	PUNCT
ma-234	227	17	≤	≤	NOUN
ma-234	227	18	2d‖f	2d‖f	NUM
ma-234	227	19	‖mα	‖mα	NUM
ma-234	227	20	q,∞	q,∞	PROPN
ma-234	227	21	.3.2	.3.2	PROPN
ma-234	227	22	.	.	PUNCT
ma-234	228	1	continuity	continuity	NOUN
ma-234	228	2	of	of	ADP
ma-234	228	3	the	the	DET
ma-234	228	4	translation	translation	NOUN
ma-234	228	5	operator	operator	NOUN
ma-234	228	6	in	in	ADP
ma-234	228	7	mα	mα	PROPN
ma-234	228	8	q	q	NOUN
ma-234	228	9	,	,	PUNCT
ma-234	228	10	p.	p.	NOUN
ma-234	228	11	this	this	DET
ma-234	228	12	subsection	subsection	NOUN
ma-234	228	13	deals	deal	VERB
ma-234	228	14	with	with	ADP
ma-234	228	15	the	the	DET
ma-234	228	16	continuity	continuity	NOUN
ma-234	228	17	ofthe	ofthe	NOUN
ma-234	228	18	translation	translation	NOUN
ma-234	228	19	operator	operator	NOUN
ma-234	228	20	in	in	ADP
ma-234	228	21	bourgain	bourgain	NOUN
ma-234	228	22	-	-	PUNCT
ma-234	228	23	morrey	morrey	NOUN
ma-234	228	24	spaces	space	NOUN
ma-234	228	25	.	.	PUNCT
ma-234	229	1	we	we	PRON
ma-234	229	2	shall	shall	AUX
ma-234	229	3	use	use	VERB
ma-234	229	4	in	in	ADP
ma-234	229	5	the	the	DET
ma-234	229	6	sequel	sequel	NOUN
ma-234	229	7	the	the	DET
ma-234	229	8	followingproperties	followingpropertie	NOUN
ma-234	229	9	.	.	PUNCT
ma-234	230	1	proposition	proposition	NOUN
ma-234	230	2	3.7	3.7	NUM
ma-234	230	3	.	.	PUNCT
ma-234	231	1	[	[	X
ma-234	231	2	8	8	NUM
ma-234	231	3	]	]	PUNCT
ma-234	231	4	let	let	VERB
ma-234	231	5	us	we	PRON
ma-234	231	6	assume	assume	VERB
ma-234	231	7	that	that	SCONJ
ma-234	231	8	1	1	NUM
ma-234	231	9	≤	≤	NUM
ma-234	231	10	q	q	ADJ
ma-234	231	11	≤	≤	NUM
ma-234	231	12	α	α	NOUN
ma-234	231	13	≤	≤	NOUN
ma-234	231	14	p	p	NOUN
ma-234	231	15	≤	≤	NOUN
ma-234	231	16	∞.	∞.	PROPN
ma-234	231	17	1	1	NUM
ma-234	231	18	)	)	PUNCT
ma-234	231	19	if	if	SCONJ
ma-234	231	20	α	α	NOUN
ma-234	231	21	<	<	X
ma-234	231	22	∞	∞	PROPN
ma-234	231	23	,	,	PUNCT
ma-234	231	24	then	then	ADV
ma-234	231	25	there	there	PRON
ma-234	231	26	exists	exist	VERB
ma-234	231	27	c1	c1	PROPN
ma-234	231	28	>	>	X
ma-234	231	29	0	0	PUNCT
ma-234	232	1	such	such	ADJ
ma-234	232	2	that	that	PRON
ma-234	232	3	for	for	ADP
ma-234	232	4	all	all	DET
ma-234	232	5	y	y	PROPN
ma-234	232	6	∈	∈	PROPN
ma-234	232	7	rd	rd	PROPN
ma-234	232	8	and	and	CCONJ
ma-234	232	9	f	f	PROPN
ma-234	232	10	∈mα	∈mα	PROPN
ma-234	232	11	q	q	PROPN
ma-234	232	12	,	,	PUNCT
ma-234	232	13	p	p	X
ma-234	232	14	,	,	PUNCT
ma-234	232	15	we	we	PRON
ma-234	232	16	have	have	VERB
ma-234	232	17	‖f	‖f	PRON
ma-234	232	18	(	(	PUNCT
ma-234	232	19	·	·	PUNCT
ma-234	232	20	−	−	NOUN
ma-234	233	1	y)‖mα	y)‖mα	NUM
ma-234	233	2	q	q	NOUN
ma-234	233	3	,	,	PUNCT
ma-234	233	4	p	p	ADJ
ma-234	233	5	≤	≤	PROPN
ma-234	233	6	c1	c1	NOUN
ma-234	233	7	‖f	‖f	PUNCT
ma-234	233	8	‖mα	‖mα	NUM
ma-234	233	9	q	q	NOUN
ma-234	233	10	,	,	PUNCT
ma-234	233	11	p	p	NOUN
ma-234	233	12	.	.	NOUN
ma-234	233	13	2	2	X
ma-234	233	14	)	)	PUNCT
ma-234	233	15	if	if	SCONJ
ma-234	233	16	q	q	X
ma-234	233	17	<	<	X
ma-234	233	18	α	α	X
ma-234	233	19	<	<	X
ma-234	233	20	p	p	X
ma-234	233	21	<	<	X
ma-234	233	22	∞	∞	PROPN
ma-234	233	23	then	then	ADV
ma-234	233	24	the	the	DET
ma-234	233	25	set	set	NOUN
ma-234	233	26	l∞c	l∞c	NOUN
ma-234	233	27	of	of	ADP
ma-234	233	28	all	all	PRON
ma-234	233	29	compactly	compactly	ADV
ma-234	233	30	supported	support	VERB
ma-234	233	31	bounded	bounded	ADJ
ma-234	233	32	functions	function	NOUN
ma-234	233	33	is	be	AUX
ma-234	233	34	dense	dense	ADJ
ma-234	233	35	in	in	ADP
ma-234	233	36	mα	mα	PROPN
ma-234	233	37	q	q	NOUN
ma-234	233	38	,	,	PUNCT
ma-234	233	39	p	p	NOUN
ma-234	233	40	.	.	NOUN
ma-234	234	1	3	3	X
ma-234	234	2	)	)	PUNCT
ma-234	234	3	if	if	SCONJ
ma-234	234	4	q	q	X
ma-234	234	5	<	<	X
ma-234	234	6	α	α	X
ma-234	234	7	<	<	X
ma-234	234	8	p	p	X
ma-234	234	9	<	<	X
ma-234	234	10	∞	∞	NOUN
ma-234	234	11	or	or	CCONJ
ma-234	234	12	p	p	NOUN
ma-234	235	1	=	=	NOUN
ma-234	235	2	∞	∞	PROPN
ma-234	235	3	then	then	ADV
ma-234	235	4	there	there	PRON
ma-234	235	5	exists	exist	VERB
ma-234	235	6	c2	c2	PROPN
ma-234	235	7	>	>	X
ma-234	235	8	0	0	NUM
ma-234	236	1	such	such	ADJ
ma-234	236	2	that	that	PRON
ma-234	236	3	for	for	ADP
ma-234	236	4	any	any	DET
ma-234	236	5	element	element	NOUN
ma-234	236	6	f	f	PROPN
ma-234	236	7	of	of	ADP
ma-234	236	8	mα	mα	PROPN
ma-234	236	9	q	q	NOUN
ma-234	236	10	,	,	PUNCT
ma-234	236	11	p	p	X
ma-234	236	12	,	,	PUNCT
ma-234	236	13	we	we	PRON
ma-234	236	14	have	have	VERB
ma-234	236	15	‖f	‖f	PUNCT
ma-234	236	16	‖mα	‖mα	NUM
ma-234	236	17	q	q	NOUN
ma-234	236	18	,	,	PUNCT
ma-234	236	19	p	p	ADJ
ma-234	236	20	≤	≤	PUNCT
ma-234	236	21	c2	c2	PROPN
ma-234	236	22	‖f	‖f	ADJ
ma-234	236	23	‖α	‖α	PROPN
ma-234	236	24	.	.	PUNCT
ma-234	237	1	a	a	DET
ma-234	237	2	classical	classical	ADJ
ma-234	237	3	property	property	NOUN
ma-234	237	4	of	of	ADP
ma-234	237	5	lebesgue	lebesgue	ADJ
ma-234	237	6	spaces	space	NOUN
ma-234	237	7	reads	read	VERB
ma-234	237	8	as	as	SCONJ
ma-234	237	9	follows	follow	VERB
ma-234	237	10	.	.	PUNCT
ma-234	238	1	lemma	lemma	PROPN
ma-234	238	2	3.8	3.8	NUM
ma-234	238	3	.	.	PUNCT
ma-234	239	1	if	if	SCONJ
ma-234	239	2	1	1	NUM
ma-234	239	3	≤	≤	NUM
ma-234	239	4	α	α	NOUN
ma-234	239	5	<	<	X
ma-234	239	6	∞	∞	PROPN
ma-234	239	7	and	and	CCONJ
ma-234	239	8	f	f	PROPN
ma-234	239	9	is	be	AUX
ma-234	239	10	in	in	ADP
ma-234	239	11	lα	lα	NOUN
ma-234	239	12	then	then	ADV
ma-234	239	13	we	we	PRON
ma-234	239	14	have	have	VERB
ma-234	239	15	lim	lim	PROPN
ma-234	239	16	y→0	y→0	PROPN
ma-234	239	17	‖f	‖f	ADP
ma-234	240	1	−	−	PROPN
ma-234	240	2	f	f	X
ma-234	240	3	(	(	PUNCT
ma-234	240	4	·	·	PUNCT
ma-234	240	5	−	−	NOUN
ma-234	240	6	y)‖α	y)‖α	NOUN
ma-234	240	7	=	=	NOUN
ma-234	240	8	0	0	PROPN
ma-234	240	9	.	.	PUNCT
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ma-234	241	3	.	.	PUNCT
ma-234	242	1	j.	j.	PROPN
ma-234	242	2	math	math	PROPN
ma-234	242	3	.	.	PUNCT
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ma-234	243	2	.	.	PUNCT
ma-234	244	1	10.28924	10.28924	NUM
ma-234	244	2	/	/	SYM
ma-234	244	3	ada	ada	PROPN
ma-234	244	4	/	/	SYM
ma-234	244	5	ma.4.16	ma.4.16	PROPN
ma-234	244	6	10this	10this	DET
ma-234	244	7	result	result	NOUN
ma-234	244	8	can	can	AUX
ma-234	244	9	be	be	AUX
ma-234	244	10	extended	extend	VERB
ma-234	244	11	to	to	ADP
ma-234	244	12	the	the	DET
ma-234	244	13	setting	setting	NOUN
ma-234	244	14	of	of	ADP
ma-234	244	15	bourgain	bourgain	NOUN
ma-234	244	16	-	-	PUNCT
ma-234	244	17	morrey	morrey	NOUN
ma-234	244	18	spaces	space	NOUN
ma-234	244	19	and	and	CCONJ
ma-234	244	20	this	this	DET
ma-234	244	21	extension	extension	NOUN
ma-234	244	22	willplay	willplay	VERB
ma-234	244	23	a	a	DET
ma-234	244	24	key	key	ADJ
ma-234	244	25	role	role	NOUN
ma-234	244	26	in	in	ADP
ma-234	244	27	the	the	DET
ma-234	244	28	proofs	proof	NOUN
ma-234	244	29	of	of	ADP
ma-234	244	30	our	our	PRON
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ma-234	244	32	.	.	PUNCT
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ma-234	245	3	.	.	PUNCT
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ma-234	246	2	1	1	NUM
ma-234	246	3	≤	≤	NOUN
ma-234	246	4	q	q	ADJ
ma-234	246	5	≤	≤	NUM
ma-234	246	6	α	α	NOUN
ma-234	246	7	≤	≤	NOUN
ma-234	247	1	p	p	DET
ma-234	247	2	<	<	X
ma-234	247	3	∞	∞	PROPN
ma-234	247	4	and	and	CCONJ
ma-234	247	5	f	f	PROPN
ma-234	247	6	be	be	AUX
ma-234	247	7	any	any	DET
ma-234	247	8	element	element	NOUN
ma-234	247	9	of	of	ADP
ma-234	247	10	mα	mα	PROPN
ma-234	247	11	q	q	NOUN
ma-234	247	12	,	,	PUNCT
ma-234	247	13	p	p	NOUN
ma-234	247	14	.	.	PUNCT
ma-234	248	1	then	then	ADV
ma-234	248	2	lim	lim	PROPN
ma-234	248	3	y→0	y→0	PROPN
ma-234	249	1	‖f	‖f	PRON
ma-234	249	2	−	−	PROPN
ma-234	249	3	f	f	X
ma-234	249	4	(	(	PUNCT
ma-234	249	5	·	·	PUNCT
ma-234	249	6	−	−	NOUN
ma-234	249	7	y)‖mα	y)‖mα	NUM
ma-234	249	8	q	q	NOUN
ma-234	249	9	,	,	PUNCT
ma-234	249	10	p	p	NOUN
ma-234	249	11	=	=	NOUN
ma-234	249	12	0	0	NUM
ma-234	249	13	.	.	PUNCT
ma-234	250	1	proof	proof	NOUN
ma-234	250	2	.	.	PUNCT
ma-234	251	1	if	if	SCONJ
ma-234	251	2	q	q	PRON
ma-234	251	3	=	=	SYM
ma-234	251	4	α	α	NOUN
ma-234	251	5	or	or	CCONJ
ma-234	251	6	α	α	NOUN
ma-234	251	7	=	=	PUNCT
ma-234	252	1	p	p	NOUN
ma-234	252	2	then	then	ADV
ma-234	252	3	mα	mα	PROPN
ma-234	252	4	q	q	NOUN
ma-234	252	5	,	,	PUNCT
ma-234	252	6	p	p	NOUN
ma-234	252	7	=	=	X
ma-234	252	8	{	{	PUNCT
ma-234	252	9	0	0	NUM
ma-234	252	10	}	}	PUNCT
ma-234	252	11	and	and	CCONJ
ma-234	252	12	therefore	therefore	ADV
ma-234	252	13	,	,	PUNCT
ma-234	252	14	we	we	PRON
ma-234	252	15	have	have	VERB
ma-234	252	16	nothing	nothing	PRON
ma-234	252	17	to	to	PART
ma-234	252	18	prove	prove	VERB
ma-234	252	19	.	.	PUNCT
ma-234	253	1	thus	thus	ADV
ma-234	253	2	wesuppose	wesuppose	ADP
ma-234	253	3	that	that	PRON
ma-234	253	4	q	q	X
ma-234	253	5	<	<	X
ma-234	253	6	α	α	X
ma-234	253	7	<	<	X
ma-234	253	8	p.	p.	NOUN
ma-234	253	9	by	by	ADP
ma-234	253	10	point	point	NOUN
ma-234	253	11	2	2	NUM
ma-234	253	12	)	)	PUNCT
ma-234	253	13	of	of	ADP
ma-234	253	14	proposition	proposition	NOUN
ma-234	253	15	3.7	3.7	NUM
ma-234	253	16	,	,	PUNCT
ma-234	253	17	there	there	PRON
ma-234	253	18	exists	exist	VERB
ma-234	253	19	a	a	DET
ma-234	253	20	sequence	sequence	NOUN
ma-234	253	21	(	(	PUNCT
ma-234	253	22	fn)n≥1	fn)n≥1	NOUN
ma-234	253	23	of	of	ADP
ma-234	253	24	elementsof	elementsof	ADJ
ma-234	253	25	l∞c	l∞c	NOUN
ma-234	253	26	such	such	ADJ
ma-234	253	27	that	that	SCONJ
ma-234	253	28	lim	lim	PROPN
ma-234	253	29	n→∞	n→∞	PRON
ma-234	254	1	‖fn	‖fn	PROPN
ma-234	255	1	−	−	PROPN
ma-234	255	2	f	f	PROPN
ma-234	255	3	‖mα	‖mα	NUM
ma-234	255	4	q	q	NOUN
ma-234	255	5	,	,	PUNCT
ma-234	255	6	p	p	NOUN
ma-234	255	7	=	=	NOUN
ma-234	255	8	0	0	NUM
ma-234	255	9	.	.	PUNCT
ma-234	256	1	moreover	moreover	ADV
ma-234	256	2	,	,	PUNCT
ma-234	256	3	according	accord	VERB
ma-234	256	4	to	to	ADP
ma-234	256	5	point	point	NOUN
ma-234	256	6	1	1	NUM
ma-234	256	7	)	)	PUNCT
ma-234	256	8	of	of	ADP
ma-234	256	9	proposition	proposition	NOUN
ma-234	256	10	3.7	3.7	NUM
ma-234	256	11	,	,	PUNCT
ma-234	256	12	thereexists	thereexist	NOUN
ma-234	256	13	c1	c1	PROPN
ma-234	256	14	>	>	X
ma-234	256	15	0	0	NUM
ma-234	256	16	such	such	ADJ
ma-234	256	17	that	that	DET
ma-234	256	18	‖(fn	‖(fn	PROPN
ma-234	256	19	−	−	PROPN
ma-234	256	20	f	f	PROPN
ma-234	256	21	)	)	PUNCT
ma-234	256	22	(	(	PUNCT
ma-234	256	23	·	·	PUNCT
ma-234	256	24	−	−	NOUN
ma-234	257	1	y)‖mα	y)‖mα	NUM
ma-234	257	2	q	q	NOUN
ma-234	257	3	,	,	PUNCT
ma-234	257	4	p	p	ADJ
ma-234	257	5	≤	≤	PROPN
ma-234	257	6	c1	c1	PROPN
ma-234	257	7	‖fn	‖fn	PROPN
ma-234	257	8	−	−	PROPN
ma-234	257	9	f	f	PROPN
ma-234	257	10	‖mα	‖mα	NUM
ma-234	257	11	q	q	NOUN
ma-234	257	12	,	,	PUNCT
ma-234	257	13	p	p	NOUN
ma-234	257	14	,	,	PUNCT
ma-234	257	15	y	y	PROPN
ma-234	257	16	∈	∈	PROPN
ma-234	257	17	rd	rd	PROPN
ma-234	257	18	,	,	PUNCT
ma-234	257	19	n	n	X
ma-234	257	20	≥	≥	NUM
ma-234	257	21	1	1	NUM
ma-234	257	22	.	.	PUNCT
ma-234	257	23	(	(	PUNCT
ma-234	257	24	∗	∗	NOUN
ma-234	257	25	)	)	PUNCT
ma-234	257	26	let	let	VERB
ma-234	257	27	ε	ε	PROPN
ma-234	257	28	>	>	X
ma-234	257	29	0	0	PUNCT
ma-234	257	30	be	be	AUX
ma-234	257	31	a	a	DET
ma-234	257	32	fixed	fixed	ADJ
ma-234	257	33	real	real	ADJ
ma-234	257	34	number	number	NOUN
ma-234	257	35	.	.	PUNCT
ma-234	258	1	there	there	PRON
ma-234	258	2	exists	exist	VERB
ma-234	258	3	an	an	DET
ma-234	258	4	integer	integer	NOUN
ma-234	258	5	nε	nε	NOUN
ma-234	258	6	such	such	ADJ
ma-234	258	7	that	that	DET
ma-234	258	8	‖fnε	‖fnε	NOUN
ma-234	258	9	−	−	X
ma-234	258	10	f	f	PROPN
ma-234	258	11	‖mα	‖mα	NUM
ma-234	258	12	q	q	NOUN
ma-234	258	13	,	,	PUNCT
ma-234	258	14	p	p	X
ma-234	258	15	<	<	X
ma-234	258	16	ε	ε	PROPN
ma-234	258	17	2(1	2(1	NUM
ma-234	258	18	+	+	CCONJ
ma-234	258	19	c1	c1	NOUN
ma-234	258	20	)	)	PUNCT
ma-234	258	21	.	.	PUNCT
ma-234	259	1	(	(	PUNCT
ma-234	259	2	∗∗	∗∗	NOUN
ma-234	259	3	)	)	PUNCT
ma-234	259	4	from	from	ADP
ma-234	259	5	(	(	PUNCT
ma-234	259	6	∗	∗	NOUN
ma-234	259	7	)	)	PUNCT
ma-234	259	8	,	,	PUNCT
ma-234	259	9	(	(	PUNCT
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ma-234	259	11	)	)	PUNCT
ma-234	259	12	,	,	PUNCT
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ma-234	259	14	1	1	NUM
ma-234	259	15	)	)	PUNCT
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ma-234	259	17	point	point	NOUN
ma-234	259	18	3	3	NUM
ma-234	259	19	)	)	PUNCT
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ma-234	259	21	proposition	proposition	NOUN
ma-234	259	22	3.7	3.7	NUM
ma-234	259	23	we	we	PRON
ma-234	259	24	have	have	VERB
ma-234	259	25	‖f	‖f	ADP
ma-234	259	26	−	−	PROPN
ma-234	259	27	f	f	SYM
ma-234	259	28	(	(	PUNCT
ma-234	259	29	·	·	PUNCT
ma-234	259	30	−	−	NOUN
ma-234	260	1	y)‖mα	y)‖mα	NUM
ma-234	260	2	q	q	NOUN
ma-234	260	3	,	,	PUNCT
ma-234	260	4	p	p	NOUN
ma-234	260	5	≤	≤	NOUN
ma-234	260	6	‖f	‖f	PRON
ma-234	260	7	−	−	PROPN
ma-234	260	8	fnε‖mα	fnε‖mα	PROPN
ma-234	261	1	q	q	NOUN
ma-234	261	2	,	,	PUNCT
ma-234	261	3	p	p	PROPN
ma-234	261	4	+	+	X
ma-234	261	5	‖fnε	‖fnε	PROPN
ma-234	261	6	−	−	PROPN
ma-234	261	7	fnε	fnε	PROPN
ma-234	261	8	(	(	PUNCT
ma-234	261	9	·	·	PUNCT
ma-234	261	10	−	−	NOUN
ma-234	262	1	y)‖mα	y)‖mα	NUM
ma-234	262	2	q	q	NOUN
ma-234	262	3	,	,	PUNCT
ma-234	262	4	p	p	X
ma-234	262	5	+	+	X
ma-234	262	6	‖(fnε	‖(fnε	PUNCT
ma-234	262	7	−	−	PROPN
ma-234	262	8	f	f	X
ma-234	262	9	)	)	PUNCT
ma-234	262	10	(	(	PUNCT
ma-234	262	11	·	·	PUNCT
ma-234	262	12	−	−	NOUN
ma-234	263	1	y)‖mα	y)‖mα	NUM
ma-234	263	2	q	q	NOUN
ma-234	263	3	,	,	PUNCT
ma-234	263	4	p	p	NOUN
ma-234	263	5	≤	≤	NOUN
ma-234	263	6	‖f	‖f	PRON
ma-234	263	7	−	−	PROPN
ma-234	263	8	fnε‖mα	fnε‖mα	PROPN
ma-234	264	1	q	q	NOUN
ma-234	264	2	,	,	PUNCT
ma-234	264	3	p	p	X
ma-234	264	4	+	+	X
ma-234	264	5	‖fnε	‖fnε	PROPN
ma-234	264	6	(	(	PUNCT
ma-234	264	7	·	·	PUNCT
ma-234	264	8	−	−	ADP
ma-234	265	1	y)−	y)−	NUM
ma-234	265	2	fnε‖mα	fnε‖mα	NOUN
ma-234	265	3	q	q	NOUN
ma-234	265	4	,	,	PUNCT
ma-234	265	5	p	p	NOUN
ma-234	265	6	+	+	X
ma-234	265	7	c1	c1	NOUN
ma-234	265	8	‖fnε	‖fnε	NOUN
ma-234	265	9	−	−	X
ma-234	265	10	f	f	PROPN
ma-234	265	11	‖mα	‖mα	NUM
ma-234	265	12	q	q	NOUN
ma-234	265	13	,	,	PUNCT
ma-234	265	14	p	p	NOUN
ma-234	265	15	≤	≤	X
ma-234	265	16	(	(	PUNCT
ma-234	265	17	1	1	NUM
ma-234	265	18	+	+	NUM
ma-234	265	19	c1	c1	NOUN
ma-234	265	20	)	)	PUNCT
ma-234	265	21	‖f	‖f	PRON
ma-234	265	22	−	−	PROPN
ma-234	265	23	fnε‖mα	fnε‖mα	PROPN
ma-234	265	24	q	q	NOUN
ma-234	265	25	,	,	PUNCT
ma-234	265	26	p	p	X
ma-234	265	27	+	+	X
ma-234	265	28	‖fnε	‖fnε	PROPN
ma-234	265	29	(	(	PUNCT
ma-234	265	30	·	·	PUNCT
ma-234	265	31	−	−	ADP
ma-234	266	1	y)−	y)−	NUM
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ma-234	266	3	q	q	NOUN
ma-234	266	4	,	,	PUNCT
ma-234	266	5	p	p	X
ma-234	266	6	<	<	X
ma-234	266	7	ε	ε	PROPN
ma-234	266	8	2	2	NUM
ma-234	266	9	+	+	CCONJ
ma-234	266	10	c2	c2	PROPN
ma-234	266	11	‖fnε	‖fnε	PROPN
ma-234	266	12	(	(	PUNCT
ma-234	266	13	·	·	PUNCT
ma-234	266	14	−	−	ADP
ma-234	267	1	y)−	y)−	PROPN
ma-234	267	2	fnε‖α	fnε‖α	PROPN
ma-234	267	3	.	.	PUNCT
ma-234	268	1	according	accord	VERB
ma-234	268	2	to	to	ADP
ma-234	268	3	lemma	lemma	PROPN
ma-234	268	4	3.8	3.8	NUM
ma-234	268	5	,	,	PUNCT
ma-234	268	6	for	for	ADP
ma-234	268	7	any	any	DET
ma-234	268	8	y	y	PROPN
ma-234	268	9	∈	∈	PROPN
ma-234	268	10	rd	rd	NOUN
ma-234	268	11	such	such	ADJ
ma-234	268	12	that	that	SCONJ
ma-234	268	13	0	0	NUM
ma-234	268	14	<	<	X
ma-234	268	15	|y	|y	NOUN
ma-234	268	16	|	|	CCONJ
ma-234	268	17	<	<	X
ma-234	268	18	1	1	NUM
ma-234	268	19	,	,	PUNCT
ma-234	268	20	we	we	PRON
ma-234	268	21	have	have	VERB
ma-234	268	22	‖fnε	‖fnε	X
ma-234	268	23	(	(	PUNCT
ma-234	268	24	·	·	PUNCT
ma-234	268	25	−	−	ADP
ma-234	269	1	y)−	y)−	PROPN
ma-234	269	2	fnε‖α	fnε‖α	PROPN
ma-234	269	3	<	<	X
ma-234	269	4	ε	ε	PROPN
ma-234	269	5	2c2and	2c2and	NUM
ma-234	269	6	therefore	therefore	ADV
ma-234	269	7	we	we	PRON
ma-234	269	8	obtain	obtain	VERB
ma-234	269	9	‖f	‖f	PUNCT
ma-234	269	10	−	−	PROPN
ma-234	269	11	f	f	SYM
ma-234	269	12	(	(	PUNCT
ma-234	269	13	·	·	PUNCT
ma-234	269	14	−	−	NOUN
ma-234	270	1	y)‖mα	y)‖mα	NUM
ma-234	270	2	q	q	NOUN
ma-234	270	3	,	,	PUNCT
ma-234	270	4	p	p	X
ma-234	270	5	<	<	X
ma-234	270	6	ε	ε	PROPN
ma-234	270	7	.	.	PUNCT
ma-234	271	1	this	this	PRON
ma-234	271	2	ends	end	VERB
ma-234	271	3	the	the	DET
ma-234	271	4	proof	proof	NOUN
ma-234	271	5	.	.	PUNCT
ma-234	272	1	�	�	PROPN
ma-234	272	2	4	4	NUM
ma-234	272	3	.	.	X
ma-234	272	4	inclusion	inclusion	NOUN
ma-234	272	5	and	and	CCONJ
ma-234	272	6	approximation	approximation	NOUN
ma-234	272	7	results	result	VERB
ma-234	272	8	4.1	4.1	NUM
ma-234	272	9	.	.	PUNCT
ma-234	273	1	inclusion	inclusion	NOUN
ma-234	273	2	of	of	ADP
ma-234	273	3	mα	mα	PROPN
ma-234	273	4	q	q	NOUN
ma-234	273	5	,	,	PUNCT
ma-234	273	6	p	p	NOUN
ma-234	273	7	in	in	ADP
ma-234	273	8	f(q	f(q	PROPN
ma-234	273	9	,	,	PUNCT
ma-234	273	10	p	p	X
ma-234	273	11	,	,	PUNCT
ma-234	273	12	α	α	NOUN
ma-234	273	13	)	)	PUNCT
ma-234	273	14	.	.	PUNCT
ma-234	274	1	this	this	DET
ma-234	274	2	subsection	subsection	NOUN
ma-234	274	3	is	be	AUX
ma-234	274	4	devoted	devote	VERB
ma-234	274	5	to	to	PART
ma-234	274	6	prove	prove	VERB
ma-234	274	7	exclusively	exclusively	ADV
ma-234	274	8	theorem	theorem	VERB
ma-234	274	9	2.3	2.3	NUM
ma-234	274	10	.	.	PUNCT
ma-234	275	1	proof	proof	NOUN
ma-234	275	2	of	of	ADP
ma-234	275	3	theorem	theorem	ADJ
ma-234	275	4	2.31	2.31	NUM
ma-234	275	5	)	)	PUNCT
ma-234	275	6	•	•	NOUN
ma-234	276	1	we	we	PRON
ma-234	276	2	recall	recall	VERB
ma-234	276	3	that	that	PRON
ma-234	276	4	mα	mα	PROPN
ma-234	276	5	q,∞	q,∞	PROPN
ma-234	276	6	=	=	PRON
ma-234	276	7	mα	mα	PROPN
ma-234	276	8	q	q	NOUN
ma-234	276	9	=	=	SYM
ma-234	276	10	f(q,∞	f(q,∞	NOUN
ma-234	276	11	,	,	PUNCT
ma-234	276	12	α	α	NOUN
ma-234	276	13	)	)	PUNCT
ma-234	276	14	.	.	PUNCT
ma-234	277	1	therefore	therefore	ADV
ma-234	277	2	,	,	PUNCT
ma-234	277	3	we	we	PRON
ma-234	277	4	have	have	VERB
ma-234	277	5	nothing	nothing	PRON
ma-234	277	6	to	to	PART
ma-234	277	7	prove	prove	VERB
ma-234	277	8	if	if	SCONJ
ma-234	277	9	p	p	PROPN
ma-234	277	10	=	=	NOUN
ma-234	277	11	∞.	∞.	PROPN
ma-234	277	12	•	•	NOUN
ma-234	277	13	if	if	SCONJ
ma-234	277	14	p	p	PROPN
ma-234	277	15	<	<	X
ma-234	277	16	∞	∞	PROPN
ma-234	277	17	and	and	CCONJ
ma-234	277	18	α	α	NOUN
ma-234	277	19	∈	∈	PROPN
ma-234	277	20	{	{	PUNCT
ma-234	277	21	q	q	NOUN
ma-234	277	22	,	,	PUNCT
ma-234	277	23	p	p	NOUN
ma-234	277	24	}	}	PUNCT
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ma-234	277	26	mα	mα	PROPN
ma-234	277	27	q	q	NOUN
ma-234	277	28	,	,	PUNCT
ma-234	277	29	p	p	NOUN
ma-234	277	30	=	=	X
ma-234	277	31	{	{	PUNCT
ma-234	277	32	0	0	NUM
ma-234	277	33	}	}	PUNCT
ma-234	277	34	.	.	PUNCT
ma-234	278	1	thus	thus	ADV
ma-234	278	2	the	the	DET
ma-234	278	3	result	result	NOUN
ma-234	278	4	is	be	AUX
ma-234	278	5	obvious	obvious	ADJ
ma-234	278	6	.	.	PUNCT
ma-234	279	1	•	•	NUM
ma-234	279	2	assume	assume	VERB
ma-234	279	3	that	that	SCONJ
ma-234	279	4	1	1	NUM
ma-234	279	5	≤	≤	NOUN
ma-234	279	6	q	q	NOUN
ma-234	279	7	<	<	X
ma-234	279	8	α	α	X
ma-234	279	9	<	<	X
ma-234	279	10	p	p	X
ma-234	279	11	<	<	X
ma-234	279	12	∞.let	∞.let	PROPN
ma-234	279	13	f	f	NOUN
ma-234	279	14	be	be	AUX
ma-234	279	15	in	in	ADP
ma-234	279	16	l1	l1	PROPN
ma-234	279	17	loc	loc	PROPN
ma-234	279	18	and	and	CCONJ
ma-234	279	19	{	{	PUNCT
ma-234	279	20	qi	qi	X
ma-234	279	21	:	:	PUNCT
ma-234	280	1	i	i	PRON
ma-234	280	2	∈	∈	PROPN
ma-234	280	3	i	i	PRON
ma-234	280	4	}	}	PUNCT
ma-234	280	5	be	be	VERB
ma-234	280	6	a	a	DET
ma-234	280	7	disjoint	disjoint	ADJ
ma-234	280	8	family	family	NOUN
ma-234	280	9	of	of	ADP
ma-234	280	10	cubes	cube	NOUN
ma-234	280	11	of	of	ADP
ma-234	280	12	rd	rd	PROPN
ma-234	280	13	.	.	PUNCT
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ma-234	281	2	eur	eur	PROPN
ma-234	281	3	.	.	PUNCT
ma-234	282	1	j.	j.	PROPN
ma-234	282	2	math	math	PROPN
ma-234	282	3	.	.	PUNCT
ma-234	283	1	anal	anal	PROPN
ma-234	283	2	.	.	PUNCT
ma-234	284	1	10.28924	10.28924	NUM
ma-234	284	2	/	/	SYM
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ma-234	284	4	/	/	SYM
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ma-234	284	6	11	11	NUM
ma-234	284	7	a	a	NOUN
ma-234	284	8	)	)	PUNCT
ma-234	284	9	let	let	VERB
ma-234	284	10	us	we	PRON
ma-234	284	11	consider	consider	VERB
ma-234	284	12	an	an	DET
ma-234	284	13	element	element	NOUN
ma-234	284	14	i	i	PRON
ma-234	284	15	of	of	ADP
ma-234	284	16	i.	i.	NOUN
ma-234	284	17	we	we	PRON
ma-234	284	18	can	can	AUX
ma-234	284	19	associate	associate	VERB
ma-234	284	20	to	to	ADP
ma-234	284	21	i	i	PRON
ma-234	284	22	an	an	DET
ma-234	284	23	element	element	NOUN
ma-234	284	24	t	t	PROPN
ma-234	284	25	of	of	ADP
ma-234	284	26	{	{	PUNCT
ma-234	284	27	−1/3	−1/3	ADJ
ma-234	284	28	,	,	PUNCT
ma-234	284	29	0	0	NUM
ma-234	284	30	,	,	PUNCT
ma-234	285	1	1/3}d	1/3}d	NUM
ma-234	285	2	andan	andan	PROPN
ma-234	285	3	element	element	PROPN
ma-234	285	4	r(i	r(i	PROPN
ma-234	285	5	,	,	PUNCT
ma-234	285	6	t	t	PROPN
ma-234	285	7	)	)	PUNCT
ma-234	285	8	of	of	ADP
ma-234	285	9	dt	dt	INTJ
ma-234	285	10	such	such	ADJ
ma-234	285	11	that	that	SCONJ
ma-234	285	12	qi	qi	PROPN
ma-234	285	13	⊂	⊂	PRON
ma-234	285	14	r(i	r(i	PROPN
ma-234	285	15	,	,	PUNCT
ma-234	285	16	t	t	PROPN
ma-234	285	17	)	)	PUNCT
ma-234	285	18	and	and	CCONJ
ma-234	285	19	`	`	PUNCT
ma-234	285	20	(	(	PUNCT
ma-234	285	21	r(i	r(i	X
ma-234	285	22	,	,	PUNCT
ma-234	285	23	t	t	PROPN
ma-234	285	24	)	)	PUNCT
ma-234	285	25	)	)	PUNCT
ma-234	285	26	≤	≤	ADV
ma-234	285	27	3	3	NUM
ma-234	285	28	`	`	PUNCT
ma-234	285	29	(	(	PUNCT
ma-234	285	30	qi	qi	X
ma-234	285	31	)	)	PUNCT
ma-234	285	32	(	(	PUNCT
ma-234	285	33	see	see	VERB
ma-234	285	34	proposition	proposition	NOUN
ma-234	285	35	3.3	3.3	NUM
ma-234	285	36	)	)	PUNCT
ma-234	285	37	.	.	PUNCT
ma-234	286	1	wehave	wehave	NOUN
ma-234	286	2	|qi	|qi	NUM
ma-234	287	1	|	|	ADV
ma-234	287	2	1	1	NUM
ma-234	287	3	α	α	NOUN
ma-234	287	4	−	−	NOUN
ma-234	287	5	1	1	NUM
ma-234	287	6	q	q	NOUN
ma-234	287	7	(	(	PUNCT
ma-234	287	8	∫	∫	PROPN
ma-234	287	9	qi	qi	PROPN
ma-234	287	10	|f	|f	PROPN
ma-234	287	11	(	(	PUNCT
ma-234	287	12	x)|qdx	x)|qdx	PROPN
ma-234	287	13	)	)	PUNCT
ma-234	287	14	1	1	NUM
ma-234	287	15	q	q	NOUN
ma-234	287	16	≤	≤	NUM
ma-234	287	17	(	(	PUNCT
ma-234	287	18	|qi	|qi	NUM
ma-234	287	19	|	|	ADV
ma-234	287	20	|r(i	|r(i	INTJ
ma-234	287	21	,	,	PUNCT
ma-234	287	22	t)|	t)|	INTJ
ma-234	287	23	)	)	PUNCT
ma-234	287	24	1	1	NUM
ma-234	287	25	α	α	NOUN
ma-234	287	26	−	−	NOUN
ma-234	287	27	1	1	NUM
ma-234	288	1	q	q	NOUN
ma-234	288	2	|r(i	|r(i	INTJ
ma-234	288	3	,	,	PUNCT
ma-234	288	4	t)|	t)|	ADV
ma-234	288	5	1	1	NUM
ma-234	288	6	α	α	NOUN
ma-234	288	7	−	−	NOUN
ma-234	288	8	1	1	NUM
ma-234	288	9	q	q	NOUN
ma-234	288	10	(	(	PUNCT
ma-234	288	11	∫	∫	PROPN
ma-234	288	12	r(i	r(i	PROPN
ma-234	288	13	,	,	PUNCT
ma-234	288	14	t	t	PROPN
ma-234	288	15	)	)	PUNCT
ma-234	288	16	|f	|f	PROPN
ma-234	289	1	(	(	PUNCT
ma-234	289	2	x)|qdx	x)|qdx	PROPN
ma-234	289	3	)	)	PUNCT
ma-234	289	4	1	1	NUM
ma-234	289	5	q	q	NOUN
ma-234	289	6	≤	≤	NUM
ma-234	289	7	3	3	NUM
ma-234	289	8	d	d	NOUN
ma-234	289	9	(	(	PUNCT
ma-234	289	10	1	1	NUM
ma-234	289	11	α	α	NOUN
ma-234	289	12	−	−	PROPN
ma-234	289	13	1	1	NUM
ma-234	289	14	q	q	NOUN
ma-234	289	15	)	)	PUNCT
ma-234	290	1	|r(i	|r(i	INTJ
ma-234	290	2	,	,	PUNCT
ma-234	290	3	t)|	t)|	ADV
ma-234	290	4	1	1	NUM
ma-234	290	5	α	α	NOUN
ma-234	290	6	−	−	NOUN
ma-234	290	7	1	1	NUM
ma-234	290	8	q	q	NOUN
ma-234	290	9	(	(	PUNCT
ma-234	290	10	∫	∫	PROPN
ma-234	290	11	r(i	r(i	PROPN
ma-234	290	12	,	,	PUNCT
ma-234	290	13	t	t	PROPN
ma-234	290	14	)	)	PUNCT
ma-234	290	15	|f	|f	PROPN
ma-234	291	1	(	(	PUNCT
ma-234	291	2	x)|qdx	x)|qdx	PROPN
ma-234	291	3	)	)	PUNCT
ma-234	291	4	1	1	NUM
ma-234	291	5	q	q	NOUN
ma-234	291	6	.	.	PUNCT
ma-234	292	1	b	b	X
ma-234	292	2	)	)	PUNCT
ma-234	292	3	let	let	VERB
ma-234	292	4	us	we	PRON
ma-234	292	5	fix	fix	VERB
ma-234	292	6	t	t	PROPN
ma-234	292	7	in	in	ADP
ma-234	292	8	{	{	PUNCT
ma-234	292	9	−1/3	−1/3	ADJ
ma-234	292	10	,	,	PUNCT
ma-234	292	11	0	0	NUM
ma-234	292	12	,	,	PUNCT
ma-234	292	13	1/3}d	1/3}d	NOUN
ma-234	292	14	and	and	CCONJ
ma-234	292	15	set	set	VERB
ma-234	292	16	rt	rt	PROPN
ma-234	292	17	=	=	PUNCT
ma-234	293	1	{	{	PUNCT
ma-234	293	2	r	r	NOUN
ma-234	293	3	∈	∈	NOUN
ma-234	293	4	dt	dt	NOUN
ma-234	293	5	:	:	PUNCT
ma-234	294	1	∃	∃	PROPN
ma-234	294	2	i	i	NOUN
ma-234	294	3	∈	∈	PROPN
ma-234	294	4	i	i	PRON
ma-234	294	5	such	such	ADJ
ma-234	294	6	that	that	SCONJ
ma-234	294	7	r(i	r(i	PROPN
ma-234	294	8	,	,	PUNCT
ma-234	294	9	t	t	PROPN
ma-234	294	10	)	)	PUNCT
ma-234	294	11	=	=	SYM
ma-234	295	1	r	r	NOUN
ma-234	295	2	}	}	PUNCT
ma-234	295	3	.	.	PUNCT
ma-234	296	1	note	note	VERB
ma-234	296	2	that	that	SCONJ
ma-234	296	3	,	,	PUNCT
ma-234	296	4	for	for	ADP
ma-234	296	5	all	all	DET
ma-234	296	6	r	r	NOUN
ma-234	296	7	∈	∈	PROPN
ma-234	296	8	rt	rt	NOUN
ma-234	296	9	,	,	PUNCT
ma-234	296	10	we	we	PRON
ma-234	296	11	have	have	VERB
ma-234	296	12	∀	∀	X
ma-234	297	1	i	i	PRON
ma-234	297	2	∈	∈	VERB
ma-234	298	1	i	i	PRON
ma-234	298	2	,	,	PUNCT
ma-234	298	3	r	r	NOUN
ma-234	298	4	=	=	SYM
ma-234	298	5	r(i	r(i	PROPN
ma-234	298	6	,	,	PUNCT
ma-234	298	7	t	t	PROPN
ma-234	298	8	)	)	PUNCT
ma-234	298	9	=	=	NOUN
ma-234	298	10	⇒	⇒	NOUN
ma-234	298	11	`	`	PUNCT
ma-234	298	12	(	(	PUNCT
ma-234	298	13	qi	qi	X
ma-234	298	14	)	)	PUNCT
ma-234	298	15	≤	≤	NOUN
ma-234	298	16	`	`	PUNCT
ma-234	298	17	(	(	PUNCT
ma-234	298	18	r	r	NOUN
ma-234	298	19	)	)	PUNCT
ma-234	298	20	≤	≤	NOUN
ma-234	298	21	3`(qi	3`(qi	NUM
ma-234	298	22	)	)	PUNCT
ma-234	299	1	=	=	NOUN
ma-234	299	2	⇒	⇒	NOUN
ma-234	299	3	|qi	|qi	X
ma-234	299	4	|	|	ADV
ma-234	299	5	≤	≤	NUM
ma-234	299	6	|r|	|r|	NOUN
ma-234	299	7	≤	≤	ADJ
ma-234	299	8	3d	3d	NUM
ma-234	299	9	|qi	|qi	NUM
ma-234	299	10	|∑	|∑	VERB
ma-234	299	11	i∈i	i∈i	ADJ
ma-234	299	12	,	,	PUNCT
ma-234	299	13	r	r	NOUN
ma-234	299	14	=	=	SYM
ma-234	299	15	r(i	r(i	NOUN
ma-234	299	16	,	,	PUNCT
ma-234	299	17	t	t	PROPN
ma-234	299	18	)	)	PUNCT
ma-234	299	19	|qi	|qi	NUM
ma-234	299	20	|	|	ADV
ma-234	299	21	≤	≤	NUM
ma-234	299	22	|r|	|r|	PROPN
ma-234	299	23	.	.	PUNCT
ma-234	300	1	this	this	PRON
ma-234	300	2	shows	show	VERB
ma-234	300	3	that	that	SCONJ
ma-234	300	4	the	the	DET
ma-234	300	5	cardinality	cardinality	NOUN
ma-234	300	6	of	of	ADP
ma-234	300	7	the	the	DET
ma-234	300	8	set	set	NOUN
ma-234	300	9	{	{	PUNCT
ma-234	300	10	i	i	NOUN
ma-234	300	11	∈	∈	PROPN
ma-234	300	12	i	i	PRON
ma-234	300	13	:	:	PUNCT
ma-234	300	14	r	r	NOUN
ma-234	300	15	=	=	SYM
ma-234	300	16	r(i	r(i	PROPN
ma-234	300	17	,	,	PUNCT
ma-234	300	18	t	t	PROPN
ma-234	300	19	)	)	PUNCT
ma-234	300	20	}	}	PUNCT
ma-234	300	21	does	do	AUX
ma-234	300	22	not	not	PART
ma-234	300	23	exceed	exceed	VERB
ma-234	300	24	3d	3d	NUM
ma-234	300	25	.c	.c	NOUN
ma-234	300	26	)	)	PUNCT
ma-234	301	1	we	we	PRON
ma-234	301	2	have∑	have∑	VERB
ma-234	301	3	i∈i	i∈i	ADV
ma-234	301	4	(	(	PUNCT
ma-234	301	5	|qi	|qi	NUM
ma-234	301	6	|	|	ADV
ma-234	302	1	1	1	NUM
ma-234	302	2	α	α	NOUN
ma-234	302	3	−	−	NOUN
ma-234	302	4	1	1	NUM
ma-234	302	5	q	q	NOUN
ma-234	302	6	(	(	PUNCT
ma-234	302	7	∫	∫	PROPN
ma-234	302	8	qi	qi	PROPN
ma-234	302	9	|f	|f	PROPN
ma-234	302	10	(	(	PUNCT
ma-234	302	11	x)|qdx	x)|qdx	PROPN
ma-234	302	12	)	)	PUNCT
ma-234	302	13	1	1	NUM
ma-234	302	14	q	q	NOUN
ma-234	302	15	)	)	PUNCT
ma-234	302	16	p	p	NOUN
ma-234	302	17	=	=	PUNCT
ma-234	302	18	∑	∑	PUNCT
ma-234	302	19	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	X
ma-234	302	20	∑	∑	PUNCT
ma-234	302	21	r∈rt	r∈rt	PROPN
ma-234	302	22	∑	∑	PUNCT
ma-234	302	23	i	i	PRON
ma-234	302	24	:	:	PUNCT
ma-234	302	25	r	r	NOUN
ma-234	302	26	=	=	SYM
ma-234	302	27	r(i	r(i	NOUN
ma-234	302	28	,	,	PUNCT
ma-234	302	29	t	t	PROPN
ma-234	302	30	)	)	PUNCT
ma-234	302	31	(	(	PUNCT
ma-234	302	32	|qi	|qi	NUM
ma-234	302	33	|	|	ADV
ma-234	303	1	1	1	NUM
ma-234	303	2	α	α	NOUN
ma-234	303	3	−	−	NOUN
ma-234	303	4	1	1	NUM
ma-234	303	5	q	q	NOUN
ma-234	303	6	(	(	PUNCT
ma-234	303	7	∫	∫	PROPN
ma-234	303	8	qi	qi	PROPN
ma-234	303	9	|f	|f	PROPN
ma-234	303	10	(	(	PUNCT
ma-234	303	11	x)|qdx	x)|qdx	PROPN
ma-234	303	12	)	)	PUNCT
ma-234	303	13	1	1	NUM
ma-234	303	14	q	q	NOUN
ma-234	303	15	)	)	PUNCT
ma-234	303	16	p	p	NOUN
ma-234	303	17	≤	≤	NOUN
ma-234	303	18	∑	∑	PUNCT
ma-234	303	19	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	X
ma-234	303	20	∑	∑	PUNCT
ma-234	303	21	r∈rt	r∈rt	PROPN
ma-234	303	22	∑	∑	PUNCT
ma-234	303	23	i	i	PRON
ma-234	303	24	:	:	PUNCT
ma-234	303	25	r	r	NOUN
ma-234	303	26	=	=	SYM
ma-234	303	27	r(i	r(i	NOUN
ma-234	303	28	,	,	PUNCT
ma-234	303	29	t	t	PROPN
ma-234	303	30	)	)	PUNCT
ma-234	303	31	(	(	PUNCT
ma-234	303	32	3	3	NUM
ma-234	303	33	d	d	NOUN
ma-234	303	34	(	(	PUNCT
ma-234	303	35	1	1	NUM
ma-234	303	36	α	α	NOUN
ma-234	303	37	−	−	PROPN
ma-234	303	38	1	1	NUM
ma-234	303	39	q	q	NOUN
ma-234	303	40	)	)	PUNCT
ma-234	303	41	p|r|	p|r|	NOUN
ma-234	303	42	1	1	NUM
ma-234	303	43	α	α	NOUN
ma-234	303	44	−	−	NOUN
ma-234	303	45	1	1	NUM
ma-234	303	46	q	q	NOUN
ma-234	303	47	(	(	PUNCT
ma-234	303	48	∫	∫	PROPN
ma-234	303	49	r	r	NOUN
ma-234	303	50	|f	|f	PROPN
ma-234	303	51	(	(	PUNCT
ma-234	303	52	x)|qdx	x)|qdx	PROPN
ma-234	303	53	)	)	PUNCT
ma-234	303	54	1	1	NUM
ma-234	303	55	q	q	NOUN
ma-234	303	56	)	)	PUNCT
ma-234	303	57	p	p	NOUN
ma-234	303	58	(	(	PUNCT
ma-234	303	59	by	by	ADP
ma-234	303	60	point	point	NOUN
ma-234	303	61	a	a	PRON
ma-234	303	62	)	)	PUNCT
ma-234	303	63	)	)	PUNCT
ma-234	304	1	≤	≤	ADV
ma-234	304	2	3	3	NUM
ma-234	304	3	d	d	NOUN
ma-234	304	4	(	(	PUNCT
ma-234	304	5	1	1	NUM
ma-234	304	6	α	α	NOUN
ma-234	304	7	−	−	PROPN
ma-234	304	8	1	1	NUM
ma-234	304	9	q	q	NOUN
ma-234	304	10	)	)	PUNCT
ma-234	305	1	p	p	X
ma-234	305	2	3d	3d	NUM
ma-234	305	3	∑	∑	PROPN
ma-234	305	4	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	X
ma-234	305	5	∑	∑	PUNCT
ma-234	305	6	r∈rt	r∈rt	PROPN
ma-234	305	7	(	(	PUNCT
ma-234	305	8	|r|	|r|	NOUN
ma-234	305	9	1	1	NUM
ma-234	305	10	α	α	NOUN
ma-234	305	11	−	−	PROPN
ma-234	305	12	1	1	NUM
ma-234	305	13	q	q	NOUN
ma-234	306	1	(	(	PUNCT
ma-234	306	2	∫	∫	PROPN
ma-234	306	3	r	r	NOUN
ma-234	306	4	|f	|f	PROPN
ma-234	306	5	(	(	PUNCT
ma-234	306	6	x)|qdx	x)|qdx	PROPN
ma-234	306	7	)	)	PUNCT
ma-234	306	8	1	1	NUM
ma-234	306	9	q	q	NOUN
ma-234	306	10	)	)	PUNCT
ma-234	306	11	p	p	NOUN
ma-234	306	12	(	(	PUNCT
ma-234	306	13	by	by	ADP
ma-234	306	14	point	point	NOUN
ma-234	306	15	b	b	NOUN
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ma-234	306	17	)	)	PUNCT
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ma-234	307	5	1	1	NUM
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ma-234	307	8	1	1	NUM
ma-234	307	9	q	q	NOUN
ma-234	307	10	)	)	PUNCT
ma-234	308	1	p	p	X
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ma-234	308	5	∑	∑	PUNCT
ma-234	308	6	r∈dt	r∈dt	PROPN
ma-234	308	7	(	(	PUNCT
ma-234	308	8	|r|	|r|	NOUN
ma-234	308	9	1	1	NUM
ma-234	308	10	α	α	NOUN
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ma-234	308	12	1	1	NUM
ma-234	308	13	q	q	NOUN
ma-234	308	14	(	(	PUNCT
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ma-234	308	18	(	(	PUNCT
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ma-234	308	20	)	)	PUNCT
ma-234	308	21	1	1	NUM
ma-234	308	22	q	q	NOUN
ma-234	308	23	)	)	PUNCT
ma-234	308	24	p	p	NOUN
ma-234	308	25	(	(	PUNCT
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ma-234	308	27	of	of	ADP
ma-234	308	28	rt	rt	PROPN
ma-234	308	29	⊂	⊂	PROPN
ma-234	308	30	dt	dt	PROPN
ma-234	308	31	)	)	PUNCT
ma-234	308	32	.	.	PUNCT
ma-234	309	1	therefore[∑	therefore[∑	PROPN
ma-234	309	2	i∈i	i∈i	ADV
ma-234	309	3	(	(	PUNCT
ma-234	309	4	|qi	|qi	NUM
ma-234	309	5	|	|	ADV
ma-234	309	6	1	1	NUM
ma-234	309	7	α	α	NOUN
ma-234	309	8	−	−	NOUN
ma-234	309	9	1	1	NUM
ma-234	309	10	q	q	NOUN
ma-234	309	11	(	(	PUNCT
ma-234	309	12	∫	∫	PROPN
ma-234	309	13	qi	qi	PROPN
ma-234	309	14	|f	|f	PROPN
ma-234	309	15	(	(	PUNCT
ma-234	309	16	x)|qdx	x)|qdx	PROPN
ma-234	309	17	)	)	PUNCT
ma-234	309	18	1	1	NUM
ma-234	309	19	q	q	NOUN
ma-234	309	20	)	)	PUNCT
ma-234	310	1	p	p	X
ma-234	310	2	]	]	X
ma-234	310	3	1	1	NUM
ma-234	310	4	p	p	NOUN
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ma-234	310	6	3	3	NUM
ma-234	310	7	d	d	NOUN
ma-234	310	8	(	(	PUNCT
ma-234	310	9	1	1	NUM
ma-234	310	10	α	α	NOUN
ma-234	310	11	−	−	NOUN
ma-234	310	12	1	1	NUM
ma-234	310	13	q	q	NOUN
ma-234	310	14	+	+	NUM
ma-234	310	15	1	1	NUM
ma-234	310	16	p	p	NOUN
ma-234	310	17	)	)	PUNCT
ma-234	310	18	∑	∑	PROPN
ma-234	310	19	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	NOUN
ma-234	310	20	‖f	‖f	ADP
ma-234	310	21	‖mα	‖mα	NUM
ma-234	310	22	q	q	NOUN
ma-234	310	23	,	,	PUNCT
ma-234	310	24	p(dt	p(dt	PROPN
ma-234	310	25	)	)	PUNCT
ma-234	310	26	.	.	PUNCT
ma-234	311	1	since	since	SCONJ
ma-234	311	2	the	the	DET
ma-234	311	3	above	above	ADJ
ma-234	311	4	inequality	inequality	NOUN
ma-234	311	5	is	be	AUX
ma-234	311	6	true	true	ADJ
ma-234	311	7	for	for	ADP
ma-234	311	8	all	all	DET
ma-234	311	9	disjoint	disjoint	NOUN
ma-234	311	10	family	family	NOUN
ma-234	311	11	{	{	PUNCT
ma-234	311	12	qi	qi	NOUN
ma-234	311	13	:	:	PUNCT
ma-234	311	14	i	i	PRON
ma-234	311	15	∈	∈	VERB
ma-234	311	16	i	i	PRON
ma-234	311	17	}	}	PUNCT
ma-234	311	18	of	of	ADP
ma-234	311	19	cubes	cube	NOUN
ma-234	311	20	of	of	ADP
ma-234	311	21	rd	rd	NOUN
ma-234	311	22	,	,	PUNCT
ma-234	311	23	we	we	PRON
ma-234	311	24	have	have	VERB
ma-234	311	25	‖f	‖f	ADP
ma-234	311	26	‖f(q	‖f(q	NOUN
ma-234	311	27	,	,	PUNCT
ma-234	311	28	p	p	X
ma-234	311	29	,	,	PUNCT
ma-234	311	30	α	α	NOUN
ma-234	311	31	)	)	PUNCT
ma-234	311	32	≤	≤	NOUN
ma-234	311	33	3	3	NUM
ma-234	311	34	d	d	NOUN
ma-234	311	35	(	(	PUNCT
ma-234	311	36	1	1	NUM
ma-234	311	37	α	α	NOUN
ma-234	311	38	−	−	NOUN
ma-234	311	39	1	1	NUM
ma-234	311	40	q	q	NOUN
ma-234	311	41	+	+	NUM
ma-234	311	42	1	1	NUM
ma-234	311	43	p	p	NOUN
ma-234	311	44	)	)	PUNCT
ma-234	311	45	∑	∑	PROPN
ma-234	311	46	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	NOUN
ma-234	311	47	‖f	‖f	ADP
ma-234	311	48	‖mα	‖mα	NUM
ma-234	311	49	q	q	NOUN
ma-234	311	50	,	,	PUNCT
ma-234	311	51	p(dt	p(dt	PROPN
ma-234	311	52	)	)	PUNCT
ma-234	311	53	.	.	PUNCT
ma-234	312	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	312	2	eur	eur	PROPN
ma-234	312	3	.	.	PUNCT
ma-234	313	1	j.	j.	PROPN
ma-234	313	2	math	math	PROPN
ma-234	313	3	.	.	PUNCT
ma-234	314	1	anal	anal	PROPN
ma-234	314	2	.	.	PUNCT
ma-234	315	1	10.28924	10.28924	NUM
ma-234	315	2	/	/	SYM
ma-234	315	3	ada	ada	PROPN
ma-234	315	4	/	/	SYM
ma-234	315	5	ma.4.16	ma.4.16	PROPN
ma-234	315	6	12therefore	12therefore	NUM
ma-234	315	7	,	,	PUNCT
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ma-234	315	9	3.6	3.6	NUM
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ma-234	315	11	that	that	SCONJ
ma-234	315	12	‖f	‖f	ADP
ma-234	315	13	‖f(q	‖f(q	NOUN
ma-234	315	14	,	,	PUNCT
ma-234	315	15	p	p	X
ma-234	315	16	,	,	PUNCT
ma-234	315	17	α	α	NOUN
ma-234	315	18	)	)	PUNCT
ma-234	315	19	≤	≤	NOUN
ma-234	315	20	3	3	NUM
ma-234	315	21	d	d	NOUN
ma-234	315	22	(	(	PUNCT
ma-234	315	23	1	1	NUM
ma-234	315	24	+	+	NUM
ma-234	315	25	1	1	NUM
ma-234	315	26	α	α	NOUN
ma-234	315	27	−	−	NOUN
ma-234	315	28	1	1	NUM
ma-234	315	29	q	q	NOUN
ma-234	315	30	+	+	NUM
ma-234	315	31	1	1	NUM
ma-234	315	32	p	p	NOUN
ma-234	315	33	)	)	PUNCT
ma-234	315	34	2	2	NUM
ma-234	315	35	d	d	NOUN
ma-234	315	36	(	(	PUNCT
ma-234	315	37	1	1	NUM
ma-234	315	38	q	q	NOUN
ma-234	315	39	−	−	PROPN
ma-234	315	40	1	1	NUM
ma-234	315	41	p	p	NOUN
ma-234	315	42	)	)	PUNCT
ma-234	315	43	‖f	‖f	PUNCT
ma-234	315	44	‖mα	‖mα	NUM
ma-234	315	45	q	q	NOUN
ma-234	315	46	,	,	PUNCT
ma-234	315	47	pand	pand	PROPN
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ma-234	315	49	mα	mα	ADP
ma-234	315	50	q	q	NOUN
ma-234	315	51	,	,	PUNCT
ma-234	315	52	p	p	PRON
ma-234	315	53	is	be	AUX
ma-234	315	54	continuously	continuously	ADV
ma-234	315	55	included	include	VERB
ma-234	315	56	in	in	ADP
ma-234	315	57	f(q	f(q	PROPN
ma-234	315	58	,	,	PUNCT
ma-234	315	59	p	p	X
ma-234	315	60	,	,	PUNCT
ma-234	315	61	α).2	α).2	PROPN
ma-234	315	62	)	)	PUNCT
ma-234	315	63	assume	assume	VERB
ma-234	316	1	that	that	SCONJ
ma-234	316	2	p	p	PROPN
ma-234	316	3	<	<	X
ma-234	316	4	∞	∞	PROPN
ma-234	316	5	and	and	CCONJ
ma-234	316	6	let	let	VERB
ma-234	316	7	f	f	PRON
ma-234	316	8	be	be	AUX
ma-234	316	9	any	any	DET
ma-234	316	10	element	element	NOUN
ma-234	316	11	of	of	ADP
ma-234	316	12	mα	mα	PROPN
ma-234	316	13	q	q	NOUN
ma-234	316	14	,	,	PUNCT
ma-234	316	15	p	p	X
ma-234	316	16	.	.	PUNCT
ma-234	317	1	by	by	ADP
ma-234	317	2	point	point	NOUN
ma-234	317	3	1	1	NUM
ma-234	317	4	)	)	PUNCT
ma-234	317	5	,	,	PUNCT
ma-234	317	6	f	f	PROPN
ma-234	317	7	is	be	AUX
ma-234	317	8	in	in	ADP
ma-234	317	9	f(q	f(q	PROPN
ma-234	317	10	,	,	PUNCT
ma-234	317	11	p	p	X
ma-234	317	12	,	,	PUNCT
ma-234	317	13	α	α	NOUN
ma-234	317	14	)	)	PUNCT
ma-234	317	15	and	and	CCONJ
ma-234	317	16	,	,	PUNCT
ma-234	317	17	forany	forany	PROPN
ma-234	317	18	y	y	PROPN
ma-234	317	19	∈	∈	PROPN
ma-234	317	20	rd	rd	PROPN
ma-234	317	21	,	,	PUNCT
ma-234	317	22	we	we	PRON
ma-234	317	23	have	have	VERB
ma-234	317	24	‖f	‖f	ADP
ma-234	318	1	−	−	PROPN
ma-234	318	2	f	f	SYM
ma-234	318	3	(	(	PUNCT
ma-234	318	4	·	·	PUNCT
ma-234	318	5	−	−	PROPN
ma-234	318	6	y)‖f(q	y)‖f(q	PROPN
ma-234	318	7	,	,	PUNCT
ma-234	318	8	p	p	X
ma-234	318	9	,	,	PUNCT
ma-234	318	10	α	α	NOUN
ma-234	318	11	)	)	PUNCT
ma-234	318	12	≤	≤	NOUN
ma-234	318	13	3	3	NUM
ma-234	318	14	d	d	NOUN
ma-234	318	15	(	(	PUNCT
ma-234	318	16	1	1	NUM
ma-234	318	17	+	+	NUM
ma-234	318	18	1	1	NUM
ma-234	318	19	α	α	NOUN
ma-234	318	20	−	−	NOUN
ma-234	318	21	1	1	NUM
ma-234	318	22	q	q	NOUN
ma-234	318	23	+	+	NUM
ma-234	318	24	1	1	NUM
ma-234	318	25	p	p	NOUN
ma-234	318	26	)	)	PUNCT
ma-234	318	27	2	2	NUM
ma-234	318	28	d	d	NOUN
ma-234	318	29	(	(	PUNCT
ma-234	318	30	1	1	NUM
ma-234	318	31	q	q	NOUN
ma-234	318	32	−	−	PROPN
ma-234	318	33	1	1	NUM
ma-234	318	34	p	p	NOUN
ma-234	318	35	)	)	PUNCT
ma-234	318	36	‖f	‖f	ADP
ma-234	318	37	−	−	PROPN
ma-234	318	38	f	f	X
ma-234	318	39	(	(	PUNCT
ma-234	318	40	·	·	PUNCT
ma-234	318	41	−	−	NOUN
ma-234	318	42	y)‖mα	y)‖mα	NUM
ma-234	318	43	q	q	NOUN
ma-234	318	44	,	,	PUNCT
ma-234	318	45	p	p	X
ma-234	318	46	.	.	PUNCT
ma-234	319	1	therefore	therefore	ADV
ma-234	319	2	,	,	PUNCT
ma-234	319	3	proposition	proposition	NOUN
ma-234	319	4	3.9	3.9	NUM
ma-234	319	5	implies	imply	VERB
ma-234	319	6	that	that	SCONJ
ma-234	319	7	lim	lim	PROPN
ma-234	319	8	y→0	y→0	PROPN
ma-234	319	9	‖f	‖f	ADP
ma-234	319	10	−	−	PROPN
ma-234	319	11	f	f	X
ma-234	319	12	(	(	PUNCT
ma-234	319	13	·	·	PUNCT
ma-234	319	14	−	−	PROPN
ma-234	319	15	y)‖f(q	y)‖f(q	PROPN
ma-234	319	16	,	,	PUNCT
ma-234	319	17	p	p	X
ma-234	319	18	,	,	PUNCT
ma-234	319	19	α	α	NOUN
ma-234	319	20	)	)	PUNCT
ma-234	319	21	=	=	SYM
ma-234	319	22	0	0	NUM
ma-234	319	23	and	and	CCONJ
ma-234	319	24	consequently	consequently	ADV
ma-234	319	25	f	f	X
ma-234	319	26	belongs	belong	VERB
ma-234	319	27	to	to	ADP
ma-234	319	28	f(q	f(q	PROPN
ma-234	319	29	,	,	PUNCT
ma-234	319	30	p	p	X
ma-234	319	31	,	,	PUNCT
ma-234	319	32	α)c	α)c	X
ma-234	319	33	.	.	PUNCT
ma-234	320	1	thus	thus	ADV
ma-234	320	2	,	,	PUNCT
ma-234	320	3	we	we	PRON
ma-234	320	4	obtain	obtain	VERB
ma-234	320	5	the	the	DET
ma-234	320	6	desired	desire	VERB
ma-234	320	7	result	result	NOUN
ma-234	320	8	.	.	PUNCT
ma-234	321	1	�	�	PROPN
ma-234	321	2	4.2	4.2	NUM
ma-234	321	3	.	.	PUNCT
ma-234	322	1	approximation	approximation	NOUN
ma-234	322	2	inmα	inmα	ADJ
ma-234	322	3	q	q	NOUN
ma-234	322	4	,	,	PUNCT
ma-234	322	5	p.	p.	NOUN
ma-234	322	6	in	in	ADP
ma-234	322	7	this	this	DET
ma-234	322	8	subsection	subsection	NOUN
ma-234	322	9	,	,	PUNCT
ma-234	322	10	we	we	PRON
ma-234	322	11	investigate	investigate	VERB
ma-234	322	12	approximation	approximation	NOUN
ma-234	322	13	of	of	ADP
ma-234	322	14	elements	element	NOUN
ma-234	322	15	of	of	ADP
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ma-234	322	17	-	-	PUNCT
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ma-234	322	23	.	.	PUNCT
ma-234	323	1	we	we	PRON
ma-234	323	2	shall	shall	AUX
ma-234	323	3	use	use	VERB
ma-234	323	4	the	the	DET
ma-234	323	5	following	follow	VERB
ma-234	323	6	result	result	NOUN
ma-234	323	7	.	.	PUNCT
ma-234	324	1	proposition	proposition	NOUN
ma-234	324	2	4.1	4.1	NUM
ma-234	324	3	.	.	PUNCT
ma-234	325	1	[	[	X
ma-234	325	2	8	8	NUM
ma-234	325	3	]	]	PUNCT
ma-234	325	4	let	let	VERB
ma-234	325	5	us	we	PRON
ma-234	325	6	assume	assume	VERB
ma-234	325	7	that	that	SCONJ
ma-234	325	8	1	1	NUM
ma-234	325	9	≤	≤	NUM
ma-234	325	10	q	q	ADJ
ma-234	325	11	≤	≤	NUM
ma-234	325	12	α	α	NOUN
ma-234	325	13	≤	≤	NOUN
ma-234	325	14	p	p	NOUN
ma-234	325	15	≤	≤	NOUN
ma-234	325	16	∞	∞	PROPN
ma-234	325	17	with	with	ADP
ma-234	325	18	α	α	PROPN
ma-234	325	19	<	<	X
ma-234	325	20	∞.	∞.	PROPN
ma-234	325	21	then	then	ADV
ma-234	325	22	there	there	PRON
ma-234	325	23	exists	exist	VERB
ma-234	325	24	c	c	NOUN
ma-234	325	25	>	>	X
ma-234	325	26	0	0	NUM
ma-234	325	27	such	such	ADJ
ma-234	325	28	that	that	PRON
ma-234	325	29	for	for	ADP
ma-234	325	30	all	all	DET
ma-234	325	31	g	g	PROPN
ma-234	325	32	∈	∈	PROPN
ma-234	325	33	l1	l1	PROPN
ma-234	325	34	and	and	CCONJ
ma-234	325	35	f	f	PROPN
ma-234	325	36	∈mα	∈mα	PROPN
ma-234	325	37	q	q	PROPN
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ma-234	325	39	p	p	X
ma-234	325	40	,	,	PUNCT
ma-234	325	41	we	we	PRON
ma-234	325	42	have	have	VERB
ma-234	325	43	‖g	‖g	PROPN
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ma-234	325	45	f	f	PROPN
ma-234	326	1	‖mα	‖mα	NUM
ma-234	326	2	q	q	NOUN
ma-234	326	3	,	,	PUNCT
ma-234	326	4	p	p	NOUN
ma-234	326	5	≤	≤	NOUN
ma-234	326	6	c	c	NOUN
ma-234	326	7	‖g‖1	‖g‖1	NOUN
ma-234	326	8	‖f	‖f	ADP
ma-234	326	9	‖mα	‖mα	NUM
ma-234	326	10	q	q	NOUN
ma-234	326	11	,	,	PUNCT
ma-234	326	12	p	p	NOUN
ma-234	326	13	.	.	PUNCT
ma-234	327	1	propositions	proposition	NOUN
ma-234	327	2	3.7	3.7	NUM
ma-234	327	3	,	,	PUNCT
ma-234	327	4	3.9	3.9	NUM
ma-234	327	5	and	and	CCONJ
ma-234	327	6	4.1	4.1	NUM
ma-234	327	7	allow	allow	VERB
ma-234	327	8	us	we	PRON
ma-234	327	9	to	to	PART
ma-234	327	10	prove	prove	VERB
ma-234	327	11	theorem	theorem	VERB
ma-234	327	12	2.4	2.4	NUM
ma-234	327	13	.	.	PUNCT
ma-234	328	1	proof	proof	NOUN
ma-234	328	2	of	of	ADP
ma-234	328	3	theorem	theorem	ADJ
ma-234	328	4	2.4	2.4	NUM
ma-234	328	5	•	•	NOUN
ma-234	328	6	(	(	PUNCT
ma-234	328	7	i)⇒	i)⇒	PROPN
ma-234	328	8	(	(	PUNCT
ma-234	328	9	i	i	PRON
ma-234	328	10	i	i	PROPN
ma-234	328	11	)	)	PUNCT
ma-234	328	12	assume	assume	VERB
ma-234	328	13	that	that	SCONJ
ma-234	328	14	f	f	PROPN
ma-234	328	15	∈mα	∈mα	NOUN
ma-234	328	16	q	q	PROPN
ma-234	328	17	,	,	PUNCT
ma-234	328	18	p	p	NOUN
ma-234	328	19	and	and	CCONJ
ma-234	328	20	n	n	PROPN
ma-234	328	21	is	be	AUX
ma-234	328	22	a	a	DET
ma-234	328	23	nonegative	nonegative	NOUN
ma-234	328	24	integer	integer	NOUN
ma-234	328	25	.	.	PUNCT
ma-234	329	1	for	for	ADP
ma-234	329	2	almost	almost	ADV
ma-234	329	3	every	every	PRON
ma-234	329	4	x	x	SYM
ma-234	329	5	∈	∈	PROPN
ma-234	329	6	rd	rd	PROPN
ma-234	329	7	,	,	PUNCT
ma-234	329	8	f	f	PROPN
ma-234	329	9	(	(	PUNCT
ma-234	329	10	x)−	x)−	PROPN
ma-234	329	11	f	f	PROPN
ma-234	329	12	∗	∗	NOUN
ma-234	329	13	φn(x	φn(x	PUNCT
ma-234	329	14	)	)	PUNCT
ma-234	329	15	=	=	SYM
ma-234	329	16	∫	∫	PROPN
ma-234	329	17	rd	rd	PROPN
ma-234	329	18	f	f	PROPN
ma-234	329	19	(	(	PUNCT
ma-234	329	20	x)φ(u)du	x)φ(u)du	PROPN
ma-234	330	1	−	−	PROPN
ma-234	331	1	∫	∫	PROPN
ma-234	332	1	rd	rd	PROPN
ma-234	333	1	f	f	PROPN
ma-234	333	2	(	(	PUNCT
ma-234	333	3	x	x	X
ma-234	333	4	−	−	PUNCT
ma-234	333	5	y)ndφ(ny)dy	y)ndφ(ny)dy	NOUN
ma-234	333	6	=	=	SYM
ma-234	333	7	∫	∫	PROPN
ma-234	333	8	rd	rd	PROPN
ma-234	333	9	f	f	PROPN
ma-234	333	10	(	(	PUNCT
ma-234	333	11	x)φ(u)du	x)φ(u)du	PROPN
ma-234	334	1	−	−	PROPN
ma-234	334	2	∫	∫	PROPN
ma-234	334	3	rd	rd	PROPN
ma-234	334	4	f	f	PROPN
ma-234	334	5	(	(	PUNCT
ma-234	334	6	x	x	X
ma-234	334	7	−	−	PUNCT
ma-234	334	8	u	u	NOUN
ma-234	334	9	n	n	NOUN
ma-234	334	10	)	)	PUNCT
ma-234	334	11	φ(u)du	φ(u)du	PROPN
ma-234	335	1	=	=	PROPN
ma-234	335	2	∫	∫	PROPN
ma-234	335	3	rd	rd	PROPN
ma-234	335	4	[	[	PUNCT
ma-234	335	5	f	f	X
ma-234	335	6	(	(	PUNCT
ma-234	335	7	x)−	x)−	PROPN
ma-234	335	8	f	f	PROPN
ma-234	335	9	(	(	PUNCT
ma-234	335	10	x	x	X
ma-234	335	11	−	−	PUNCT
ma-234	335	12	u	u	NOUN
ma-234	335	13	n	n	PROPN
ma-234	335	14	)	)	PUNCT
ma-234	335	15	]	]	PUNCT
ma-234	336	1	φ(u)du	φ(u)du	PROPN
ma-234	336	2	.	.	PROPN
ma-234	336	3	therefore	therefore	ADV
ma-234	336	4	,	,	PUNCT
ma-234	336	5	for	for	ADP
ma-234	336	6	any	any	DET
ma-234	336	7	dyadic	dyadic	ADJ
ma-234	336	8	cube	cube	NOUN
ma-234	336	9	qk	qk	PROPN
ma-234	336	10	,	,	PUNCT
ma-234	336	11	m	m	VERB
ma-234	336	12	(	(	PUNCT
ma-234	336	13	(	(	PUNCT
ma-234	336	14	k	k	X
ma-234	336	15	,	,	PUNCT
ma-234	336	16	m	m	NOUN
ma-234	336	17	)	)	PUNCT
ma-234	336	18	∈	∈	PROPN
ma-234	336	19	zd	zd	PROPN
ma-234	336	20	×	×	PROPN
ma-234	336	21	z	z	PROPN
ma-234	336	22	)	)	PUNCT
ma-234	336	23	,	,	PUNCT
ma-234	336	24	the	the	DET
ma-234	336	25	minkowski	minkowski	PROPN
ma-234	336	26	inequality	inequality	NOUN
ma-234	336	27	implies	imply	VERB
ma-234	336	28	that,∥∥(f	that,∥∥(f	NOUN
ma-234	336	29	−	−	PROPN
ma-234	336	30	f	f	PROPN
ma-234	336	31	∗	∗	NOUN
ma-234	336	32	φn)χqk	φn)χqk	NOUN
ma-234	336	33	,	,	PUNCT
ma-234	336	34	m	m	VERB
ma-234	336	35	∥∥	∥∥	X
ma-234	336	36	q	q	PROPN
ma-234	336	37	≤	≤	NUM
ma-234	336	38	∫	∫	PROPN
ma-234	336	39	rd	rd	PROPN
ma-234	337	1	∥∥∥[f	∥∥∥[f	NOUN
ma-234	337	2	−	−	PROPN
ma-234	337	3	f	f	X
ma-234	337	4	(	(	PUNCT
ma-234	337	5	·	·	PUNCT
ma-234	337	6	−	−	PUNCT
ma-234	337	7	u	u	PROPN
ma-234	337	8	n	n	PROPN
ma-234	337	9	)	)	PUNCT
ma-234	337	10	]	]	PUNCT
ma-234	338	1	χqk	χqk	NOUN
ma-234	338	2	,	,	PUNCT
ma-234	338	3	m	m	PROPN
ma-234	338	4	∥∥∥	∥∥∥	PROPN
ma-234	338	5	q	q	X
ma-234	338	6	φ(u)du	φ(u)du	PROPN
ma-234	338	7	and	and	CCONJ
ma-234	338	8	so	so	ADV
ma-234	338	9	‖f	‖f	ADP
ma-234	338	10	−	−	PROPN
ma-234	338	11	f	f	PROPN
ma-234	338	12	∗	∗	X
ma-234	338	13	φn‖mα	φn‖mα	PUNCT
ma-234	339	1	q	q	NOUN
ma-234	339	2	,	,	PUNCT
ma-234	339	3	p	p	PROPN
ma-234	339	4	≤	≤	NUM
ma-234	339	5	∫	∫	PROPN
ma-234	339	6	rd	rd	PROPN
ma-234	339	7	∥∥∥f	∥∥∥f	PROPN
ma-234	339	8	−	−	PROPN
ma-234	339	9	f	f	PROPN
ma-234	339	10	(	(	PUNCT
ma-234	339	11	·	·	PUNCT
ma-234	339	12	−	−	PUNCT
ma-234	339	13	u	u	NOUN
ma-234	339	14	n	n	NOUN
ma-234	339	15	)	)	PUNCT
ma-234	339	16	∥∥∥	∥∥∥	PROPN
ma-234	339	17	mα	mα	PROPN
ma-234	340	1	q	q	NOUN
ma-234	340	2	,	,	PUNCT
ma-234	340	3	p	p	PROPN
ma-234	340	4	φ(u)du	φ(u)du	PROPN
ma-234	340	5	.	.	PROPN
ma-234	341	1	according	accord	VERB
ma-234	341	2	to	to	ADP
ma-234	341	3	proposition	proposition	NOUN
ma-234	341	4	3.9	3.9	NUM
ma-234	341	5	,	,	PUNCT
ma-234	341	6	we	we	PRON
ma-234	341	7	have	have	VERB
ma-234	341	8	lim	lim	PROPN
ma-234	341	9	n→∞	n→∞	NUM
ma-234	341	10	∥∥∥f	∥∥∥f	PROPN
ma-234	341	11	−	−	PROPN
ma-234	341	12	f	f	PROPN
ma-234	341	13	(	(	PUNCT
ma-234	341	14	·	·	PUNCT
ma-234	341	15	−	−	PUNCT
ma-234	341	16	u	u	NOUN
ma-234	341	17	n	n	NOUN
ma-234	341	18	)	)	PUNCT
ma-234	341	19	∥∥∥	∥∥∥	PROPN
ma-234	341	20	mα	mα	PROPN
ma-234	342	1	q	q	NOUN
ma-234	342	2	,	,	PUNCT
ma-234	342	3	p	p	NOUN
ma-234	342	4	φ(u	φ(u	NOUN
ma-234	342	5	)	)	PUNCT
ma-234	342	6	=	=	SYM
ma-234	342	7	0	0	NUM
ma-234	342	8	,	,	PUNCT
ma-234	342	9	u	u	PROPN
ma-234	342	10	∈	∈	PROPN
ma-234	342	11	rd	rd	PROPN
ma-234	342	12	.	.	PUNCT
ma-234	343	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	343	2	eur	eur	PROPN
ma-234	343	3	.	.	PUNCT
ma-234	344	1	j.	j.	PROPN
ma-234	344	2	math	math	PROPN
ma-234	344	3	.	.	PUNCT
ma-234	345	1	anal	anal	PROPN
ma-234	345	2	.	.	PUNCT
ma-234	346	1	10.28924	10.28924	NUM
ma-234	346	2	/	/	SYM
ma-234	346	3	ada	ada	PROPN
ma-234	346	4	/	/	SYM
ma-234	346	5	ma.4.16	ma.4.16	PROPN
ma-234	346	6	13furthermore	13furthermore	NUM
ma-234	346	7	,	,	PUNCT
ma-234	346	8	by	by	ADP
ma-234	346	9	minkowski	minkowski	PROPN
ma-234	346	10	’s	’s	PART
ma-234	346	11	inequality	inequality	NOUN
ma-234	346	12	and	and	CCONJ
ma-234	346	13	point	point	NOUN
ma-234	346	14	1	1	NUM
ma-234	346	15	)	)	PUNCT
ma-234	346	16	of	of	ADP
ma-234	346	17	proposition	proposition	NOUN
ma-234	346	18	3.7	3.7	NUM
ma-234	346	19	,	,	PUNCT
ma-234	347	1	we	we	PRON
ma-234	347	2	have∥∥∥f	have∥∥∥f	PROPN
ma-234	347	3	−	−	PROPN
ma-234	347	4	f	f	PROPN
ma-234	347	5	(	(	PUNCT
ma-234	347	6	·	·	PUNCT
ma-234	347	7	−	−	PUNCT
ma-234	347	8	u	u	NOUN
ma-234	347	9	n	n	NOUN
ma-234	347	10	)	)	PUNCT
ma-234	347	11	∥∥∥	∥∥∥	PROPN
ma-234	347	12	mα	mα	PROPN
ma-234	348	1	q	q	NOUN
ma-234	348	2	,	,	PUNCT
ma-234	348	3	p	p	NOUN
ma-234	348	4	φ(u	φ(u	NOUN
ma-234	348	5	)	)	PUNCT
ma-234	348	6	≤	≤	NOUN
ma-234	348	7	(	(	PUNCT
ma-234	348	8	1	1	NUM
ma-234	348	9	+	+	NUM
ma-234	348	10	c1	c1	NOUN
ma-234	348	11	)	)	PUNCT
ma-234	348	12	‖f	‖f	PUNCT
ma-234	349	1	‖mα	‖mα	NUM
ma-234	349	2	q	q	NOUN
ma-234	349	3	,	,	PUNCT
ma-234	349	4	p	p	NOUN
ma-234	349	5	φ(u	φ(u	NOUN
ma-234	349	6	)	)	PUNCT
ma-234	349	7	,	,	PUNCT
ma-234	349	8	u	u	PROPN
ma-234	349	9	∈	∈	PROPN
ma-234	349	10	rd	rd	PROPN
ma-234	349	11	,	,	PUNCT
ma-234	349	12	n	n	X
ma-234	349	13	≥	≥	NUM
ma-234	349	14	1	1	NUM
ma-234	349	15	.	.	PUNCT
ma-234	350	1	thus	thus	ADV
ma-234	350	2	,	,	PUNCT
ma-234	350	3	an	an	DET
ma-234	350	4	application	application	NOUN
ma-234	350	5	of	of	ADP
ma-234	350	6	the	the	DET
ma-234	350	7	dominated	dominate	VERB
ma-234	350	8	convergence	convergence	NOUN
ma-234	350	9	theorem	theorem	NOUN
ma-234	350	10	gives	give	VERB
ma-234	350	11	lim	lim	PROPN
ma-234	350	12	n→∞	n→∞	NUM
ma-234	350	13	‖f	‖f	PUNCT
ma-234	350	14	−	−	PROPN
ma-234	350	15	f	f	PROPN
ma-234	350	16	∗	∗	X
ma-234	350	17	φn‖mα	φn‖mα	PUNCT
ma-234	350	18	q	q	NOUN
ma-234	350	19	,	,	PUNCT
ma-234	350	20	p	p	X
ma-234	350	21	=	=	NOUN
ma-234	350	22	0	0	NUM
ma-234	350	23	.	.	NOUN
ma-234	351	1	•	•	NUM
ma-234	351	2	(	(	PUNCT
ma-234	352	1	i	i	PRON
ma-234	352	2	i)⇒	i)⇒	PROPN
ma-234	352	3	(	(	PUNCT
ma-234	352	4	i	i	PRON
ma-234	352	5	i	i	PROPN
ma-234	352	6	i	i	VERB
ma-234	352	7	)	)	PUNCT
ma-234	352	8	assume	assume	VERB
ma-234	352	9	that	that	SCONJ
ma-234	352	10	the	the	DET
ma-234	352	11	assertion	assertion	NOUN
ma-234	352	12	(	(	PUNCT
ma-234	352	13	i	i	PRON
ma-234	352	14	i	i	PROPN
ma-234	352	15	)	)	PUNCT
ma-234	352	16	holds.let	holds.let	X
ma-234	352	17	us	we	PRON
ma-234	352	18	fix	fix	VERB
ma-234	352	19	an	an	DET
ma-234	352	20	integer	integer	NOUN
ma-234	352	21	n	n	PRON
ma-234	352	22	≥	≥	NOUN
ma-234	352	23	1	1	NUM
ma-234	352	24	and	and	CCONJ
ma-234	352	25	β	β	X
ma-234	352	26	∈	∈	PROPN
ma-234	352	27	nd	nd	INTJ
ma-234	352	28	.	.	PUNCT
ma-234	353	1	since	since	SCONJ
ma-234	353	2	φn	φn	ADP
ma-234	353	3	∈	∈	PROPN
ma-234	353	4	c∞c	c∞c	VERB
ma-234	353	5	,	,	PUNCT
ma-234	353	6	f	f	PROPN
ma-234	353	7	∗φn	∗φn	PROPN
ma-234	353	8	belongs	belong	VERB
ma-234	353	9	to	to	ADP
ma-234	353	10	c∞	c∞	PROPN
ma-234	353	11	and	and	CCONJ
ma-234	353	12	by	by	ADP
ma-234	353	13	proposition4.1	proposition4.1	NOUN
ma-234	353	14	,	,	PUNCT
ma-234	353	15	f	f	PROPN
ma-234	353	16	∗	∗	NOUN
ma-234	353	17	φn	φn	VERB
ma-234	353	18	is	be	AUX
ma-234	353	19	in	in	ADP
ma-234	353	20	mα	mα	PROPN
ma-234	353	21	q	q	NOUN
ma-234	353	22	,	,	PUNCT
ma-234	353	23	p	p	X
ma-234	353	24	.	.	PUNCT
ma-234	354	1	furthermore	furthermore	ADV
ma-234	354	2	,	,	PUNCT
ma-234	354	3	it	it	PRON
ma-234	354	4	is	be	AUX
ma-234	354	5	well	well	ADV
ma-234	354	6	known	know	VERB
ma-234	354	7	that	that	SCONJ
ma-234	354	8	∂β	∂β	PROPN
ma-234	354	9	(	(	PUNCT
ma-234	354	10	f	f	PROPN
ma-234	354	11	∗	∗	X
ma-234	354	12	φn	φn	NOUN
ma-234	354	13	)	)	PUNCT
ma-234	354	14	=	=	SYM
ma-234	354	15	f	f	PROPN
ma-234	354	16	∗	∗	NOUN
ma-234	354	17	∂βφn	∂βφn	X
ma-234	354	18	and	and	CCONJ
ma-234	354	19	by	by	ADP
ma-234	354	20	notingthat	notingthat	PROPN
ma-234	354	21	∂βφn	∂βφn	PUNCT
ma-234	354	22	∈	∈	PROPN
ma-234	354	23	l1	l1	PROPN
ma-234	354	24	,	,	PUNCT
ma-234	354	25	proposition	proposition	NOUN
ma-234	354	26	4.1	4.1	NUM
ma-234	354	27	implies	imply	VERB
ma-234	354	28	that	that	SCONJ
ma-234	354	29	∂β	∂β	PROPN
ma-234	354	30	(	(	PUNCT
ma-234	354	31	f	f	PROPN
ma-234	354	32	∗	∗	PROPN
ma-234	354	33	φn	φn	NOUN
ma-234	354	34	)	)	PUNCT
ma-234	354	35	belongs	belong	VERB
ma-234	354	36	to	to	ADP
ma-234	354	37	mα	mα	PROPN
ma-234	354	38	q	q	NOUN
ma-234	354	39	,	,	PUNCT
ma-234	354	40	p	p	NOUN
ma-234	354	41	.	.	PUNCT
ma-234	355	1	thus	thus	ADV
ma-234	355	2	f	f	PROPN
ma-234	355	3	∗	∗	NOUN
ma-234	355	4	φn	φn	PROPN
ma-234	355	5	belongs	belong	VERB
ma-234	355	6	to	to	ADP
ma-234	355	7	c∞mα	c∞mα	NOUN
ma-234	355	8	q	q	NOUN
ma-234	355	9	,	,	PUNCT
ma-234	355	10	p	p	NOUN
ma-234	355	11	and	and	CCONJ
ma-234	355	12	since	since	ADV
ma-234	355	13	,	,	PUNCT
ma-234	355	14	by	by	ADP
ma-234	355	15	hypothesis	hypothesis	NOUN
ma-234	355	16	,	,	PUNCT
ma-234	355	17	lim	lim	PROPN
ma-234	355	18	n→∞	n→∞	NUM
ma-234	355	19	‖f	‖f	PUNCT
ma-234	355	20	−	−	PROPN
ma-234	355	21	f	f	PROPN
ma-234	355	22	∗	∗	X
ma-234	355	23	φn‖mα	φn‖mα	PUNCT
ma-234	355	24	q	q	NOUN
ma-234	355	25	,	,	PUNCT
ma-234	355	26	p	p	X
ma-234	355	27	=	=	NOUN
ma-234	355	28	0	0	NUM
ma-234	355	29	,	,	PUNCT
ma-234	355	30	we	we	PRON
ma-234	355	31	can	can	AUX
ma-234	355	32	conclude	conclude	VERB
ma-234	355	33	that	that	SCONJ
ma-234	355	34	f	f	PROPN
ma-234	355	35	belongs	belong	VERB
ma-234	355	36	to	to	ADP
ma-234	355	37	the	the	DET
ma-234	355	38	closure	closure	NOUN
ma-234	355	39	in	in	ADP
ma-234	355	40	mα	mα	PROPN
ma-234	355	41	q	q	NOUN
ma-234	355	42	,	,	PUNCT
ma-234	355	43	p	p	NOUN
ma-234	355	44	of	of	ADP
ma-234	355	45	c∞mα	c∞mα	NOUN
ma-234	355	46	q	q	NOUN
ma-234	355	47	,	,	PUNCT
ma-234	355	48	p	p	NOUN
ma-234	355	49	.	.	PUNCT
ma-234	356	1	•	•	NUM
ma-234	356	2	(	(	PUNCT
ma-234	356	3	i	i	PRON
ma-234	356	4	i	i	PRON
ma-234	356	5	i)⇒	i)⇒	PROPN
ma-234	356	6	(	(	PUNCT
ma-234	356	7	i	i	NOUN
ma-234	356	8	)	)	PUNCT
ma-234	356	9	since	since	SCONJ
ma-234	356	10	c∞mα	c∞mα	NOUN
ma-234	356	11	q	q	X
ma-234	356	12	,	,	PUNCT
ma-234	356	13	p	p	PRON
ma-234	356	14	is	be	AUX
ma-234	356	15	a	a	DET
ma-234	356	16	subset	subset	NOUN
ma-234	356	17	of	of	ADP
ma-234	356	18	mα	mα	PROPN
ma-234	356	19	q	q	NOUN
ma-234	356	20	,	,	PUNCT
ma-234	356	21	p	p	X
ma-234	356	22	,	,	PUNCT
ma-234	356	23	it	it	PRON
ma-234	356	24	is	be	AUX
ma-234	356	25	obvious	obvious	ADJ
ma-234	356	26	that	that	SCONJ
ma-234	356	27	its	its	PRON
ma-234	356	28	closure	closure	NOUN
ma-234	356	29	in	in	ADP
ma-234	356	30	mα	mα	PROPN
ma-234	356	31	q	q	NOUN
ma-234	356	32	,	,	PUNCT
ma-234	356	33	p	p	NOUN
ma-234	356	34	is	be	AUX
ma-234	356	35	included	include	VERB
ma-234	356	36	in	in	ADP
ma-234	356	37	mα	mα	PROPN
ma-234	356	38	q	q	NOUN
ma-234	356	39	,	,	PUNCT
ma-234	356	40	p	p	NOUN
ma-234	356	41	and	and	CCONJ
ma-234	356	42	therefore	therefore	ADV
ma-234	356	43	the	the	DET
ma-234	356	44	claim	claim	NOUN
ma-234	356	45	follows	follow	VERB
ma-234	356	46	.	.	PUNCT
ma-234	357	1	the	the	DET
ma-234	357	2	proof	proof	NOUN
ma-234	357	3	is	be	AUX
ma-234	357	4	complete	complete	ADJ
ma-234	357	5	.	.	PUNCT
ma-234	358	1	�	�	PROPN
ma-234	358	2	we	we	PRON
ma-234	358	3	recall	recall	VERB
ma-234	358	4	the	the	DET
ma-234	358	5	following	follow	VERB
ma-234	358	6	well	well	ADV
ma-234	358	7	known	know	VERB
ma-234	358	8	result	result	NOUN
ma-234	358	9	in	in	ADP
ma-234	358	10	lebesgue	lebesgue	ADJ
ma-234	358	11	spaces	space	NOUN
ma-234	358	12	.	.	PUNCT
ma-234	359	1	lemma	lemma	PROPN
ma-234	359	2	4.2	4.2	NUM
ma-234	359	3	.	.	PUNCT
ma-234	360	1	[	[	X
ma-234	360	2	1	1	X
ma-234	360	3	]	]	X
ma-234	360	4	if	if	SCONJ
ma-234	360	5	1	1	NUM
ma-234	360	6	≤	≤	NUM
ma-234	360	7	α	α	NOUN
ma-234	360	8	<	<	X
ma-234	360	9	∞	∞	PROPN
ma-234	360	10	and	and	CCONJ
ma-234	360	11	f	f	PROPN
ma-234	360	12	is	be	AUX
ma-234	360	13	in	in	ADP
ma-234	360	14	lα	lα	NOUN
ma-234	360	15	then	then	ADV
ma-234	360	16	we	we	PRON
ma-234	360	17	have	have	VERB
ma-234	360	18	lim	lim	PROPN
ma-234	360	19	n→∞	n→∞	NUM
ma-234	360	20	‖f	‖f	PRON
ma-234	360	21	χen‖α	χen‖α	NUM
ma-234	360	22	=	=	SYM
ma-234	360	23	0	0	NUM
ma-234	360	24	,	,	PUNCT
ma-234	360	25	where	where	SCONJ
ma-234	360	26	(	(	PUNCT
ma-234	360	27	en)n≥1	en)n≥1	VERB
ma-234	360	28	is	be	AUX
ma-234	360	29	a	a	DET
ma-234	360	30	nonincreasing	nonincrease	VERB
ma-234	360	31	sequence	sequence	NOUN
ma-234	360	32	of	of	ADP
ma-234	360	33	measurable	measurable	ADJ
ma-234	360	34	subsets	subset	NOUN
ma-234	360	35	of	of	ADP
ma-234	360	36	rd	rd	NOUN
ma-234	360	37	satisfying	satisfy	VERB
ma-234	360	38	∣∣∣∣∣∣⋂n≥1	∣∣∣∣∣∣⋂n≥1	NOUN
ma-234	360	39	en	en	X
ma-234	360	40	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ma-234	360	41	=	=	SYM
ma-234	360	42	0	0	PROPN
ma-234	360	43	.	.	PUNCT
ma-234	361	1	the	the	DET
ma-234	361	2	next	next	ADJ
ma-234	361	3	proposition	proposition	NOUN
ma-234	361	4	shows	show	VERB
ma-234	361	5	that	that	SCONJ
ma-234	361	6	an	an	DET
ma-234	361	7	analogous	analogous	ADJ
ma-234	361	8	result	result	NOUN
ma-234	361	9	holds	hold	VERB
ma-234	361	10	for	for	ADP
ma-234	361	11	bourgain	bourgain	NOUN
ma-234	361	12	-	-	PUNCT
ma-234	361	13	morrey	morrey	NOUN
ma-234	361	14	spaces	space	NOUN
ma-234	361	15	.	.	PUNCT
ma-234	362	1	proposition	proposition	NOUN
ma-234	362	2	4.3	4.3	NUM
ma-234	362	3	.	.	PUNCT
ma-234	363	1	let	let	VERB
ma-234	363	2	1	1	NUM
ma-234	363	3	≤	≤	NOUN
ma-234	363	4	q	q	ADJ
ma-234	363	5	≤	≤	NUM
ma-234	363	6	α	α	NOUN
ma-234	363	7	≤	≤	NOUN
ma-234	364	1	p	p	X
ma-234	364	2	<	<	X
ma-234	364	3	∞	∞	PROPN
ma-234	364	4	,	,	PUNCT
ma-234	364	5	f	f	PROPN
ma-234	364	6	be	be	VERB
ma-234	364	7	any	any	DET
ma-234	364	8	element	element	NOUN
ma-234	364	9	of	of	ADP
ma-234	364	10	mα	mα	PROPN
ma-234	364	11	q	q	NOUN
ma-234	364	12	,	,	PUNCT
ma-234	364	13	p	p	NOUN
ma-234	364	14	and	and	CCONJ
ma-234	364	15	(	(	PUNCT
ma-234	364	16	en)n≥1	en)n≥1	AUX
ma-234	364	17	be	be	AUX
ma-234	364	18	a	a	DET
ma-234	364	19	nonincreasing	nonincrease	VERB
ma-234	364	20	sequence	sequence	NOUN
ma-234	364	21	of	of	ADP
ma-234	364	22	measurable	measurable	ADJ
ma-234	364	23	subsets	subset	NOUN
ma-234	364	24	of	of	ADP
ma-234	364	25	rd	rd	NOUN
ma-234	364	26	satisfying	satisfy	VERB
ma-234	364	27	∣∣∣∣∣∣⋂n≥1	∣∣∣∣∣∣⋂n≥1	NOUN
ma-234	364	28	en	en	X
ma-234	364	29	∣∣∣∣∣∣	∣∣∣∣∣∣	PROPN
ma-234	364	30	=	=	SYM
ma-234	364	31	0	0	PROPN
ma-234	364	32	.	.	PUNCT
ma-234	365	1	then	then	ADV
ma-234	365	2	lim	lim	PROPN
ma-234	365	3	n→∞	n→∞	NUM
ma-234	365	4	‖f	‖f	PUNCT
ma-234	365	5	χen‖mα	χen‖mα	PROPN
ma-234	365	6	q	q	NOUN
ma-234	365	7	,	,	PUNCT
ma-234	365	8	p	p	NOUN
ma-234	365	9	=	=	NOUN
ma-234	365	10	0	0	NUM
ma-234	365	11	.	.	PUNCT
ma-234	366	1	proof	proof	NOUN
ma-234	366	2	.	.	PUNCT
ma-234	367	1	if	if	SCONJ
ma-234	367	2	q	q	PRON
ma-234	367	3	=	=	SYM
ma-234	367	4	α	α	NOUN
ma-234	367	5	or	or	CCONJ
ma-234	367	6	α	α	NOUN
ma-234	367	7	=	=	PUNCT
ma-234	368	1	p	p	NOUN
ma-234	368	2	then	then	ADV
ma-234	368	3	mα	mα	PROPN
ma-234	368	4	q	q	NOUN
ma-234	368	5	,	,	PUNCT
ma-234	368	6	p	p	NOUN
ma-234	368	7	=	=	X
ma-234	368	8	{	{	PUNCT
ma-234	368	9	0	0	NUM
ma-234	368	10	}	}	PUNCT
ma-234	368	11	and	and	CCONJ
ma-234	368	12	therefore	therefore	ADV
ma-234	368	13	we	we	PRON
ma-234	368	14	have	have	VERB
ma-234	368	15	nothing	nothing	PRON
ma-234	368	16	to	to	PART
ma-234	368	17	prove	prove	VERB
ma-234	368	18	.	.	PUNCT
ma-234	369	1	hence	hence	ADV
ma-234	369	2	wesuppose	wesuppose	VERB
ma-234	369	3	that	that	SCONJ
ma-234	369	4	1	1	NUM
ma-234	369	5	≤	≤	NOUN
ma-234	369	6	q	q	NOUN
ma-234	369	7	<	<	X
ma-234	369	8	α	α	X
ma-234	369	9	<	<	X
ma-234	369	10	p	p	X
ma-234	369	11	<	<	X
ma-234	369	12	∞.	∞.	PROPN
ma-234	369	13	by	by	ADP
ma-234	369	14	point	point	NOUN
ma-234	369	15	2	2	NUM
ma-234	369	16	)	)	PUNCT
ma-234	369	17	of	of	ADP
ma-234	369	18	proposition	proposition	NOUN
ma-234	369	19	3.7	3.7	NUM
ma-234	369	20	,	,	PUNCT
ma-234	369	21	there	there	PRON
ma-234	369	22	exists	exist	VERB
ma-234	369	23	a	a	DET
ma-234	369	24	sequence	sequence	NOUN
ma-234	369	25	(	(	PUNCT
ma-234	369	26	fn)n≥1of	fn)n≥1of	NOUN
ma-234	369	27	elements	element	NOUN
ma-234	369	28	of	of	ADP
ma-234	369	29	l∞c	l∞c	NOUN
ma-234	370	1	such	such	ADJ
ma-234	370	2	that	that	SCONJ
ma-234	370	3	lim	lim	PROPN
ma-234	370	4	n→∞	n→∞	PRON
ma-234	370	5	‖fn	‖fn	PROPN
ma-234	370	6	−	−	PROPN
ma-234	370	7	f	f	PROPN
ma-234	370	8	‖mα	‖mα	NUM
ma-234	370	9	q	q	NOUN
ma-234	370	10	,	,	PUNCT
ma-234	370	11	p	p	X
ma-234	370	12	=	=	X
ma-234	370	13	0.let	0.let	NOUN
ma-234	370	14	ε	ε	PROPN
ma-234	370	15	>	>	X
ma-234	370	16	0	0	PUNCT
ma-234	370	17	be	be	AUX
ma-234	370	18	a	a	DET
ma-234	370	19	fixed	fixed	ADJ
ma-234	370	20	real	real	ADJ
ma-234	370	21	number	number	NOUN
ma-234	370	22	.	.	PUNCT
ma-234	371	1	from	from	ADP
ma-234	371	2	what	what	PRON
ma-234	371	3	precedes	precede	VERB
ma-234	371	4	,	,	PUNCT
ma-234	371	5	there	there	PRON
ma-234	371	6	exists	exist	VERB
ma-234	371	7	an	an	DET
ma-234	371	8	integer	integer	NOUN
ma-234	371	9	nε	nε	NOUN
ma-234	371	10	such	such	ADJ
ma-234	371	11	that	that	DET
ma-234	371	12	‖fnε	‖fnε	NOUN
ma-234	371	13	−	−	X
ma-234	372	1	f	f	PROPN
ma-234	372	2	‖mα	‖mα	NUM
ma-234	372	3	q	q	NOUN
ma-234	372	4	,	,	PUNCT
ma-234	372	5	p	p	X
ma-234	372	6	<	<	X
ma-234	372	7	ε	ε	PROPN
ma-234	372	8	2	2	NUM
ma-234	372	9	.	.	PUNCT
ma-234	373	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	373	2	eur	eur	PROPN
ma-234	373	3	.	.	PUNCT
ma-234	374	1	j.	j.	PROPN
ma-234	374	2	math	math	PROPN
ma-234	374	3	.	.	PUNCT
ma-234	375	1	anal	anal	PROPN
ma-234	375	2	.	.	PUNCT
ma-234	376	1	10.28924	10.28924	NUM
ma-234	376	2	/	/	SYM
ma-234	376	3	ada	ada	PROPN
ma-234	376	4	/	/	SYM
ma-234	376	5	ma.4.16	ma.4.16	PROPN
ma-234	376	6	14this	14this	NUM
ma-234	376	7	and	and	CCONJ
ma-234	376	8	point	point	NOUN
ma-234	376	9	3	3	NUM
ma-234	376	10	)	)	PUNCT
ma-234	376	11	of	of	ADP
ma-234	376	12	proposition	proposition	NOUN
ma-234	376	13	3.7	3.7	NUM
ma-234	376	14	imply	imply	VERB
ma-234	376	15	that	that	SCONJ
ma-234	376	16	,	,	PUNCT
ma-234	376	17	for	for	ADP
ma-234	376	18	any	any	DET
ma-234	376	19	n	n	PRON
ma-234	376	20	≥	≥	NOUN
ma-234	376	21	1	1	NUM
ma-234	376	22	,	,	PUNCT
ma-234	376	23	‖f	‖f	ADP
ma-234	376	24	χen‖mα	χen‖mα	PROPN
ma-234	376	25	q	q	NOUN
ma-234	376	26	,	,	PUNCT
ma-234	376	27	p	p	NOUN
ma-234	376	28	≤	≤	NUM
ma-234	376	29	‖(f	‖(f	NOUN
ma-234	376	30	−	−	ADP
ma-234	376	31	fnε)χen‖mα	fnε)χen‖mα	PROPN
ma-234	376	32	q	q	NOUN
ma-234	376	33	,	,	PUNCT
ma-234	376	34	p	p	X
ma-234	377	1	+	+	NOUN
ma-234	377	2	‖fnεχen‖mα	‖fnεχen‖mα	PRON
ma-234	377	3	q	q	NOUN
ma-234	377	4	,	,	PUNCT
ma-234	377	5	p	p	NOUN
ma-234	377	6	≤	≤	NOUN
ma-234	377	7	‖f	‖f	PRON
ma-234	377	8	−	−	PROPN
ma-234	377	9	fnε‖mα	fnε‖mα	PROPN
ma-234	377	10	q	q	NOUN
ma-234	377	11	,	,	PUNCT
ma-234	377	12	p	p	X
ma-234	377	13	+	+	NOUN
ma-234	377	14	‖fnεχen‖mα	‖fnεχen‖mα	PRON
ma-234	377	15	q	q	NOUN
ma-234	377	16	,	,	PUNCT
ma-234	377	17	p	p	NOUN
ma-234	377	18	≤	≤	NUM
ma-234	377	19	ε	ε	PROPN
ma-234	377	20	2	2	NUM
ma-234	377	21	+	+	CCONJ
ma-234	377	22	c2	c2	PROPN
ma-234	377	23	‖fnεχen‖α	‖fnεχen‖α	NOUN
ma-234	377	24	.since	.since	NOUN
ma-234	377	25	fnε	fnε	PROPN
ma-234	377	26	∈	∈	PROPN
ma-234	377	27	lα	lα	PROPN
ma-234	377	28	,	,	PUNCT
ma-234	377	29	lemma	lemma	PROPN
ma-234	377	30	4.2	4.2	NUM
ma-234	377	31	implies	imply	VERB
ma-234	377	32	that	that	SCONJ
ma-234	377	33	there	there	PRON
ma-234	377	34	exists	exist	VERB
ma-234	377	35	an	an	DET
ma-234	377	36	integer	integer	PROPN
ma-234	377	37	n0	n0	X
ma-234	377	38	≥	≥	NUM
ma-234	377	39	1	1	NUM
ma-234	377	40	such	such	ADJ
ma-234	377	41	that	that	SCONJ
ma-234	378	1	n	n	NUM
ma-234	378	2	≥	≥	NOUN
ma-234	378	3	n0	n0	NOUN
ma-234	379	1	=	=	NOUN
ma-234	379	2	⇒	⇒	NOUN
ma-234	379	3	‖fnεχen‖α	‖fnεχen‖α	VERB
ma-234	379	4	<	<	X
ma-234	379	5	ε	ε	PROPN
ma-234	379	6	2c2	2c2	NUM
ma-234	379	7	.	.	PUNCT
ma-234	380	1	therefore	therefore	ADV
ma-234	380	2	n	n	NUM
ma-234	380	3	≥	≥	NOUN
ma-234	380	4	n0	n0	NUM
ma-234	380	5	=	=	NOUN
ma-234	380	6	⇒	⇒	NOUN
ma-234	380	7	‖f	‖f	PUNCT
ma-234	380	8	χen‖mα	χen‖mα	PROPN
ma-234	380	9	q	q	NOUN
ma-234	380	10	,	,	PUNCT
ma-234	380	11	p	p	X
ma-234	380	12	<	<	X
ma-234	380	13	ε.this	ε.this	PRON
ma-234	380	14	provides	provide	VERB
ma-234	380	15	the	the	DET
ma-234	380	16	desired	desire	VERB
ma-234	380	17	result	result	NOUN
ma-234	380	18	.	.	PUNCT
ma-234	381	1	�	�	PROPN
ma-234	382	1	[	[	X
ma-234	382	2	1	1	NUM
ma-234	382	3	,	,	PUNCT
ma-234	382	4	proposition	proposition	NOUN
ma-234	382	5	3.6	3.6	NUM
ma-234	382	6	]	]	PUNCT
ma-234	382	7	asserts	assert	VERB
ma-234	382	8	that	that	SCONJ
ma-234	382	9	proposition	proposition	NOUN
ma-234	382	10	4.3	4.3	NUM
ma-234	382	11	is	be	AUX
ma-234	382	12	equivalent	equivalent	ADJ
ma-234	382	13	to	to	ADP
ma-234	382	14	the	the	DET
ma-234	382	15	following	following	NOUN
ma-234	382	16	dominated	dominate	VERB
ma-234	382	17	con	con	PROPN
ma-234	382	18	-	-	PUNCT
ma-234	382	19	vergence	vergence	NOUN
ma-234	382	20	theorem	theorem	VERB
ma-234	382	21	.	.	PUNCT
ma-234	383	1	proposition	proposition	NOUN
ma-234	383	2	4.4	4.4	NUM
ma-234	383	3	.	.	PUNCT
ma-234	384	1	let	let	VERB
ma-234	384	2	1	1	NUM
ma-234	384	3	≤	≤	NOUN
ma-234	384	4	q	q	ADJ
ma-234	384	5	≤	≤	NUM
ma-234	384	6	α	α	NOUN
ma-234	384	7	≤	≤	NOUN
ma-234	385	1	p	p	DET
ma-234	385	2	<	<	X
ma-234	385	3	∞	∞	PROPN
ma-234	385	4	and	and	CCONJ
ma-234	385	5	f	f	PROPN
ma-234	385	6	be	be	AUX
ma-234	385	7	any	any	DET
ma-234	385	8	element	element	NOUN
ma-234	385	9	ofmα	ofmα	ADJ
ma-234	385	10	q	q	NOUN
ma-234	385	11	,	,	PUNCT
ma-234	385	12	p	p	X
ma-234	385	13	.	.	PUNCT
ma-234	386	1	if	if	SCONJ
ma-234	386	2	(	(	PUNCT
ma-234	386	3	fn)n≥1	fn)n≥1	NOUN
ma-234	386	4	is	be	AUX
ma-234	386	5	a	a	DET
ma-234	386	6	sequence	sequence	NOUN
ma-234	386	7	of	of	ADP
ma-234	386	8	measurable	measurable	ADJ
ma-234	386	9	functions	function	NOUN
ma-234	386	10	satisfying	satisfy	VERB
ma-234	386	11	|fn|	|fn|	PROPN
ma-234	386	12	≤	≤	NUM
ma-234	386	13	|f	|f	PUNCT
ma-234	387	1	|	|	ADV
ma-234	387	2	for	for	ADP
ma-234	387	3	all	all	DET
ma-234	387	4	n	n	PRON
ma-234	387	5	≥	≥	NOUN
ma-234	387	6	1	1	NUM
ma-234	387	7	and	and	CCONJ
ma-234	387	8	lim	lim	PROPN
ma-234	387	9	n→∞	n→∞	PRON
ma-234	388	1	fn	fn	NOUN
ma-234	388	2	=	=	NOUN
ma-234	388	3	g	g	NOUN
ma-234	388	4	almost	almost	ADV
ma-234	388	5	everywhere	everywhere	ADV
ma-234	388	6	,	,	PUNCT
ma-234	388	7	for	for	ADP
ma-234	388	8	some	some	DET
ma-234	388	9	measurable	measurable	ADJ
ma-234	388	10	function	function	NOUN
ma-234	388	11	g	g	NOUN
ma-234	388	12	,	,	PUNCT
ma-234	388	13	then	then	ADV
ma-234	388	14	lim	lim	PROPN
ma-234	388	15	n→∞	n→∞	X
ma-234	389	1	‖fn	‖fn	PROPN
ma-234	389	2	−	−	PROPN
ma-234	389	3	g‖mα	g‖mα	NOUN
ma-234	389	4	q	q	NOUN
ma-234	389	5	,	,	PUNCT
ma-234	389	6	p	p	NOUN
ma-234	389	7	=	=	NOUN
ma-234	389	8	0	0	X
ma-234	389	9	.	.	PUNCT
ma-234	390	1	proposition	proposition	NOUN
ma-234	390	2	4.4	4.4	NUM
ma-234	390	3	yields	yield	NOUN
ma-234	390	4	obviously	obviously	ADV
ma-234	390	5	what	what	PRON
ma-234	390	6	follows	follow	VERB
ma-234	390	7	.	.	PUNCT
ma-234	391	1	lemma	lemma	PROPN
ma-234	391	2	4.5	4.5	NUM
ma-234	391	3	.	.	PUNCT
ma-234	392	1	let	let	VERB
ma-234	392	2	1	1	NUM
ma-234	392	3	≤	≤	NOUN
ma-234	392	4	q	q	ADJ
ma-234	392	5	≤	≤	NUM
ma-234	392	6	α	α	NOUN
ma-234	392	7	≤	≤	NOUN
ma-234	393	1	p	p	NOUN
ma-234	393	2	<	<	X
ma-234	393	3	∞.	∞.	PROPN
ma-234	393	4	then	then	ADV
ma-234	393	5	for	for	ADP
ma-234	393	6	any	any	DET
ma-234	393	7	element	element	NOUN
ma-234	393	8	f	f	PROPN
ma-234	393	9	of	of	ADP
ma-234	393	10	mα	mα	PROPN
ma-234	393	11	q	q	NOUN
ma-234	393	12	,	,	PUNCT
ma-234	393	13	p	p	X
ma-234	393	14	,	,	PUNCT
ma-234	393	15	we	we	PRON
ma-234	393	16	have	have	VERB
ma-234	393	17	lim	lim	PROPN
ma-234	393	18	n→∞	n→∞	NUM
ma-234	393	19	∥∥f	∥∥f	PROPN
ma-234	393	20	−	−	PROPN
ma-234	393	21	f	f	PROPN
ma-234	393	22	χq(0,n	χq(0,n	ADV
ma-234	393	23	)	)	PUNCT
ma-234	393	24	∥∥	∥∥	PROPN
ma-234	393	25	mα	mα	PROPN
ma-234	394	1	q	q	ADJ
ma-234	394	2	,	,	PUNCT
ma-234	394	3	p	p	X
ma-234	394	4	=	=	NOUN
ma-234	394	5	0	0	X
ma-234	394	6	.	.	PUNCT
ma-234	395	1	we	we	PRON
ma-234	395	2	are	be	AUX
ma-234	395	3	now	now	ADV
ma-234	395	4	ready	ready	ADJ
ma-234	395	5	to	to	PART
ma-234	395	6	prove	prove	VERB
ma-234	395	7	theorem	theorem	VERB
ma-234	395	8	2.5	2.5	NUM
ma-234	395	9	.	.	PUNCT
ma-234	396	1	proof	proof	NOUN
ma-234	396	2	of	of	ADP
ma-234	396	3	theorem	theorem	NOUN
ma-234	396	4	2.5for	2.5for	NUM
ma-234	396	5	any	any	DET
ma-234	396	6	integer	integer	NOUN
ma-234	396	7	n	n	PRON
ma-234	396	8	≥	≥	NOUN
ma-234	396	9	1	1	NUM
ma-234	396	10	,	,	PUNCT
ma-234	396	11	we	we	PRON
ma-234	396	12	have	have	AUX
ma-234	396	13	,	,	PUNCT
ma-234	396	14	by	by	ADP
ma-234	396	15	proposition	proposition	NOUN
ma-234	396	16	4.1	4.1	NUM
ma-234	396	17	,	,	PUNCT
ma-234	396	18	‖f	‖f	ADP
ma-234	396	19	−	−	PROPN
ma-234	397	1	(	(	PUNCT
ma-234	397	2	f	f	NOUN
ma-234	397	3	ωn	ωn	PROPN
ma-234	397	4	)	)	PUNCT
ma-234	397	5	∗	∗	NOUN
ma-234	397	6	φn‖mα	φn‖mα	PUNCT
ma-234	397	7	q	q	NOUN
ma-234	397	8	,	,	PUNCT
ma-234	397	9	p	p	NOUN
ma-234	397	10	≤	≤	NOUN
ma-234	397	11	‖f	‖f	ADP
ma-234	397	12	−	−	PROPN
ma-234	397	13	f	f	PROPN
ma-234	397	14	∗	∗	X
ma-234	397	15	φn‖mα	φn‖mα	PUNCT
ma-234	398	1	q	q	NOUN
ma-234	398	2	,	,	PUNCT
ma-234	398	3	p	p	NOUN
ma-234	398	4	+	+	NUM
ma-234	398	5	‖(f	‖(f	NOUN
ma-234	398	6	−	−	NOUN
ma-234	399	1	f	f	NOUN
ma-234	399	2	ωn	ωn	PROPN
ma-234	399	3	)	)	PUNCT
ma-234	399	4	∗	∗	NOUN
ma-234	399	5	φn‖mα	φn‖mα	PUNCT
ma-234	400	1	q	q	NOUN
ma-234	400	2	,	,	PUNCT
ma-234	400	3	p	p	NOUN
ma-234	400	4	≤	≤	NOUN
ma-234	400	5	‖f	‖f	ADP
ma-234	400	6	−	−	PROPN
ma-234	400	7	f	f	PROPN
ma-234	400	8	∗	∗	X
ma-234	400	9	φn‖mα	φn‖mα	PUNCT
ma-234	400	10	q	q	NOUN
ma-234	400	11	,	,	PUNCT
ma-234	400	12	p	p	X
ma-234	400	13	+	+	X
ma-234	400	14	c	c	NOUN
ma-234	400	15	‖f	‖f	PRON
ma-234	400	16	−	−	PROPN
ma-234	400	17	f	f	PROPN
ma-234	400	18	ωn‖mα	ωn‖mα	PROPN
ma-234	400	19	q	q	NOUN
ma-234	400	20	,	,	PUNCT
ma-234	400	21	p	p	NOUN
ma-234	400	22	‖φn‖1	‖φn‖1	PROPN
ma-234	400	23	≤	≤	NOUN
ma-234	400	24	‖f	‖f	ADP
ma-234	400	25	−	−	PROPN
ma-234	400	26	f	f	PROPN
ma-234	400	27	∗	∗	X
ma-234	400	28	φn‖mα	φn‖mα	PUNCT
ma-234	400	29	q	q	NOUN
ma-234	400	30	,	,	PUNCT
ma-234	400	31	p	p	X
ma-234	400	32	+	+	X
ma-234	400	33	c	c	NOUN
ma-234	400	34	‖f	‖f	PRON
ma-234	400	35	−	−	PROPN
ma-234	400	36	f	f	PROPN
ma-234	400	37	ωn‖mα	ωn‖mα	PROPN
ma-234	400	38	q	q	PROPN
ma-234	400	39	,	,	PUNCT
ma-234	400	40	p	p	X
ma-234	400	41	.	.	PUNCT
ma-234	401	1	notice	notice	VERB
ma-234	401	2	that	that	SCONJ
ma-234	401	3	,	,	PUNCT
ma-234	401	4	for	for	ADP
ma-234	401	5	any	any	DET
ma-234	401	6	integer	integer	NOUN
ma-234	401	7	n	n	PRON
ma-234	401	8	≥	≥	NOUN
ma-234	401	9	1	1	NUM
ma-234	401	10	,	,	PUNCT
ma-234	401	11	|f	|f	PROPN
ma-234	401	12	−	−	PROPN
ma-234	402	1	f	f	PROPN
ma-234	402	2	ωn|	ωn|	PROPN
ma-234	402	3	≤	≤	PROPN
ma-234	402	4	∣∣f	∣∣f	NOUN
ma-234	402	5	−	−	PROPN
ma-234	402	6	f	f	PROPN
ma-234	402	7	χq(0,n	χq(0,n	ADV
ma-234	402	8	)	)	PUNCT
ma-234	402	9	∣∣	∣∣	PROPN
ma-234	403	1	and	and	CCONJ
ma-234	403	2	therefore	therefore	ADV
ma-234	403	3	we	we	PRON
ma-234	403	4	obtain	obtain	VERB
ma-234	403	5	‖f	‖f	PRON
ma-234	403	6	−	−	PROPN
ma-234	404	1	(	(	PUNCT
ma-234	404	2	f	f	NOUN
ma-234	404	3	ωn	ωn	PROPN
ma-234	404	4	)	)	PUNCT
ma-234	404	5	∗	∗	NOUN
ma-234	404	6	φn‖mα	φn‖mα	PUNCT
ma-234	404	7	q	q	NOUN
ma-234	404	8	,	,	PUNCT
ma-234	404	9	p	p	NOUN
ma-234	404	10	≤	≤	NOUN
ma-234	404	11	‖f	‖f	ADP
ma-234	404	12	−	−	PROPN
ma-234	404	13	f	f	PROPN
ma-234	404	14	∗	∗	X
ma-234	404	15	φn‖mα	φn‖mα	PUNCT
ma-234	404	16	q	q	NOUN
ma-234	404	17	,	,	PUNCT
ma-234	404	18	p	p	NOUN
ma-234	404	19	+	+	NUM
ma-234	404	20	c	c	NOUN
ma-234	404	21	∥∥f	∥∥f	NOUN
ma-234	404	22	−	−	PROPN
ma-234	404	23	f	f	PROPN
ma-234	404	24	χq(0,n	χq(0,n	ADV
ma-234	404	25	)	)	PUNCT
ma-234	405	1	∥∥	∥∥	PROPN
ma-234	405	2	mα	mα	PROPN
ma-234	406	1	q	q	ADJ
ma-234	406	2	,	,	PUNCT
ma-234	406	3	p	p	NOUN
ma-234	406	4	.	.	PUNCT
ma-234	407	1	thus	thus	ADV
ma-234	407	2	,	,	PUNCT
ma-234	407	3	it	it	PRON
ma-234	407	4	follows	follow	VERB
ma-234	407	5	from	from	ADP
ma-234	407	6	theorem	theorem	ADJ
ma-234	407	7	2.4	2.4	NUM
ma-234	407	8	and	and	CCONJ
ma-234	407	9	lemma	lemma	PROPN
ma-234	407	10	4.5	4.5	NUM
ma-234	407	11	that	that	PRON
ma-234	407	12	lim	lim	PROPN
ma-234	407	13	n→∞	n→∞	X
ma-234	407	14	‖f	‖f	PUNCT
ma-234	407	15	−	−	PROPN
ma-234	408	1	(	(	PUNCT
ma-234	408	2	f	f	NOUN
ma-234	408	3	ωn	ωn	PROPN
ma-234	408	4	)	)	PUNCT
ma-234	408	5	∗	∗	NOUN
ma-234	408	6	φn‖mα	φn‖mα	PUNCT
ma-234	408	7	q	q	NOUN
ma-234	408	8	,	,	PUNCT
ma-234	408	9	p	p	X
ma-234	408	10	=	=	NOUN
ma-234	408	11	0	0	PROPN
ma-234	408	12	.	.	PUNCT
ma-234	409	1	this	this	PRON
ma-234	409	2	finishes	finish	VERB
ma-234	409	3	the	the	DET
ma-234	409	4	proof	proof	NOUN
ma-234	409	5	.	.	PUNCT
ma-234	410	1	�	�	PROPN
ma-234	410	2	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	410	3	eur	eur	PROPN
ma-234	410	4	.	.	PUNCT
ma-234	411	1	j.	j.	PROPN
ma-234	411	2	math	math	PROPN
ma-234	411	3	.	.	PUNCT
ma-234	412	1	anal	anal	PROPN
ma-234	412	2	.	.	PUNCT
ma-234	413	1	10.28924	10.28924	NUM
ma-234	413	2	/	/	SYM
ma-234	413	3	ada	ada	PROPN
ma-234	413	4	/	/	SYM
ma-234	413	5	ma.4.16	ma.4.16	PROPN
ma-234	413	6	155	155	NUM
ma-234	413	7	.	.	PUNCT
ma-234	413	8	fractional	fractional	ADJ
ma-234	413	9	operators	operator	NOUN
ma-234	413	10	in	in	ADP
ma-234	413	11	mα	mα	PROPN
ma-234	413	12	q	q	NOUN
ma-234	413	13	,	,	PUNCT
ma-234	413	14	pthis	pthis	ADJ
ma-234	413	15	section	section	NOUN
ma-234	413	16	is	be	AUX
ma-234	413	17	devoted	devote	VERB
ma-234	413	18	to	to	PART
ma-234	413	19	prove	prove	VERB
ma-234	413	20	theorem	theorem	VERB
ma-234	413	21	2.1	2.1	NUM
ma-234	413	22	and	and	CCONJ
ma-234	413	23	theorem	theorem	VERB
ma-234	413	24	2.2	2.2	NUM
ma-234	413	25	.	.	PUNCT
ma-234	414	1	in	in	ADP
ma-234	414	2	order	order	NOUN
ma-234	414	3	to	to	PART
ma-234	414	4	do	do	AUX
ma-234	414	5	this	this	PRON
ma-234	414	6	we	we	PRON
ma-234	414	7	need	need	VERB
ma-234	414	8	somepreparatory	somepreparatory	NOUN
ma-234	414	9	lemmas.let	lemmas.let	X
ma-234	414	10	0	0	PUNCT
ma-234	414	11	<	<	X
ma-234	414	12	γ	γ	X
ma-234	414	13	<	<	X
ma-234	414	14	1	1	NUM
ma-234	414	15	and	and	CCONJ
ma-234	414	16	d	d	AUX
ma-234	414	17	be	be	AUX
ma-234	414	18	a	a	DET
ma-234	414	19	dyadic	dyadic	ADJ
ma-234	414	20	grid	grid	NOUN
ma-234	414	21	.	.	PUNCT
ma-234	415	1	the	the	DET
ma-234	415	2	dyadic	dyadic	ADJ
ma-234	415	3	fractional	fractional	ADJ
ma-234	415	4	maximal	maximal	ADJ
ma-234	415	5	operator	operator	NOUN
ma-234	415	6	mdγ	mdγ	NOUN
ma-234	415	7	is	be	AUX
ma-234	415	8	definedby	definedby	ADJ
ma-234	415	9	mdγ	mdγ	NOUN
ma-234	415	10	f	f	PROPN
ma-234	415	11	(	(	PUNCT
ma-234	415	12	x	x	X
ma-234	415	13	)	)	PUNCT
ma-234	415	14	=	=	SYM
ma-234	415	15	sup	sup	NOUN
ma-234	415	16	{	{	PUNCT
ma-234	415	17	|q|γ−1	|q|γ−1	NUM
ma-234	415	18	∫	∫	PROPN
ma-234	415	19	q	q	X
ma-234	415	20	|f	|f	PROPN
ma-234	416	1	(	(	PUNCT
ma-234	416	2	y)|dy	y)|dy	NOUN
ma-234	416	3	/	/	SYM
ma-234	416	4	q	q	NOUN
ma-234	416	5	∈	∈	PROPN
ma-234	416	6	d	d	NOUN
ma-234	416	7	,	,	PUNCT
ma-234	416	8	x	x	SYM
ma-234	416	9	∈	∈	PROPN
ma-234	416	10	q	q	X
ma-234	416	11	}	}	PUNCT
ma-234	416	12	,	,	PUNCT
ma-234	416	13	f	f	PROPN
ma-234	416	14	∈	∈	PROPN
ma-234	416	15	l1	l1	PROPN
ma-234	416	16	loc	loc	PROPN
ma-234	416	17	,	,	PUNCT
ma-234	416	18	x	x	PROPN
ma-234	416	19	∈	∈	PROPN
ma-234	416	20	rd	rd	NOUN
ma-234	416	21	.	.	PUNCT
ma-234	417	1	the	the	DET
ma-234	417	2	following	follow	VERB
ma-234	417	3	lemma	lemma	PROPN
ma-234	417	4	is	be	AUX
ma-234	417	5	a	a	DET
ma-234	417	6	consequence	consequence	NOUN
ma-234	417	7	of	of	ADP
ma-234	417	8	proposition	proposition	NOUN
ma-234	417	9	3.3	3.3	NUM
ma-234	417	10	.	.	PUNCT
ma-234	418	1	lemma	lemma	PROPN
ma-234	418	2	5.1	5.1	NUM
ma-234	418	3	.	.	PUNCT
ma-234	419	1	let	let	VERB
ma-234	419	2	0	0	NUM
ma-234	419	3	<	<	X
ma-234	419	4	γ	γ	X
ma-234	419	5	<	<	X
ma-234	419	6	1	1	NUM
ma-234	419	7	.	.	PUNCT
ma-234	420	1	for	for	ADP
ma-234	420	2	any	any	DET
ma-234	420	3	element	element	NOUN
ma-234	420	4	f	f	PROPN
ma-234	420	5	of	of	ADP
ma-234	420	6	l1	l1	PROPN
ma-234	420	7	loc	loc	PROPN
ma-234	420	8	we	we	PRON
ma-234	420	9	have	have	VERB
ma-234	420	10	:	:	PUNCT
ma-234	420	11	mγf	mγf	X
ma-234	420	12	(	(	PUNCT
ma-234	420	13	x	x	NOUN
ma-234	420	14	)	)	PUNCT
ma-234	420	15	≤	≤	NOUN
ma-234	420	16	3d(1−γ	3d(1−γ	NUM
ma-234	420	17	)	)	PUNCT
ma-234	420	18	max	max	PROPN
ma-234	420	19	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	PROPN
ma-234	420	20	md	md	PROPN
ma-234	420	21	t	t	PROPN
ma-234	420	22	γ	γ	PROPN
ma-234	420	23	f	f	PROPN
ma-234	420	24	(	(	PUNCT
ma-234	420	25	x	x	X
ma-234	420	26	)	)	PUNCT
ma-234	420	27	,	,	PUNCT
ma-234	420	28	x	x	PUNCT
ma-234	420	29	∈	∈	PROPN
ma-234	420	30	rd	rd	NOUN
ma-234	420	31	.	.	PUNCT
ma-234	421	1	proof	proof	NOUN
ma-234	421	2	.	.	PUNCT
ma-234	422	1	let	let	VERB
ma-234	422	2	us	we	PRON
ma-234	422	3	consider	consider	VERB
ma-234	422	4	an	an	DET
ma-234	422	5	element	element	NOUN
ma-234	422	6	(	(	PUNCT
ma-234	422	7	f	f	X
ma-234	422	8	,	,	PUNCT
ma-234	422	9	x	x	NOUN
ma-234	422	10	)	)	PUNCT
ma-234	422	11	of	of	ADP
ma-234	422	12	l1	l1	PROPN
ma-234	422	13	loc	loc	PROPN
ma-234	422	14	×	×	PROPN
ma-234	422	15	rd	rd	PROPN
ma-234	422	16	and	and	CCONJ
ma-234	422	17	a	a	DET
ma-234	422	18	cube	cube	NOUN
ma-234	422	19	q	q	PROPN
ma-234	422	20	of	of	ADP
ma-234	422	21	rd	rd	NOUN
ma-234	422	22	containing	contain	VERB
ma-234	422	23	x	x	X
ma-234	422	24	.	.	PUNCT
ma-234	423	1	byproposition	byproposition	NOUN
ma-234	423	2	3.3	3.3	NUM
ma-234	423	3	,	,	PUNCT
ma-234	423	4	there	there	PRON
ma-234	423	5	exist	exist	VERB
ma-234	423	6	an	an	DET
ma-234	423	7	element	element	NOUN
ma-234	423	8	t	t	PROPN
ma-234	423	9	of	of	ADP
ma-234	423	10	{	{	PUNCT
ma-234	423	11	−1/3	−1/3	ADJ
ma-234	423	12	,	,	PUNCT
ma-234	423	13	0	0	NUM
ma-234	423	14	,	,	PUNCT
ma-234	423	15	1/3}d	1/3}d	NOUN
ma-234	423	16	and	and	CCONJ
ma-234	423	17	a	a	DET
ma-234	423	18	cube	cube	NOUN
ma-234	423	19	qt	qt	NOUN
ma-234	423	20	of	of	ADP
ma-234	423	21	dt	dt	PROPN
ma-234	423	22	such	such	ADJ
ma-234	423	23	that	that	DET
ma-234	423	24	q	q	NOUN
ma-234	423	25	isincluded	isinclude	VERB
ma-234	423	26	in	in	ADP
ma-234	423	27	qt	qt	NOUN
ma-234	423	28	and	and	CCONJ
ma-234	423	29	`	`	PUNCT
ma-234	423	30	(	(	PUNCT
ma-234	423	31	qt	qt	NOUN
ma-234	423	32	)	)	PUNCT
ma-234	423	33	≤	≤	NOUN
ma-234	423	34	3	3	NUM
ma-234	423	35	`	`	PUNCT
ma-234	423	36	(	(	PUNCT
ma-234	423	37	q	q	NOUN
ma-234	423	38	)	)	PUNCT
ma-234	423	39	.	.	PUNCT
ma-234	424	1	thus	thus	ADV
ma-234	424	2	,	,	PUNCT
ma-234	424	3	we	we	PRON
ma-234	424	4	have	have	VERB
ma-234	424	5	|q|γ−1	|q|γ−1	NUM
ma-234	424	6	∫	∫	PROPN
ma-234	424	7	q	q	NOUN
ma-234	424	8	|f	|f	PROPN
ma-234	424	9	(	(	PUNCT
ma-234	424	10	y)|dy	y)|dy	NOUN
ma-234	424	11	=	=	SYM
ma-234	424	12	`	`	PUNCT
ma-234	424	13	(	(	PUNCT
ma-234	424	14	q)d(γ−1	q)d(γ−1	NUM
ma-234	424	15	)	)	PUNCT
ma-234	424	16	∫	∫	PROPN
ma-234	425	1	q	q	PROPN
ma-234	425	2	|f	|f	PROPN
ma-234	425	3	(	(	PUNCT
ma-234	425	4	y)|dy	y)|dy	NOUN
ma-234	425	5	≤	≤	NOUN
ma-234	425	6	[	[	PUNCT
ma-234	425	7	1	1	NUM
ma-234	425	8	3	3	NUM
ma-234	425	9	`	`	PUNCT
ma-234	425	10	(	(	PUNCT
ma-234	425	11	qt	qt	NOUN
ma-234	425	12	)	)	PUNCT
ma-234	425	13	]	]	PUNCT
ma-234	425	14	d(γ−1	d(γ−1	PROPN
ma-234	425	15	)	)	PUNCT
ma-234	425	16	∫	∫	PROPN
ma-234	425	17	q	q	PROPN
ma-234	425	18	|f	|f	PROPN
ma-234	425	19	(	(	PUNCT
ma-234	425	20	y)|dy	y)|dy	NOUN
ma-234	425	21	≤	≤	NUM
ma-234	425	22	3d(1−γ)|qt	3d(1−γ)|qt	NUM
ma-234	425	23	|γ−1	|γ−1	ADJ
ma-234	425	24	∫	∫	PROPN
ma-234	425	25	qt	qt	PROPN
ma-234	425	26	|f	|f	PROPN
ma-234	426	1	(	(	PUNCT
ma-234	426	2	y)|dy	y)|dy	NOUN
ma-234	426	3	≤	≤	PROPN
ma-234	427	1	3d(1−γ)md	3d(1−γ)md	PROPN
ma-234	427	2	t	t	PROPN
ma-234	427	3	γ	γ	X
ma-234	427	4	f	f	PROPN
ma-234	427	5	(	(	PUNCT
ma-234	427	6	x	x	NOUN
ma-234	427	7	)	)	PUNCT
ma-234	427	8	.	.	PUNCT
ma-234	428	1	consequently	consequently	ADV
ma-234	428	2	mγf	mγf	ADV
ma-234	428	3	(	(	PUNCT
ma-234	428	4	x	x	NOUN
ma-234	428	5	)	)	PUNCT
ma-234	428	6	≤	≤	NOUN
ma-234	428	7	3d(1−γ	3d(1−γ	NUM
ma-234	428	8	)	)	PUNCT
ma-234	428	9	max	max	PROPN
ma-234	428	10	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	PROPN
ma-234	428	11	md	md	PROPN
ma-234	428	12	t	t	PROPN
ma-234	428	13	γ	γ	PROPN
ma-234	428	14	f	f	PROPN
ma-234	428	15	(	(	PUNCT
ma-234	428	16	x	x	NOUN
ma-234	428	17	)	)	PUNCT
ma-234	428	18	.	.	PUNCT
ma-234	429	1	the	the	DET
ma-234	429	2	proof	proof	NOUN
ma-234	429	3	is	be	AUX
ma-234	429	4	complete	complete	ADJ
ma-234	429	5	.	.	PUNCT
ma-234	430	1	�	�	PROPN
ma-234	430	2	recall	recall	VERB
ma-234	430	3	that	that	SCONJ
ma-234	430	4	the	the	DET
ma-234	430	5	density	density	NOUN
ma-234	430	6	of	of	ADP
ma-234	430	7	l∞c	l∞c	NOUN
ma-234	430	8	in	in	ADP
ma-234	430	9	mα	mα	PROPN
ma-234	430	10	q	q	NOUN
ma-234	430	11	,	,	PUNCT
ma-234	430	12	p	p	X
ma-234	430	13	(	(	PUNCT
ma-234	430	14	see	see	NOUN
ma-234	430	15	point	point	NOUN
ma-234	430	16	2	2	NUM
ma-234	430	17	)	)	PUNCT
ma-234	430	18	of	of	ADP
ma-234	430	19	proposition	proposition	NOUN
ma-234	430	20	3.7	3.7	NUM
ma-234	430	21	)	)	PUNCT
ma-234	430	22	has	have	AUX
ma-234	430	23	been	be	AUX
ma-234	430	24	proved	prove	VERB
ma-234	430	25	in	in	ADP
ma-234	430	26	[	[	X
ma-234	430	27	8].here	8].here	NUM
ma-234	430	28	,	,	PUNCT
ma-234	430	29	we	we	PRON
ma-234	430	30	improve	improve	VERB
ma-234	430	31	this	this	DET
ma-234	430	32	result	result	NOUN
ma-234	430	33	which	which	PRON
ma-234	430	34	will	will	AUX
ma-234	430	35	play	play	VERB
ma-234	430	36	a	a	DET
ma-234	430	37	key	key	ADJ
ma-234	430	38	role	role	NOUN
ma-234	430	39	in	in	ADP
ma-234	430	40	the	the	DET
ma-234	430	41	proof	proof	NOUN
ma-234	430	42	of	of	ADP
ma-234	430	43	lemma	lemma	PROPN
ma-234	430	44	5.3	5.3	NUM
ma-234	430	45	.	.	PUNCT
ma-234	431	1	lemma	lemma	PROPN
ma-234	431	2	5.2	5.2	NUM
ma-234	431	3	.	.	PUNCT
ma-234	432	1	let	let	VERB
ma-234	432	2	1	1	NUM
ma-234	432	3	≤	≤	NOUN
ma-234	432	4	q	q	ADJ
ma-234	432	5	≤	≤	NUM
ma-234	432	6	α	α	NOUN
ma-234	432	7	≤	≤	NOUN
ma-234	433	1	p	p	DET
ma-234	433	2	<	<	X
ma-234	433	3	∞	∞	PROPN
ma-234	433	4	and	and	CCONJ
ma-234	433	5	f	f	PROPN
ma-234	433	6	be	be	AUX
ma-234	433	7	any	any	DET
ma-234	433	8	element	element	NOUN
ma-234	433	9	ofmα	ofmα	ADJ
ma-234	433	10	q	q	NOUN
ma-234	433	11	,	,	PUNCT
ma-234	433	12	p	p	NOUN
ma-234	433	13	.	.	PUNCT
ma-234	434	1	then	then	ADV
ma-234	434	2	there	there	PRON
ma-234	434	3	exists	exist	VERB
ma-234	434	4	a	a	DET
ma-234	434	5	sequence	sequence	NOUN
ma-234	434	6	(	(	PUNCT
ma-234	434	7	fn)n≥1	fn)n≥1	NOUN
ma-234	434	8	of	of	ADP
ma-234	434	9	elements	element	NOUN
ma-234	434	10	of	of	ADP
ma-234	434	11	l∞c	l∞c	NOUN
ma-234	434	12	∩mα	∩mα	PROPN
ma-234	434	13	q	q	PROPN
ma-234	434	14	,	,	PUNCT
ma-234	434	15	p	p	X
ma-234	434	16	such	such	ADJ
ma-234	434	17	that	that	SCONJ
ma-234	434	18	(	(	PUNCT
ma-234	434	19	|fn|)n≥1	|fn|)n≥1	PROPN
ma-234	434	20	↑	↑	NOUN
ma-234	434	21	|f	|f	PROPN
ma-234	435	1	|	|	ADV
ma-234	435	2	almost	almost	ADV
ma-234	435	3	everywhere	everywhere	ADV
ma-234	435	4	and	and	CCONJ
ma-234	435	5	lim	lim	PROPN
ma-234	435	6	n→∞	n→∞	NUM
ma-234	435	7	‖f	‖f	PUNCT
ma-234	435	8	−	−	PROPN
ma-234	435	9	fn‖mα	fn‖mα	NOUN
ma-234	435	10	q	q	NOUN
ma-234	435	11	,	,	PUNCT
ma-234	435	12	p	p	NOUN
ma-234	435	13	=	=	NOUN
ma-234	435	14	0	0	NUM
ma-234	435	15	.	.	PUNCT
ma-234	435	16	proof	proof	NOUN
ma-234	435	17	.	.	PUNCT
ma-234	436	1	let	let	VERB
ma-234	436	2	us	we	PRON
ma-234	436	3	set	set	VERB
ma-234	436	4	,	,	PUNCT
ma-234	436	5	for	for	ADP
ma-234	436	6	any	any	DET
ma-234	436	7	integer	integer	NOUN
ma-234	436	8	n	n	PRON
ma-234	436	9	≥	≥	NOUN
ma-234	436	10	1	1	NUM
ma-234	436	11	,	,	PUNCT
ma-234	437	1	fn	fn	NOUN
ma-234	437	2	=	=	SYM
ma-234	437	3	sgn(f	sgn(f	PROPN
ma-234	437	4	)	)	PUNCT
ma-234	437	5	min	min	NOUN
ma-234	438	1	(	(	PUNCT
ma-234	438	2	|f	|f	PROPN
ma-234	438	3	|	|	ADV
ma-234	438	4	,	,	PUNCT
ma-234	438	5	nχq(0,2n	nχq(0,2n	PROPN
ma-234	438	6	)	)	PUNCT
ma-234	438	7	)	)	PUNCT
ma-234	438	8	,	,	PUNCT
ma-234	438	9	where	where	SCONJ
ma-234	438	10	,	,	PUNCT
ma-234	438	11	for	for	ADP
ma-234	438	12	any	any	DET
ma-234	438	13	x	x	SYM
ma-234	438	14	∈	∈	PROPN
ma-234	438	15	rd	rd	PROPN
ma-234	438	16	,	,	PUNCT
ma-234	438	17	sgn(f	sgn(f	PROPN
ma-234	438	18	)	)	PUNCT
ma-234	438	19	(	(	PUNCT
ma-234	438	20	x	x	X
ma-234	438	21	)	)	PUNCT
ma-234	438	22	=	=	PRON
ma-234	438	23	{	{	PUNCT
ma-234	438	24	f	f	X
ma-234	438	25	(	(	PUNCT
ma-234	438	26	x	x	PROPN
ma-234	438	27	)	)	PUNCT
ma-234	438	28	|f	|f	PROPN
ma-234	439	1	(	(	PUNCT
ma-234	439	2	x)|	x)|	PROPN
ma-234	439	3	if	if	SCONJ
ma-234	439	4	f	f	PROPN
ma-234	439	5	(	(	PUNCT
ma-234	439	6	x	x	X
ma-234	439	7	)	)	PUNCT
ma-234	439	8	6=	6=	ADP
ma-234	439	9	0	0	NUM
ma-234	439	10	0	0	NUM
ma-234	440	1	if	if	SCONJ
ma-234	440	2	f	f	PROPN
ma-234	440	3	(	(	PUNCT
ma-234	440	4	x	x	X
ma-234	440	5	)	)	PUNCT
ma-234	440	6	=	=	SYM
ma-234	440	7	0	0	X
ma-234	440	8	.	.	PUNCT
ma-234	441	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	441	2	eur	eur	PROPN
ma-234	441	3	.	.	PUNCT
ma-234	442	1	j.	j.	PROPN
ma-234	442	2	math	math	PROPN
ma-234	442	3	.	.	PUNCT
ma-234	443	1	anal	anal	PROPN
ma-234	443	2	.	.	PUNCT
ma-234	444	1	10.28924	10.28924	NUM
ma-234	444	2	/	/	SYM
ma-234	444	3	ada	ada	PROPN
ma-234	444	4	/	/	SYM
ma-234	444	5	ma.4.16	ma.4.16	PROPN
ma-234	444	6	16it	16it	PROPN
ma-234	444	7	is	be	AUX
ma-234	444	8	easy	easy	ADJ
ma-234	444	9	to	to	PART
ma-234	444	10	see	see	VERB
ma-234	444	11	that	that	PRON
ma-234	444	12	(	(	PUNCT
ma-234	444	13	fn)n≥1	fn)n≥1	NOUN
ma-234	444	14	is	be	AUX
ma-234	444	15	a	a	DET
ma-234	444	16	sequence	sequence	NOUN
ma-234	444	17	of	of	ADP
ma-234	444	18	elements	element	NOUN
ma-234	444	19	of	of	ADP
ma-234	444	20	l∞c	l∞c	NOUN
ma-234	444	21	∩mα	∩mα	PROPN
ma-234	444	22	q	q	NOUN
ma-234	444	23	,	,	PUNCT
ma-234	444	24	p	p	NOUN
ma-234	444	25	satisfying	satisfy	VERB
ma-234	444	26	(	(	PUNCT
ma-234	444	27	|fn|)n≥1	|fn|)n≥1	PROPN
ma-234	444	28	↑	↑	PROPN
ma-234	444	29	|f	|f	PROPN
ma-234	444	30	|almost	|almost	NOUN
ma-234	444	31	everywhere	everywhere	ADV
ma-234	444	32	and	and	CCONJ
ma-234	444	33			PUNCT
ma-234	444	34	|fn|	|fn|	NOUN
ma-234	444	35	≤	≤	NOUN
ma-234	444	36	|f	|f	PUNCT
ma-234	445	1	|	|	ADV
ma-234	445	2	,	,	PUNCT
ma-234	445	3	n	n	PRON
ma-234	445	4	≥	≥	NOUN
ma-234	445	5	1	1	NUM
ma-234	445	6	lim	lim	NOUN
ma-234	445	7	n→∞	n→∞	PRON
ma-234	445	8	fn	fn	NOUN
ma-234	445	9	=	=	SYM
ma-234	445	10	f	f	PROPN
ma-234	445	11	almost	almost	ADV
ma-234	445	12	everywhere.therefore	everywhere.therefore	ADJ
ma-234	445	13	,	,	PUNCT
ma-234	445	14	an	an	DET
ma-234	445	15	application	application	NOUN
ma-234	445	16	of	of	ADP
ma-234	445	17	proposition	proposition	NOUN
ma-234	445	18	4.4	4.4	NUM
ma-234	445	19	leads	lead	VERB
ma-234	445	20	to	to	ADP
ma-234	445	21	lim	lim	PROPN
ma-234	445	22	n→∞	n→∞	NUM
ma-234	445	23	‖f	‖f	ADP
ma-234	445	24	−	−	NOUN
ma-234	445	25	fn‖mα	fn‖mα	NOUN
ma-234	445	26	q	q	NOUN
ma-234	445	27	,	,	PUNCT
ma-234	445	28	p	p	NOUN
ma-234	445	29	=	=	NOUN
ma-234	445	30	0	0	PROPN
ma-234	445	31	.	.	PUNCT
ma-234	446	1	this	this	PRON
ma-234	446	2	ends	end	VERB
ma-234	446	3	the	the	DET
ma-234	446	4	proof	proof	NOUN
ma-234	446	5	.	.	PUNCT
ma-234	447	1	�	�	PROPN
ma-234	447	2	as	as	ADP
ma-234	447	3	a	a	DET
ma-234	447	4	consequence	consequence	NOUN
ma-234	447	5	of	of	ADP
ma-234	447	6	lemma	lemma	PROPN
ma-234	447	7	5.2	5.2	NUM
ma-234	447	8	,	,	PUNCT
ma-234	447	9	the	the	DET
ma-234	447	10	following	follow	VERB
ma-234	447	11	result	result	NOUN
ma-234	447	12	holds	hold	VERB
ma-234	447	13	true	true	ADJ
ma-234	447	14	.	.	PUNCT
ma-234	448	1	lemma	lemma	PROPN
ma-234	448	2	5.3	5.3	NUM
ma-234	448	3	.	.	PUNCT
ma-234	449	1	let	let	VERB
ma-234	449	2	us	we	PRON
ma-234	449	3	assume	assume	VERB
ma-234	449	4	that	that	SCONJ
ma-234	449	5	0	0	NUM
ma-234	449	6	<	<	X
ma-234	449	7	γ	γ	X
ma-234	449	8	<	<	X
ma-234	449	9	1	1	NUM
ma-234	449	10	α	α	NOUN
ma-234	449	11	≤	≤	NUM
ma-234	449	12	1	1	NUM
ma-234	449	13	and	and	CCONJ
ma-234	449	14	1	1	NUM
ma-234	449	15	p	p	NOUN
ma-234	449	16	=	=	NOUN
ma-234	450	1	1	1	NUM
ma-234	450	2	α	α	NOUN
ma-234	450	3	−	−	PROPN
ma-234	450	4	γ	γ	PROPN
ma-234	450	5	.	.	PROPN
ma-234	450	6	then	then	ADV
ma-234	450	7	for	for	ADP
ma-234	450	8	any	any	DET
ma-234	450	9	dyadic	dyadic	ADJ
ma-234	450	10	grid	grid	NOUN
ma-234	450	11	d	d	NOUN
ma-234	450	12	and	and	CCONJ
ma-234	450	13	any	any	DET
ma-234	450	14	element	element	NOUN
ma-234	450	15	f	f	PROPN
ma-234	450	16	of	of	ADP
ma-234	450	17	mα	mα	PROPN
ma-234	450	18	1,p	1,p	PROPN
ma-234	450	19	,	,	PUNCT
ma-234	450	20	we	we	PRON
ma-234	450	21	have	have	VERB
ma-234	450	22	‖mdγ	‖mdγ	PROPN
ma-234	450	23	f	f	PROPN
ma-234	450	24	‖p	‖p	PROPN
ma-234	450	25	≤	≤	ADJ
ma-234	450	26	2	2	NUM
ma-234	450	27	‖f	‖f	ADP
ma-234	450	28	‖mα	‖mα	NUM
ma-234	450	29	1,p	1,p	NOUN
ma-234	450	30	.	.	PUNCT
ma-234	451	1	proof	proof	NOUN
ma-234	451	2	.	.	PUNCT
ma-234	452	1	if	if	SCONJ
ma-234	452	2	α	α	PRON
ma-234	452	3	=	=	NOUN
ma-234	452	4	1	1	NUM
ma-234	452	5	then	then	ADV
ma-234	452	6	mα	mα	PROPN
ma-234	452	7	q	q	NOUN
ma-234	452	8	,	,	PUNCT
ma-234	452	9	p	p	NOUN
ma-234	452	10	=	=	X
ma-234	452	11	{	{	PUNCT
ma-234	452	12	0	0	NUM
ma-234	452	13	}	}	PUNCT
ma-234	452	14	and	and	CCONJ
ma-234	452	15	so	so	ADV
ma-234	452	16	the	the	DET
ma-234	452	17	result	result	NOUN
ma-234	452	18	is	be	AUX
ma-234	452	19	obvious	obvious	ADJ
ma-234	452	20	.	.	PUNCT
ma-234	453	1	thus	thus	ADV
ma-234	453	2	,	,	PUNCT
ma-234	453	3	we	we	PRON
ma-234	453	4	assume	assume	VERB
ma-234	453	5	that	that	SCONJ
ma-234	453	6	α	α	PRON
ma-234	453	7	>	>	X
ma-234	453	8	1.let	1.let	PROPN
ma-234	453	9	f	f	NOUN
ma-234	453	10	be	be	VERB
ma-234	453	11	any	any	DET
ma-234	453	12	element	element	NOUN
ma-234	453	13	of	of	ADP
ma-234	453	14	mα	mα	PROPN
ma-234	453	15	1,p	1,p	PROPN
ma-234	453	16	and	and	CCONJ
ma-234	453	17	d	d	NOUN
ma-234	453	18	be	be	AUX
ma-234	453	19	a	a	DET
ma-234	453	20	dyadic	dyadic	ADJ
ma-234	453	21	grid.1	grid.1	NOUN
ma-234	453	22	)	)	PUNCT
ma-234	453	23	assume	assume	VERB
ma-234	453	24	that	that	SCONJ
ma-234	453	25	f	f	PROPN
ma-234	453	26	also	also	ADV
ma-234	453	27	belongs	belong	VERB
ma-234	453	28	to	to	ADP
ma-234	453	29	l∞.a	l∞.a	PROPN
ma-234	453	30	)	)	PUNCT
ma-234	453	31	we	we	PRON
ma-234	453	32	have	have	AUX
ma-234	453	33	,	,	PUNCT
ma-234	453	34	for	for	ADP
ma-234	453	35	all	all	DET
ma-234	453	36	cube	cube	NOUN
ma-234	453	37	q	q	PROPN
ma-234	453	38	of	of	ADP
ma-234	453	39	rd	rd	PROPN
ma-234	453	40	,	,	PUNCT
ma-234	453	41	|q|γ−1	|q|γ−1	X
ma-234	453	42	∫	∫	PROPN
ma-234	453	43	q	q	X
ma-234	454	1	|f	|f	PROPN
ma-234	454	2	(	(	PUNCT
ma-234	454	3	y)|dy	y)|dy	NOUN
ma-234	454	4	≤	≤	PROPN
ma-234	454	5	|q|γ‖f	|q|γ‖f	NUM
ma-234	454	6	‖∞	‖∞	NOUN
ma-234	454	7	and	and	CCONJ
ma-234	454	8	|q|γ−1	|q|γ−1	NUM
ma-234	454	9	∫	∫	PROPN
ma-234	454	10	q	q	X
ma-234	454	11	|f	|f	PROPN
ma-234	454	12	(	(	PUNCT
ma-234	454	13	y)|dy	y)|dy	NOUN
ma-234	454	14	=	=	PUNCT
ma-234	454	15	|q|γ−	|q|γ−	PUNCT
ma-234	454	16	1	1	NUM
ma-234	454	17	α	α	NUM
ma-234	454	18	|q|	|q|	NUM
ma-234	454	19	1	1	NUM
ma-234	454	20	α	α	NUM
ma-234	454	21	−1	−1	NOUN
ma-234	454	22	∫	∫	PROPN
ma-234	454	23	q	q	PROPN
ma-234	454	24	|f	|f	PROPN
ma-234	454	25	(	(	PUNCT
ma-234	454	26	y)|dy	y)|dy	NOUN
ma-234	454	27	=	=	PUNCT
ma-234	454	28	|q|γ−	|q|γ−	VERB
ma-234	454	29	1	1	NUM
ma-234	454	30	α	α	NOUN
ma-234	454	31	‖f	‖f	ADP
ma-234	454	32	‖mα	‖mα	NUM
ma-234	454	33	1	1	NUM
ma-234	454	34	.	.	PUNCT
ma-234	455	1	consequently	consequently	ADV
ma-234	455	2	,	,	PUNCT
ma-234	455	3	lim	lim	PROPN
ma-234	455	4	`	`	PUNCT
ma-234	455	5	(	(	PUNCT
ma-234	455	6	q)→∞	q)→∞	X
ma-234	455	7	|q|γ−1	|q|γ−1	X
ma-234	455	8	∫	∫	NOUN
ma-234	455	9	q	q	X
ma-234	455	10	|f	|f	PROPN
ma-234	455	11	(	(	PUNCT
ma-234	455	12	y)|dy	y)|dy	NOUN
ma-234	455	13	=	=	SYM
ma-234	455	14	0	0	PUNCT
ma-234	455	15	(	(	PUNCT
ma-234	455	16	∗	∗	NOUN
ma-234	455	17	)	)	PUNCT
ma-234	455	18	and	and	CCONJ
ma-234	455	19	,	,	PUNCT
ma-234	455	20	for	for	ADP
ma-234	455	21	all	all	DET
ma-234	455	22	q	q	PROPN
ma-234	455	23	∈	∈	PROPN
ma-234	455	24	q	q	NOUN
ma-234	455	25	,	,	PUNCT
ma-234	455	26	|q|γ−1	|q|γ−1	X
ma-234	455	27	∫	∫	PROPN
ma-234	455	28	q	q	X
ma-234	455	29	|f	|f	PROPN
ma-234	455	30	(	(	PUNCT
ma-234	455	31	y)|dy	y)|dy	NOUN
ma-234	455	32	≤	≤	PROPN
ma-234	455	33	{	{	PUNCT
ma-234	455	34	‖f	‖f	DET
ma-234	455	35	‖∞	‖∞	NOUN
ma-234	455	36	if	if	SCONJ
ma-234	455	37	`	`	PUNCT
ma-234	455	38	(	(	PUNCT
ma-234	455	39	q	q	X
ma-234	455	40	)	)	PUNCT
ma-234	455	41	≥	≥	NOUN
ma-234	455	42	1	1	NUM
ma-234	455	43	‖f	‖f	ADP
ma-234	455	44	‖mα	‖mα	NUM
ma-234	455	45	1	1	NUM
ma-234	455	46	if	if	SCONJ
ma-234	455	47	`	`	PUNCT
ma-234	455	48	(	(	PUNCT
ma-234	455	49	q	q	NOUN
ma-234	455	50	)	)	PUNCT
ma-234	455	51	≤	≤	NUM
ma-234	455	52	1	1	NUM
ma-234	455	53	.	.	PUNCT
ma-234	456	1	thus	thus	ADV
ma-234	456	2	,	,	PUNCT
ma-234	456	3	for	for	ADP
ma-234	456	4	all	all	DET
ma-234	456	5	x	x	SYM
ma-234	456	6	∈	∈	PROPN
ma-234	456	7	rd	rd	PROPN
ma-234	456	8	,	,	PUNCT
ma-234	456	9	mγf	mγf	X
ma-234	456	10	(	(	PUNCT
ma-234	456	11	x	x	NOUN
ma-234	456	12	)	)	PUNCT
ma-234	456	13	≤	≤	NUM
ma-234	456	14	m	m	VERB
ma-234	456	15	with	with	ADP
ma-234	456	16	m	m	PROPN
ma-234	456	17	=	=	SYM
ma-234	456	18	max	max	PROPN
ma-234	456	19	(	(	PUNCT
ma-234	456	20	‖f	‖f	ADJ
ma-234	456	21	‖∞	‖∞	NOUN
ma-234	456	22	,	,	PUNCT
ma-234	456	23	‖f	‖f	ADP
ma-234	456	24	‖mα	‖mα	NUM
ma-234	456	25	1	1	NUM
ma-234	456	26	)	)	PUNCT
ma-234	456	27	.	.	PUNCT
ma-234	457	1	since	since	SCONJ
ma-234	457	2	mdγ	mdγ	PROPN
ma-234	457	3	f	f	PROPN
ma-234	457	4	≤mγf	≤mγf	NUM
ma-234	457	5	,	,	PUNCT
ma-234	457	6	we	we	PRON
ma-234	457	7	have	have	AUX
ma-234	457	8	,	,	PUNCT
ma-234	457	9	for	for	ADP
ma-234	457	10	any	any	DET
ma-234	457	11	x	x	SYM
ma-234	457	12	∈	∈	PROPN
ma-234	457	13	rd	rd	PROPN
ma-234	457	14	,	,	PUNCT
ma-234	457	15	mdγ	mdγ	NOUN
ma-234	457	16	f	f	PROPN
ma-234	457	17	(	(	PUNCT
ma-234	457	18	x	x	NOUN
ma-234	457	19	)	)	PUNCT
ma-234	457	20	≤	≤	ADJ
ma-234	457	21	m.	m.	NOUN
ma-234	457	22	(	(	PUNCT
ma-234	457	23	∗∗	∗∗	NOUN
ma-234	457	24	)	)	PUNCT
ma-234	457	25	b	b	X
ma-234	457	26	)	)	PUNCT
ma-234	457	27	assume	assume	VERB
ma-234	457	28	that	that	SCONJ
ma-234	457	29	f	f	PROPN
ma-234	457	30	6=	6=	PROPN
ma-234	457	31	0	0	NUM
ma-234	457	32	.	.	PUNCT
ma-234	458	1	by	by	ADP
ma-234	458	2	(	(	PUNCT
ma-234	458	3	∗∗	∗∗	PROPN
ma-234	458	4	)	)	PUNCT
ma-234	458	5	,	,	PUNCT
ma-234	458	6	we	we	PRON
ma-234	458	7	have	have	VERB
ma-234	458	8	,	,	PUNCT
ma-234	458	9	for	for	ADP
ma-234	458	10	all	all	DET
ma-234	458	11	x	x	SYM
ma-234	458	12	∈	∈	PROPN
ma-234	458	13	rd	rd	PROPN
ma-234	458	14	,	,	PUNCT
ma-234	458	15	mdγ	mdγ	NOUN
ma-234	458	16	f	f	PROPN
ma-234	458	17	(	(	PUNCT
ma-234	458	18	x	x	X
ma-234	458	19	)	)	PUNCT
ma-234	458	20	∈	∈	PROPN
ma-234	458	21	(	(	PUNCT
ma-234	458	22	0,m	0,m	NOUN
ma-234	458	23	]	]	X
ma-234	458	24	.	.	PUNCT
ma-234	459	1	(	(	PUNCT
ma-234	459	2	∗	∗	NOUN
ma-234	459	3	∗	∗	NOUN
ma-234	459	4	∗	∗	NOUN
ma-234	459	5	)	)	PUNCT
ma-234	460	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	460	2	eur	eur	PROPN
ma-234	460	3	.	.	PUNCT
ma-234	461	1	j.	j.	PROPN
ma-234	461	2	math	math	PROPN
ma-234	461	3	.	.	PUNCT
ma-234	462	1	anal	anal	PROPN
ma-234	462	2	.	.	PUNCT
ma-234	463	1	10.28924	10.28924	NUM
ma-234	463	2	/	/	SYM
ma-234	463	3	ada	ada	PROPN
ma-234	463	4	/	/	SYM
ma-234	463	5	ma.4.16	ma.4.16	PROPN
ma-234	463	6	17(i	17(i	NUM
ma-234	463	7	)	)	PUNCT
ma-234	463	8	let	let	VERB
ma-234	463	9	us	we	PRON
ma-234	463	10	consider	consider	VERB
ma-234	463	11	an	an	DET
ma-234	463	12	integer	integer	NOUN
ma-234	463	13	j	j	PROPN
ma-234	463	14	≥	≥	NUM
ma-234	463	15	0	0	NUM
ma-234	463	16	and	and	CCONJ
ma-234	463	17	set	set	X
ma-234	463	18	ej	ej	PROPN
ma-234	464	1	=	=	PRON
ma-234	464	2	{	{	PUNCT
ma-234	464	3	x	x	PUNCT
ma-234	464	4	∈	∈	PROPN
ma-234	464	5	rd	rd	NOUN
ma-234	464	6	:	:	PUNCT
ma-234	464	7	mdγ	mdγ	NOUN
ma-234	464	8	f	f	PROPN
ma-234	464	9	(	(	PUNCT
ma-234	464	10	x	x	X
ma-234	464	11	)	)	PUNCT
ma-234	464	12	∈	∈	PROPN
ma-234	464	13	(	(	PUNCT
ma-234	464	14	2−j−1	2−j−1	NUM
ma-234	464	15	m	m	NOUN
ma-234	464	16	,	,	PUNCT
ma-234	464	17	2−jm	2−jm	NUM
ma-234	464	18	]	]	PUNCT
ma-234	464	19	}	}	PUNCT
ma-234	464	20	dj	dj	NOUN
ma-234	464	21	=	=	NOUN
ma-234	464	22	{	{	PUNCT
ma-234	464	23	q	q	PUNCT
ma-234	464	24	∈	∈	PROPN
ma-234	465	1	d	d	X
ma-234	465	2	:	:	PUNCT
ma-234	465	3	|q|γ−1	|q|γ−1	NUM
ma-234	465	4	∫	∫	PROPN
ma-234	465	5	q	q	X
ma-234	465	6	|f	|f	PROPN
ma-234	465	7	(	(	PUNCT
ma-234	465	8	y)|dy	y)|dy	NOUN
ma-234	465	9	∈	∈	PROPN
ma-234	465	10	(	(	PUNCT
ma-234	465	11	2−j−1	2−j−1	NUM
ma-234	465	12	m	m	NOUN
ma-234	465	13	,	,	PUNCT
ma-234	465	14	2−jm	2−jm	NUM
ma-234	465	15	]	]	PUNCT
ma-234	465	16	}	}	PUNCT
ma-234	465	17	.	.	PUNCT
ma-234	466	1	we	we	PRON
ma-234	466	2	have	have	VERB
ma-234	466	3	,	,	PUNCT
ma-234	466	4	for	for	ADP
ma-234	466	5	all	all	DET
ma-234	466	6	x	x	SYM
ma-234	466	7	∈	∈	PROPN
ma-234	466	8	rd	rd	NOUN
ma-234	466	9	,	,	PUNCT
ma-234	466	10	x	x	PROPN
ma-234	466	11	∈	∈	PROPN
ma-234	466	12	ej	ej	VERB
ma-234	466	13	⇐	⇐	ADJ
ma-234	466	14	⇒	⇒	NOUN
ma-234	466	15	∃qx	∃qx	X
ma-234	466	16	∈	∈	NOUN
ma-234	466	17	dj	dj	NOUN
ma-234	466	18	:	:	PUNCT
ma-234	466	19	x	x	X
ma-234	466	20	∈	∈	PROPN
ma-234	466	21	qx	qx	VERB
ma-234	466	22	⇐	⇐	ADJ
ma-234	466	23	⇒	⇒	NOUN
ma-234	466	24	x	x	SYM
ma-234	466	25	∈	∈	PROPN
ma-234	466	26	⋃	⋃	VERB
ma-234	466	27	q∈dj	q∈dj	ADJ
ma-234	466	28	q.	q.	NOUN
ma-234	466	29	thus	thus	ADV
ma-234	466	30	ej	ej	ADP
ma-234	467	1	=	=	SYM
ma-234	467	2	⋃	⋃	ADP
ma-234	467	3	q∈dj	q∈dj	PROPN
ma-234	467	4	q.	q.	NOUN
ma-234	467	5	note	note	VERB
ma-234	467	6	that	that	SCONJ
ma-234	467	7	,	,	PUNCT
ma-234	467	8	by	by	ADP
ma-234	467	9	(	(	PUNCT
ma-234	467	10	∗	∗	NOUN
ma-234	467	11	)	)	PUNCT
ma-234	467	12	,	,	PUNCT
ma-234	467	13	sup	sup	NOUN
ma-234	467	14	{	{	PUNCT
ma-234	467	15	`	`	PUNCT
ma-234	467	16	(	(	PUNCT
ma-234	467	17	q	q	X
ma-234	467	18	)	)	PUNCT
ma-234	467	19	:	:	PUNCT
ma-234	467	20	q	q	PUNCT
ma-234	467	21	∈	∈	NOUN
ma-234	467	22	dj	dj	NOUN
ma-234	467	23	}	}	PUNCT
ma-234	467	24	<	<	X
ma-234	467	25	∞	∞	NUM
ma-234	467	26	and	and	CCONJ
ma-234	467	27	let	let	VERB
ma-234	467	28	us	we	PRON
ma-234	467	29	denote	denote	VERB
ma-234	467	30	by	by	ADP
ma-234	467	31	∆j	∆j	NOUN
ma-234	467	32	the	the	DET
ma-234	467	33	set	set	NOUN
ma-234	467	34	of	of	ADP
ma-234	467	35	maximal	maximal	ADJ
ma-234	467	36	elements(for	elements(for	VERB
ma-234	467	37	the	the	DET
ma-234	467	38	inclusion	inclusion	NOUN
ma-234	467	39	)	)	PUNCT
ma-234	467	40	of	of	ADP
ma-234	467	41	dj	dj	NOUN
ma-234	467	42	.	.	PUNCT
ma-234	468	1	it	it	PRON
ma-234	468	2	is	be	AUX
ma-234	468	3	easy	easy	ADJ
ma-234	468	4	to	to	PART
ma-234	468	5	see	see	VERB
ma-234	468	6	that	that	PUNCT
ma-234	468	7	⋃	⋃	NOUN
ma-234	468	8	q∈∆j	q∈∆j	X
ma-234	468	9	q	q	NOUN
ma-234	468	10	=	=	PUNCT
ma-234	468	11	⋃	⋃	VERB
ma-234	468	12	q∈dj	q∈dj	NOUN
ma-234	468	13	q	q	NOUN
ma-234	468	14	=	=	PUNCT
ma-234	468	15	ej	ej	X
ma-234	468	16	∀	∀	NOUN
ma-234	468	17	q′	q′	NOUN
ma-234	468	18	,	,	PUNCT
ma-234	468	19	q′′	q′′	SCONJ
ma-234	468	20	∈	∈	PROPN
ma-234	468	21	∆j	∆j	NOUN
ma-234	468	22	,	,	PUNCT
ma-234	468	23	q	q	PROPN
ma-234	468	24	′	′	NOUN
ma-234	468	25	6=	6=	ADP
ma-234	468	26	q′′	q′′	SCONJ
ma-234	468	27	=	=	SYM
ma-234	468	28	⇒	⇒	NOUN
ma-234	468	29	|q′	|q′	NOUN
ma-234	468	30	∩q′′|	∩q′′|	VERB
ma-234	468	31	=	=	PUNCT
ma-234	468	32	0.moreover	0.moreover	NOUN
ma-234	468	33	,	,	PUNCT
ma-234	468	34	by	by	ADP
ma-234	468	35	the	the	DET
ma-234	468	36	definition	definition	NOUN
ma-234	468	37	of	of	ADP
ma-234	468	38	ej	ej	PROPN
ma-234	468	39	,	,	PUNCT
ma-234	468	40	we	we	PRON
ma-234	468	41	have	have	AUX
ma-234	468	42	,	,	PUNCT
ma-234	468	43	for	for	ADP
ma-234	468	44	all	all	DET
ma-234	468	45	x	x	SYM
ma-234	468	46	∈	∈	PROPN
ma-234	468	47	q	q	NOUN
ma-234	468	48	∈	∈	PROPN
ma-234	468	49	∆j	∆j	PROPN
ma-234	468	50	,	,	PUNCT
ma-234	468	51	|q|γ−1	|q|γ−1	X
ma-234	468	52	∫	∫	PROPN
ma-234	468	53	q	q	X
ma-234	468	54	|f	|f	PROPN
ma-234	468	55	(	(	PUNCT
ma-234	468	56	y)|dy	y)|dy	NOUN
ma-234	468	57	≤mdγ	≤mdγ	PROPN
ma-234	468	58	f	f	X
ma-234	468	59	(	(	PUNCT
ma-234	468	60	x	x	NOUN
ma-234	468	61	)	)	PUNCT
ma-234	468	62	≤	≤	NOUN
ma-234	469	1	2−jm	2−jm	NUM
ma-234	469	2	<	<	SYM
ma-234	469	3	2	2	NUM
ma-234	469	4	|q|γ−1	|q|γ−1	NUM
ma-234	469	5	∫	∫	PROPN
ma-234	469	6	q	q	X
ma-234	469	7	|f	|f	PROPN
ma-234	469	8	(	(	PUNCT
ma-234	469	9	y)|dy	y)|dy	NOUN
ma-234	469	10	and	and	CCONJ
ma-234	469	11	therefore	therefore	ADV
ma-234	469	12	,	,	PUNCT
ma-234	469	13	for	for	ADP
ma-234	469	14	all	all	DET
ma-234	469	15	q	q	PROPN
ma-234	469	16	∈	∈	PROPN
ma-234	469	17	∆j	∆j	PROPN
ma-234	469	18	,	,	PUNCT
ma-234	469	19	∫	∫	PROPN
ma-234	469	20	q	q	X
ma-234	470	1	[	[	PUNCT
ma-234	470	2	mdγ	mdγ	NOUN
ma-234	470	3	f	f	X
ma-234	470	4	(	(	PUNCT
ma-234	470	5	x	x	X
ma-234	470	6	)	)	PUNCT
ma-234	470	7	]	]	X
ma-234	470	8	p	p	X
ma-234	470	9	dx	dx	PROPN
ma-234	470	10	≤	≤	NUM
ma-234	470	11	2p	2p	NUM
ma-234	470	12	|q|	|q|	X
ma-234	470	13	[	[	PUNCT
ma-234	470	14	|q|γ−1	|q|γ−1	NUM
ma-234	470	15	∫	∫	PROPN
ma-234	470	16	q	q	PROPN
ma-234	470	17	|f	|f	PROPN
ma-234	470	18	(	(	PUNCT
ma-234	470	19	y)|dy	y)|dy	NOUN
ma-234	470	20	]	]	X
ma-234	470	21	p	p	X
ma-234	470	22	=	=	NOUN
ma-234	470	23	2p	2p	NOUN
ma-234	470	24	[	[	PUNCT
ma-234	470	25	|q|	|q|	NUM
ma-234	470	26	1	1	NUM
ma-234	470	27	α	α	NUM
ma-234	470	28	−1	−1	NOUN
ma-234	470	29	∫	∫	PROPN
ma-234	470	30	q	q	PROPN
ma-234	470	31	|f	|f	PROPN
ma-234	470	32	(	(	PUNCT
ma-234	470	33	y)|dy	y)|dy	NOUN
ma-234	470	34	]	]	X
ma-234	470	35	p	p	X
ma-234	470	36	.	.	PUNCT
ma-234	471	1	so	so	ADV
ma-234	471	2	we	we	PRON
ma-234	471	3	obtain	obtain	VERB
ma-234	471	4	∫	∫	PROPN
ma-234	471	5	ej	ej	PROPN
ma-234	471	6	[	[	PUNCT
ma-234	471	7	mdγ	mdγ	NOUN
ma-234	471	8	f	f	X
ma-234	471	9	(	(	PUNCT
ma-234	471	10	x	x	X
ma-234	471	11	)	)	PUNCT
ma-234	471	12	]	]	X
ma-234	471	13	p	p	X
ma-234	471	14	dx	dx	PROPN
ma-234	471	15	=	=	SYM
ma-234	471	16	∑	∑	PUNCT
ma-234	471	17	q∈∆j	q∈∆j	PROPN
ma-234	471	18	∫	∫	PROPN
ma-234	471	19	q	q	X
ma-234	472	1	[	[	PUNCT
ma-234	472	2	mdγ	mdγ	NOUN
ma-234	472	3	f	f	X
ma-234	472	4	(	(	PUNCT
ma-234	472	5	x	x	X
ma-234	472	6	)	)	PUNCT
ma-234	472	7	]	]	X
ma-234	472	8	p	p	X
ma-234	472	9	dx	dx	PROPN
ma-234	472	10	≤	≤	NUM
ma-234	472	11	2p	2p	NUM
ma-234	472	12	∑	∑	PUNCT
ma-234	472	13	q∈∆j	q∈∆j	NOUN
ma-234	472	14	[	[	PUNCT
ma-234	472	15	|q|	|q|	NUM
ma-234	472	16	1	1	NUM
ma-234	472	17	α	α	NUM
ma-234	472	18	−1	−1	NOUN
ma-234	472	19	∫	∫	PROPN
ma-234	472	20	q	q	PROPN
ma-234	472	21	|f	|f	PROPN
ma-234	472	22	(	(	PUNCT
ma-234	472	23	y)|dy	y)|dy	NOUN
ma-234	472	24	]	]	X
ma-234	472	25	p	p	X
ma-234	472	26	.	.	PUNCT
ma-234	473	1	(	(	PUNCT
ma-234	473	2	∗	∗	NOUN
ma-234	473	3	∗	∗	X
ma-234	473	4	∗∗	∗∗	PROPN
ma-234	473	5	)	)	PUNCT
ma-234	473	6	(	(	PUNCT
ma-234	473	7	ii	ii	NOUN
ma-234	473	8	)	)	PUNCT
ma-234	473	9	by	by	ADP
ma-234	473	10	(	(	PUNCT
ma-234	473	11	∗	∗	NOUN
ma-234	473	12	∗	∗	NOUN
ma-234	473	13	∗	∗	NOUN
ma-234	473	14	)	)	PUNCT
ma-234	473	15	,	,	PUNCT
ma-234	473	16	we	we	PRON
ma-234	473	17	have	have	VERB
ma-234	473	18	⋃	⋃	NOUN
ma-234	473	19	j≥0	j≥0	ADJ
ma-234	473	20	ej	ej	PROPN
ma-234	473	21	=	=	PROPN
ma-234	473	22	rd	rd	PROPN
ma-234	473	23	.	.	PUNCT
ma-234	474	1	meanwhile	meanwhile	ADV
ma-234	474	2	,	,	PUNCT
ma-234	474	3	the	the	DET
ma-234	474	4	definition	definition	NOUN
ma-234	474	5	of	of	ADP
ma-234	474	6	ej(j	ej(j	NUM
ma-234	474	7	≥	≥	NOUN
ma-234	474	8	0	0	NUM
ma-234	474	9	)	)	PUNCT
ma-234	474	10	shows	show	VERB
ma-234	474	11	that	that	SCONJ
ma-234	474	12	∀	∀	NOUN
ma-234	474	13	j	j	PROPN
ma-234	474	14	′	′	NOUN
ma-234	474	15	,	,	PUNCT
ma-234	474	16	j	j	PROPN
ma-234	474	17	′′	′′	PROPN
ma-234	474	18	∈	∈	PROPN
ma-234	474	19	n	n	CCONJ
ma-234	474	20	,	,	PUNCT
ma-234	474	21	with	with	ADP
ma-234	474	22	j	j	PROPN
ma-234	474	23	′	′	NUM
ma-234	474	24	6=	6=	PROPN
ma-234	474	25	j	j	PROPN
ma-234	474	26	′′,ej	′′,ej	PROPN
ma-234	474	27	′	′	NUM
ma-234	474	28	∩	∩	NOUN
ma-234	474	29	ej	ej	PART
ma-234	474	30	′′	′′	PROPN
ma-234	474	31	=	=	PRON
ma-234	474	32	∅.	∅.	ADV
ma-234	474	33	hence	hence	ADV
ma-234	474	34	we	we	PRON
ma-234	474	35	have	have	VERB
ma-234	474	36	∫	∫	PROPN
ma-234	474	37	rd	rd	PROPN
ma-234	474	38	[	[	PUNCT
ma-234	474	39	mdγ	mdγ	NOUN
ma-234	474	40	f	f	X
ma-234	474	41	(	(	PUNCT
ma-234	474	42	x	x	X
ma-234	474	43	)	)	PUNCT
ma-234	474	44	]	]	X
ma-234	474	45	p	p	X
ma-234	474	46	dx	dx	PROPN
ma-234	475	1	=	=	SYM
ma-234	475	2	∑	∑	PUNCT
ma-234	475	3	j≥0	j≥0	PROPN
ma-234	475	4	∫	∫	PROPN
ma-234	475	5	ej	ej	PROPN
ma-234	475	6	[	[	PUNCT
ma-234	475	7	mdγ	mdγ	NOUN
ma-234	475	8	f	f	X
ma-234	475	9	(	(	PUNCT
ma-234	475	10	x	x	X
ma-234	475	11	)	)	PUNCT
ma-234	475	12	]	]	X
ma-234	475	13	p	p	X
ma-234	475	14	dx	dx	PROPN
ma-234	475	15	and	and	CCONJ
ma-234	475	16	therefore	therefore	ADV
ma-234	475	17	,	,	PUNCT
ma-234	475	18	by	by	ADP
ma-234	475	19	(	(	PUNCT
ma-234	475	20	∗	∗	NOUN
ma-234	475	21	∗	∗	NOUN
ma-234	475	22	∗∗	∗∗	PROPN
ma-234	475	23	)	)	PUNCT
ma-234	475	24	,	,	PUNCT
ma-234	475	25	‖mdγ	‖mdγ	PROPN
ma-234	475	26	f	f	PROPN
ma-234	475	27	‖p	‖p	PROPN
ma-234	475	28	≤	≤	ADV
ma-234	475	29	2	2	NUM
ma-234	475	30	∑	∑	NOUN
ma-234	475	31	j≥0	j≥0	PROPN
ma-234	475	32	∑	∑	ADP
ma-234	475	33	q∈∆j	q∈∆j	NOUN
ma-234	475	34	[	[	PUNCT
ma-234	475	35	|q|	|q|	NUM
ma-234	475	36	1	1	NUM
ma-234	475	37	α	α	NUM
ma-234	475	38	−1	−1	NOUN
ma-234	475	39	∫	∫	PROPN
ma-234	475	40	q	q	PROPN
ma-234	475	41	|f	|f	PROPN
ma-234	475	42	(	(	PUNCT
ma-234	475	43	y)|dy	y)|dy	NOUN
ma-234	475	44	]	]	PUNCT
ma-234	475	45	p	p	PROPN
ma-234	475	46	1	1	NUM
ma-234	475	47	p	p	NOUN
ma-234	475	48	.	.	PUNCT
ma-234	476	1	(	(	PUNCT
ma-234	476	2	∗	∗	NOUN
ma-234	476	3	∗	∗	NOUN
ma-234	476	4	∗	∗	NOUN
ma-234	476	5	∗	∗	NOUN
ma-234	476	6	∗	∗	NOUN
ma-234	476	7	)	)	PUNCT
ma-234	477	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	477	2	eur	eur	PROPN
ma-234	477	3	.	.	PUNCT
ma-234	478	1	j.	j.	PROPN
ma-234	478	2	math	math	PROPN
ma-234	478	3	.	.	PUNCT
ma-234	479	1	anal	anal	PROPN
ma-234	479	2	.	.	PUNCT
ma-234	480	1	10.28924	10.28924	NUM
ma-234	480	2	/	/	SYM
ma-234	480	3	ada	ada	PROPN
ma-234	480	4	/	/	SYM
ma-234	480	5	ma.4.16	ma.4.16	PROPN
ma-234	481	1	18note	18note	NUM
ma-234	481	2	that	that	PRON
ma-234	481	3	,	,	PUNCT
ma-234	481	4	if	if	SCONJ
ma-234	481	5	j	j	PROPN
ma-234	481	6	′	′	NOUN
ma-234	481	7	and	and	CCONJ
ma-234	481	8	j	j	PROPN
ma-234	481	9	′′	′′	PROPN
ma-234	481	10	are	be	AUX
ma-234	481	11	two	two	NUM
ma-234	481	12	integer	integer	NOUN
ma-234	481	13	such	such	ADJ
ma-234	481	14	that	that	PRON
ma-234	481	15	j	j	PROPN
ma-234	481	16	′′	′′	PROPN
ma-234	481	17	>	>	X
ma-234	481	18	j	j	PROPN
ma-234	481	19	′	′	NUM
ma-234	481	20	≥	≥	NOUN
ma-234	481	21	0	0	PUNCT
ma-234	482	1	then	then	ADV
ma-234	482	2	∀	∀	X
ma-234	482	3	(	(	PUNCT
ma-234	482	4	q′	q′	NOUN
ma-234	482	5	,	,	PUNCT
ma-234	482	6	q′′	q′′	ADV
ma-234	482	7	)	)	PUNCT
ma-234	482	8	∈	∈	PROPN
ma-234	482	9	∆j	∆j	NOUN
ma-234	483	1	′	′	NUM
ma-234	483	2	×	×	NOUN
ma-234	483	3	∆j	∆j	PROPN
ma-234	483	4	′′	′′	PROPN
ma-234	483	5	,	,	PUNCT
ma-234	483	6	q	q	PROPN
ma-234	483	7	′	′	NOUN
ma-234	483	8	6=	6=	ADP
ma-234	484	1	q′′	q′′	ADV
ma-234	484	2	and	and	CCONJ
ma-234	484	3	so	so	ADV
ma-234	484	4	{	{	PUNCT
ma-234	484	5	qk	qk	INTJ
ma-234	484	6	,	,	PUNCT
ma-234	484	7	m	m	VERB
ma-234	484	8	:	:	PUNCT
ma-234	484	9	qk	qk	X
ma-234	484	10	,	,	PUNCT
ma-234	484	11	m	m	PROPN
ma-234	484	12	∈	∈	NOUN
ma-234	484	13	∆j	∆j	NOUN
ma-234	484	14	′	′	ADJ
ma-234	484	15	}	}	PUNCT
ma-234	484	16	∩	∩	NOUN
ma-234	484	17	{	{	PUNCT
ma-234	484	18	qk	qk	NOUN
ma-234	484	19	,	,	PUNCT
ma-234	484	20	m	m	VERB
ma-234	484	21	:	:	PUNCT
ma-234	484	22	qk	qk	X
ma-234	484	23	,	,	PUNCT
ma-234	484	24	m	m	PROPN
ma-234	484	25	∈	∈	PROPN
ma-234	484	26	∆j	∆j	PROPN
ma-234	484	27	′′	′′	PROPN
ma-234	484	28	}	}	PUNCT
ma-234	484	29	=	=	VERB
ma-234	484	30	∅	∅	NOUN
ma-234	484	31	,	,	PUNCT
ma-234	484	32	k	k	PROPN
ma-234	484	33	∈	∈	PROPN
ma-234	484	34	zd	zd	PROPN
ma-234	484	35	,	,	PUNCT
ma-234	484	36	m	m	VERB
ma-234	484	37	∈	∈	NOUN
ma-234	484	38	z.therefore∑	z.therefore∑	NOUN
ma-234	484	39	j≥0	j≥0	PROPN
ma-234	484	40	∑	∑	PUNCT
ma-234	484	41	q∈∆j	q∈∆j	NOUN
ma-234	484	42	[	[	PUNCT
ma-234	484	43	|q|	|q|	NUM
ma-234	484	44	1	1	NUM
ma-234	484	45	α	α	NUM
ma-234	484	46	−1	−1	NOUN
ma-234	484	47	∫	∫	PROPN
ma-234	484	48	q	q	PROPN
ma-234	484	49	|f	|f	PROPN
ma-234	484	50	(	(	PUNCT
ma-234	484	51	y)|dy	y)|dy	NOUN
ma-234	484	52	]	]	X
ma-234	484	53	p	p	X
ma-234	484	54	=	=	X
ma-234	484	55	∑	∑	PUNCT
ma-234	484	56	m∈z	m∈z	NOUN
ma-234	484	57	∑	∑	PROPN
ma-234	484	58	k∈zd	k∈zd	NOUN
ma-234	484	59	:	:	PUNCT
ma-234	484	60	qk	qk	INTJ
ma-234	484	61	,	,	PUNCT
ma-234	484	62	m∈	m∈	PROPN
ma-234	484	63	⋃	⋃	NOUN
ma-234	484	64	j≥0	j≥0	ADJ
ma-234	484	65	∆j	∆j	PROPN
ma-234	484	66	[	[	PUNCT
ma-234	484	67	|qk	|qk	NUM
ma-234	484	68	,	,	PUNCT
ma-234	484	69	m|	m|	NOUN
ma-234	484	70	1	1	NUM
ma-234	484	71	α	α	NOUN
ma-234	484	72	−1	−1	NOUN
ma-234	484	73	∫	∫	PROPN
ma-234	484	74	qk	qk	PROPN
ma-234	484	75	,	,	PUNCT
ma-234	484	76	m	m	PROPN
ma-234	484	77	|f	|f	PROPN
ma-234	484	78	(	(	PUNCT
ma-234	484	79	y)|dy	y)|dy	NOUN
ma-234	484	80	]	]	PUNCT
ma-234	484	81	p	p	X
ma-234	484	82	≤	≤	NOUN
ma-234	484	83	‖f	‖f	DET
ma-234	484	84	‖pmα	‖pmα	PROPN
ma-234	484	85	1,p	1,p	PROPN
ma-234	484	86	.	.	PUNCT
ma-234	485	1	this	this	DET
ma-234	485	2	inequality	inequality	NOUN
ma-234	485	3	combined	combine	VERB
ma-234	485	4	with	with	ADP
ma-234	485	5	(	(	PUNCT
ma-234	485	6	∗	∗	NOUN
ma-234	485	7	∗	∗	NOUN
ma-234	485	8	∗	∗	NOUN
ma-234	485	9	∗	∗	NOUN
ma-234	485	10	∗	∗	NOUN
ma-234	485	11	)	)	PUNCT
ma-234	485	12	gives	give	VERB
ma-234	485	13	‖mdγ	‖mdγ	PROPN
ma-234	485	14	f	f	PROPN
ma-234	485	15	‖p	‖p	PROPN
ma-234	485	16	≤	≤	ADJ
ma-234	485	17	2	2	NUM
ma-234	485	18	‖f	‖f	ADP
ma-234	485	19	‖mα	‖mα	NUM
ma-234	485	20	1,p	1,p	NOUN
ma-234	485	21	.	.	PUNCT
ma-234	486	1	2	2	NUM
ma-234	486	2	)	)	PUNCT
ma-234	486	3	by	by	ADP
ma-234	486	4	lemma	lemma	PROPN
ma-234	486	5	5.2	5.2	NUM
ma-234	486	6	,	,	PUNCT
ma-234	486	7	there	there	PRON
ma-234	486	8	exists	exist	VERB
ma-234	486	9	a	a	DET
ma-234	486	10	sequence	sequence	NOUN
ma-234	486	11	(	(	PUNCT
ma-234	486	12	fn)n≥1	fn)n≥1	NOUN
ma-234	486	13	of	of	ADP
ma-234	486	14	elements	element	NOUN
ma-234	486	15	of	of	ADP
ma-234	486	16	l∞c	l∞c	NOUN
ma-234	486	17	∩mα	∩mα	PROPN
ma-234	486	18	q	q	PROPN
ma-234	486	19	,	,	PUNCT
ma-234	486	20	p	p	X
ma-234	486	21	such	such	ADJ
ma-234	486	22	that	that	SCONJ
ma-234	486	23	(	(	PUNCT
ma-234	486	24	|fn|)n≥1	|fn|)n≥1	PROPN
ma-234	486	25	↑	↑	NOUN
ma-234	486	26	|f	|f	PROPN
ma-234	486	27	|almost	|almost	NOUN
ma-234	486	28	everywhere	everywhere	ADV
ma-234	486	29	and	and	CCONJ
ma-234	486	30	lim	lim	PROPN
ma-234	486	31	n→∞	n→∞	NUM
ma-234	486	32	‖f	‖f	PUNCT
ma-234	486	33	−	−	PROPN
ma-234	486	34	fn‖mα	fn‖mα	NOUN
ma-234	486	35	q	q	NOUN
ma-234	486	36	,	,	PUNCT
ma-234	486	37	p	p	NOUN
ma-234	486	38	=	=	NOUN
ma-234	486	39	0	0	NUM
ma-234	486	40	.	.	PUNCT
ma-234	487	1	thus	thus	ADV
ma-234	487	2	,	,	PUNCT
ma-234	487	3	the	the	DET
ma-234	487	4	result	result	NOUN
ma-234	487	5	obtained	obtain	VERB
ma-234	487	6	in	in	ADP
ma-234	487	7	point	point	NOUN
ma-234	487	8	1	1	NUM
ma-234	487	9	)	)	PUNCT
ma-234	487	10	implies	imply	VERB
ma-234	487	11	that	that	PROPN
ma-234	487	12	∥∥mdγ	∥∥mdγ	SYM
ma-234	487	13	fn∥∥p	fn∥∥p	ADJ
ma-234	487	14	≤	≤	ADV
ma-234	487	15	2	2	NUM
ma-234	487	16	‖fn‖mα	‖fn‖mα	NUM
ma-234	487	17	1,p	1,p	PROPN
ma-234	487	18	≤	≤	ADV
ma-234	487	19	2	2	NUM
ma-234	487	20	‖f	‖f	ADP
ma-234	487	21	‖mα	‖mα	NUM
ma-234	487	22	1,p	1,p	PROPN
ma-234	487	23	,	,	PUNCT
ma-234	487	24	n	n	X
ma-234	487	25	≥	≥	NUM
ma-234	487	26	1	1	NUM
ma-234	487	27	0	0	NUM
ma-234	487	28	≤	≤	NUM
ma-234	487	29	(	(	PUNCT
ma-234	487	30	mdγ	mdγ	NOUN
ma-234	487	31	fn	fn	NOUN
ma-234	487	32	)	)	PUNCT
ma-234	487	33	n≥1	n≥1	NOUN
ma-234	487	34	↑mdγ	↑mdγ	NOUN
ma-234	487	35	f	f	NOUN
ma-234	487	36	and	and	CCONJ
ma-234	487	37	so	so	ADV
ma-234	487	38	(	(	PUNCT
ma-234	487	39	∥∥mdγ	∥∥mdγ	PUNCT
ma-234	487	40	fn∥∥p)n≥1	fn∥∥p)n≥1	PROPN
ma-234	487	41	↑	↑	NOUN
ma-234	487	42	∥∥mdγ	∥∥mdγ	PUNCT
ma-234	488	1	f	f	PROPN
ma-234	488	2	∥∥pand	∥∥pand	PROPN
ma-234	488	3	therefore	therefore	ADV
ma-234	488	4	‖mdγ	‖mdγ	PROPN
ma-234	488	5	f	f	PROPN
ma-234	488	6	‖p	‖p	PROPN
ma-234	488	7	≤	≤	ADJ
ma-234	488	8	2	2	NUM
ma-234	488	9	‖f	‖f	ADP
ma-234	488	10	‖mα	‖mα	NUM
ma-234	488	11	1,p	1,p	NOUN
ma-234	488	12	.the	.the	DET
ma-234	488	13	proof	proof	NOUN
ma-234	488	14	is	be	AUX
ma-234	488	15	complete	complete	ADJ
ma-234	488	16	.	.	PUNCT
ma-234	489	1	�	�	PROPN
ma-234	489	2	now	now	ADV
ma-234	489	3	we	we	PRON
ma-234	489	4	prove	prove	VERB
ma-234	489	5	theorem	theorem	VERB
ma-234	489	6	2.1	2.1	NUM
ma-234	489	7	thanks	thank	NOUN
ma-234	489	8	to	to	ADP
ma-234	489	9	lemma	lemma	PROPN
ma-234	489	10	5.1	5.1	NUM
ma-234	489	11	,	,	PUNCT
ma-234	489	12	lemma	lemma	PROPN
ma-234	489	13	5.3	5.3	NUM
ma-234	489	14	and	and	CCONJ
ma-234	489	15	corollary	corollary	ADJ
ma-234	489	16	3.6	3.6	NUM
ma-234	489	17	.	.	PUNCT
ma-234	490	1	proof	proof	NOUN
ma-234	490	2	of	of	ADP
ma-234	490	3	theorem	theorem	ADJ
ma-234	490	4	2.1let	2.1let	NUM
ma-234	490	5	f	f	AUX
ma-234	490	6	be	be	AUX
ma-234	490	7	in	in	ADP
ma-234	490	8	mα	mα	PROPN
ma-234	490	9	1,p	1,p	PROPN
ma-234	490	10	.	.	PUNCT
ma-234	491	1	by	by	ADP
ma-234	491	2	lemma	lemma	PROPN
ma-234	491	3	5.1	5.1	NUM
ma-234	491	4	and	and	CCONJ
ma-234	491	5	lemma	lemma	PROPN
ma-234	491	6	5.3	5.3	NUM
ma-234	491	7	,	,	PUNCT
ma-234	491	8	we	we	PRON
ma-234	491	9	have	have	VERB
ma-234	491	10	‖mγf	‖mγf	NOUN
ma-234	491	11	‖p	‖p	PROPN
ma-234	491	12	≤	≤	NOUN
ma-234	491	13	3d(1−γ	3d(1−γ	NUM
ma-234	491	14	)	)	PUNCT
ma-234	491	15	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-234	492	1	max	max	PROPN
ma-234	492	2	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	PROPN
ma-234	492	3	md	md	PROPN
ma-234	492	4	t	t	PROPN
ma-234	492	5	γ	γ	X
ma-234	492	6	f	f	PROPN
ma-234	492	7	∥∥∥∥∥	∥∥∥∥∥	PROPN
ma-234	492	8	p	p	PROPN
ma-234	492	9	≤	≤	PROPN
ma-234	492	10	3d(1−γ	3d(1−γ	NUM
ma-234	492	11	)	)	PUNCT
ma-234	492	12	∑	∑	PUNCT
ma-234	492	13	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	X
ma-234	492	14	∥∥∥mdtγ	∥∥∥mdtγ	X
ma-234	492	15	f	f	PROPN
ma-234	492	16	∥∥∥	∥∥∥	PROPN
ma-234	492	17	p	p	PROPN
ma-234	492	18	≤	≤	NOUN
ma-234	492	19	3d(1−γ	3d(1−γ	NUM
ma-234	492	20	)	)	PUNCT
ma-234	492	21	∑	∑	PUNCT
ma-234	492	22	t∈{−1/3,0,1/3}d	t∈{−1/3,0,1/3}d	NOUN
ma-234	492	23	2	2	NUM
ma-234	492	24	‖f	‖f	ADP
ma-234	492	25	‖mα	‖mα	NUM
ma-234	492	26	1,p(dt	1,p(dt	NUM
ma-234	492	27	)	)	PUNCT
ma-234	492	28	.	.	PUNCT
ma-234	493	1	note	note	VERB
ma-234	493	2	that	that	SCONJ
ma-234	493	3	the	the	DET
ma-234	493	4	hypotheses	hypothesis	NOUN
ma-234	493	5	imply	imply	VERB
ma-234	493	6	that	that	SCONJ
ma-234	493	7	p	p	X
ma-234	493	8	<	<	X
ma-234	493	9	∞	∞	NUM
ma-234	493	10	and	and	CCONJ
ma-234	493	11	so	so	ADV
ma-234	493	12	corollary	corollary	ADJ
ma-234	493	13	3.6	3.6	NUM
ma-234	493	14	leads	lead	NOUN
ma-234	493	15	to	to	ADP
ma-234	493	16	‖mγf	‖mγf	PROPN
ma-234	493	17	‖p	‖p	PROPN
ma-234	493	18	≤	≤	NUM
ma-234	494	1	3d(1−γ)2	3d(1−γ)2	NUM
ma-234	494	2	d	d	NOUN
ma-234	494	3	(	(	PUNCT
ma-234	494	4	2−	2−	NUM
ma-234	494	5	1	1	NUM
ma-234	494	6	p	p	NOUN
ma-234	494	7	)	)	PUNCT
ma-234	494	8	‖f	‖f	PUNCT
ma-234	494	9	‖mα	‖mα	NUM
ma-234	494	10	1,p	1,p	NOUN
ma-234	494	11	]	]	PUNCT
ma-234	494	12	(	(	PUNCT
ma-234	494	13	{	{	PUNCT
ma-234	494	14	−1/3	−1/3	ADJ
ma-234	494	15	,	,	PUNCT
ma-234	494	16	0	0	NUM
ma-234	494	17	,	,	PUNCT
ma-234	494	18	1/3}d	1/3}d	NUM
ma-234	494	19	)	)	PUNCT
ma-234	494	20	=	=	SYM
ma-234	495	1	2	2	NUM
ma-234	495	2	d	d	NOUN
ma-234	495	3	(	(	PUNCT
ma-234	495	4	2−	2−	NUM
ma-234	495	5	1	1	NUM
ma-234	495	6	p	p	NOUN
ma-234	495	7	)	)	PUNCT
ma-234	495	8	3d(2−γ	3d(2−γ	NUM
ma-234	495	9	)	)	PUNCT
ma-234	495	10	‖f	‖f	PUNCT
ma-234	495	11	‖mα	‖mα	NUM
ma-234	495	12	1,p	1,p	NOUN
ma-234	495	13	.	.	PUNCT
ma-234	496	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	496	2	eur	eur	PROPN
ma-234	496	3	.	.	PUNCT
ma-234	497	1	j.	j.	PROPN
ma-234	497	2	math	math	PROPN
ma-234	497	3	.	.	PUNCT
ma-234	498	1	anal	anal	PROPN
ma-234	498	2	.	.	PUNCT
ma-234	499	1	10.28924	10.28924	NUM
ma-234	499	2	/	/	SYM
ma-234	499	3	ada	ada	PROPN
ma-234	499	4	/	/	SYM
ma-234	499	5	ma.4.16	ma.4.16	PROPN
ma-234	499	6	19the	19the	DET
ma-234	499	7	proof	proof	NOUN
ma-234	499	8	is	be	AUX
ma-234	499	9	complete	complete	ADJ
ma-234	499	10	.	.	PUNCT
ma-234	500	1	�	�	PROPN
ma-234	500	2	as	as	ADP
ma-234	500	3	an	an	DET
ma-234	500	4	immediate	immediate	ADJ
ma-234	500	5	consequence	consequence	NOUN
ma-234	500	6	of	of	ADP
ma-234	500	7	theorem	theorem	NOUN
ma-234	500	8	2.1	2.1	NUM
ma-234	500	9	we	we	PRON
ma-234	500	10	can	can	AUX
ma-234	500	11	now	now	ADV
ma-234	500	12	prove	prove	VERB
ma-234	500	13	theorem	theorem	ADJ
ma-234	500	14	2.2	2.2	NUM
ma-234	500	15	.	.	PUNCT
ma-234	501	1	proof	proof	NOUN
ma-234	501	2	of	of	ADP
ma-234	501	3	theorem	theorem	ADJ
ma-234	501	4	2.2let	2.2let	NUM
ma-234	501	5	f	f	AUX
ma-234	501	6	be	be	AUX
ma-234	501	7	in	in	ADP
ma-234	501	8	mα	mα	PROPN
ma-234	501	9	1,p	1,p	PROPN
ma-234	501	10	.	.	PUNCT
ma-234	502	1	note	note	VERB
ma-234	502	2	that	that	SCONJ
ma-234	502	3	the	the	DET
ma-234	502	4	hypotheses	hypothesis	NOUN
ma-234	502	5	imply	imply	VERB
ma-234	502	6	that	that	SCONJ
ma-234	502	7	p	p	X
ma-234	502	8	<	<	X
ma-234	502	9	∞.	∞.	PROPN
ma-234	502	10	by	by	ADP
ma-234	502	11	[	[	X
ma-234	502	12	12	12	NUM
ma-234	502	13	,	,	PUNCT
ma-234	502	14	theorem	theorem	VERB
ma-234	502	15	1	1	NUM
ma-234	502	16	]	]	PUNCT
ma-234	502	17	,	,	PUNCT
ma-234	502	18	we	we	PRON
ma-234	502	19	have	have	VERB
ma-234	502	20	‖iγf	‖iγf	PROPN
ma-234	502	21	‖p	‖p	PROPN
ma-234	502	22	≤	≤	PROPN
ma-234	502	23	d	d	PROPN
ma-234	502	24	‖mγf	‖mγf	PROPN
ma-234	502	25	‖p	‖p	NOUN
ma-234	502	26	,	,	PUNCT
ma-234	502	27	where	where	SCONJ
ma-234	502	28	d	d	NOUN
ma-234	502	29	is	be	AUX
ma-234	502	30	a	a	DET
ma-234	502	31	real	real	ADV
ma-234	502	32	constant	constant	ADJ
ma-234	502	33	not	not	PART
ma-234	502	34	depending	depend	VERB
ma-234	502	35	on	on	ADP
ma-234	502	36	f	f	PROPN
ma-234	502	37	.	.	PUNCT
ma-234	503	1	therefore	therefore	ADV
ma-234	503	2	,	,	PUNCT
ma-234	503	3	theorem	theorem	VERB
ma-234	503	4	2.1	2.1	NUM
ma-234	503	5	provides	provide	VERB
ma-234	503	6	the	the	DET
ma-234	503	7	desiredinequality	desiredinequality	NOUN
ma-234	503	8	.	.	PUNCT
ma-234	504	1	�	�	PROPN
ma-234	504	2	6	6	NUM
ma-234	504	3	.	.	PUNCT
ma-234	504	4	application	application	NOUN
ma-234	504	5	theorem	theorem	VERB
ma-234	504	6	2.2	2.2	NUM
ma-234	504	7	,	,	PUNCT
ma-234	504	8	theorem	theorem	VERB
ma-234	504	9	2.5	2.5	NUM
ma-234	504	10	and	and	CCONJ
ma-234	504	11	the	the	DET
ma-234	504	12	boundedness	boundedness	NOUN
ma-234	504	13	properties	property	NOUN
ma-234	504	14	of	of	ADP
ma-234	504	15	riesz	riesz	NOUN
ma-234	504	16	tranforms	tranform	NOUN
ma-234	504	17	in	in	ADP
ma-234	504	18	lebesgue	lebesgue	NOUN
ma-234	504	19	spaceslead	spaceslead	NOUN
ma-234	504	20	to	to	ADP
ma-234	504	21	the	the	DET
ma-234	504	22	following	following	ADJ
ma-234	504	23	result	result	NOUN
ma-234	504	24	,	,	PUNCT
ma-234	504	25	which	which	PRON
ma-234	504	26	contains	contain	VERB
ma-234	504	27	theorem	theorem	ADJ
ma-234	504	28	2.6	2.6	NUM
ma-234	504	29	.	.	PUNCT
ma-234	505	1	proposition	proposition	NOUN
ma-234	505	2	6.1	6.1	NUM
ma-234	505	3	.	.	PUNCT
ma-234	506	1	let	let	VERB
ma-234	506	2	us	we	PRON
ma-234	506	3	assume	assume	VERB
ma-234	506	4	that	that	SCONJ
ma-234	506	5	d	d	PROPN
ma-234	506	6	≥	≥	NUM
ma-234	506	7	3	3	NUM
ma-234	506	8	,	,	PUNCT
ma-234	506	9	1	1	NUM
ma-234	506	10	≤	≤	NUM
ma-234	506	11	q	q	ADJ
ma-234	506	12	≤	≤	NUM
ma-234	506	13	α	α	NOUN
ma-234	506	14	<	<	X
ma-234	506	15	d	d	X
ma-234	506	16	,	,	PUNCT
ma-234	506	17	1	1	NUM
ma-234	506	18	p	p	NOUN
ma-234	506	19	=	=	NOUN
ma-234	507	1	1	1	NUM
ma-234	507	2	α	α	NOUN
ma-234	507	3	−	−	NOUN
ma-234	507	4	1	1	NUM
ma-234	507	5	d	d	NOUN
ma-234	507	6	and	and	CCONJ
ma-234	507	7	f	f	PROPN
ma-234	507	8	is	be	AUX
ma-234	507	9	an	an	DET
ma-234	507	10	element	element	NOUN
ma-234	507	11	of	of	ADP
ma-234	507	12	the	the	DET
ma-234	507	13	bourgain	bourgain	NOUN
ma-234	507	14	-	-	PUNCT
ma-234	507	15	morrey	morrey	NOUN
ma-234	507	16	space	space	NOUN
ma-234	507	17	mα	mα	PROPN
ma-234	507	18	q	q	NOUN
ma-234	507	19	,	,	PUNCT
ma-234	507	20	p	p	NOUN
ma-234	507	21	.	.	PUNCT
ma-234	508	1	then	then	ADV
ma-234	508	2	1	1	X
ma-234	508	3	)	)	PUNCT
ma-234	508	4	for	for	ADP
ma-234	508	5	1	1	NUM
ma-234	508	6	≤	≤	NUM
ma-234	508	7	j	j	PROPN
ma-234	508	8	≤	≤	PROPN
ma-234	508	9	d	d	PROPN
ma-234	508	10	,	,	PUNCT
ma-234	508	11	the	the	DET
ma-234	508	12	function	function	NOUN
ma-234	508	13	fj	fj	PROPN
ma-234	508	14	=	=	SYM
ma-234	508	15	rj	rj	PROPN
ma-234	508	16	(	(	PUNCT
ma-234	508	17	i	i	NOUN
ma-234	508	18	1	1	NUM
ma-234	508	19	d	d	PROPN
ma-234	508	20	f	f	PROPN
ma-234	508	21	)	)	PUNCT
ma-234	508	22	belongs	belong	VERB
ma-234	508	23	to	to	ADP
ma-234	508	24	lp	lp	NOUN
ma-234	508	25	2	2	NUM
ma-234	508	26	)	)	PUNCT
ma-234	508	27	there	there	PRON
ma-234	508	28	exists	exist	VERB
ma-234	508	29	a	a	DET
ma-234	508	30	real	real	ADJ
ma-234	508	31	constant	constant	ADJ
ma-234	508	32	cd	cd	NOUN
ma-234	508	33	depending	depend	VERB
ma-234	508	34	only	only	ADV
ma-234	508	35	on	on	ADP
ma-234	508	36	d	d	PROPN
ma-234	508	37	and	and	CCONJ
ma-234	508	38	such	such	ADJ
ma-234	508	39	that	that	SCONJ
ma-234	508	40	f	f	PROPN
ma-234	509	1	=	=	PUNCT
ma-234	509	2	(	(	PUNCT
ma-234	509	3	cd	cd	PROPN
ma-234	509	4	fj	fj	PROPN
ma-234	509	5	)	)	PUNCT
ma-234	509	6	1≤j≤d	1≤j≤d	PROPN
ma-234	509	7	is	be	AUX
ma-234	509	8	a	a	DET
ma-234	509	9	solution	solution	NOUN
ma-234	509	10	in	in	ADP
ma-234	509	11	(	(	PUNCT
ma-234	509	12	lp)d	lp)d	PROPN
ma-234	509	13	of	of	ADP
ma-234	509	14	the	the	DET
ma-234	509	15	equation	equation	NOUN
ma-234	509	16	(	(	PUNCT
ma-234	509	17	11	11	NUM
ma-234	509	18	)	)	PUNCT
ma-234	509	19	.	.	PUNCT
ma-234	510	1	proof	proof	NOUN
ma-234	510	2	.	.	PUNCT
ma-234	511	1	note	note	VERB
ma-234	511	2	that	that	SCONJ
ma-234	511	3	,	,	PUNCT
ma-234	511	4	the	the	DET
ma-234	511	5	hypotheses	hypothesis	NOUN
ma-234	511	6	imply	imply	VERB
ma-234	511	7	that	that	SCONJ
ma-234	511	8	1	1	NUM
ma-234	511	9	≤	≤	NUM
ma-234	511	10	q	q	ADJ
ma-234	511	11	≤	≤	NUM
ma-234	511	12	α	α	NOUN
ma-234	511	13	<	<	X
ma-234	511	14	p	p	X
ma-234	511	15	<	<	X
ma-234	511	16	∞.1	∞.1	NOUN
ma-234	511	17	)	)	PUNCT
ma-234	511	18	since	since	SCONJ
ma-234	511	19	mα	mα	PROPN
ma-234	511	20	q	q	NOUN
ma-234	511	21	,	,	PUNCT
ma-234	511	22	p	p	PROPN
ma-234	511	23	⊂	⊂	PROPN
ma-234	511	24	mα	mα	PROPN
ma-234	511	25	1,p	1,p	PROPN
ma-234	511	26	(	(	PUNCT
ma-234	511	27	see	see	VERB
ma-234	511	28	(	(	PUNCT
ma-234	511	29	1	1	NUM
ma-234	511	30	)	)	PUNCT
ma-234	511	31	)	)	PUNCT
ma-234	511	32	,	,	PUNCT
ma-234	511	33	theorem	theorem	VERB
ma-234	511	34	2.2	2.2	NUM
ma-234	511	35	implies	imply	VERB
ma-234	511	36	that	that	SCONJ
ma-234	511	37	i	i	PRON
ma-234	511	38	1	1	NUM
ma-234	511	39	d	d	NOUN
ma-234	511	40	f	f	X
ma-234	511	41	∈	∈	PROPN
ma-234	511	42	lp	lp	NOUN
ma-234	511	43	.	.	PUNCT
ma-234	512	1	furthermore	furthermore	ADV
ma-234	512	2	,	,	PUNCT
ma-234	512	3	it	it	PRON
ma-234	512	4	is	be	AUX
ma-234	512	5	wellknown	wellknown	ADJ
ma-234	512	6	that	that	SCONJ
ma-234	512	7	the	the	DET
ma-234	512	8	riesz	riesz	NOUN
ma-234	512	9	tranform	tranform	VERB
ma-234	512	10	rj	rj	PROPN
ma-234	512	11	is	be	AUX
ma-234	512	12	bounded	bound	VERB
ma-234	512	13	on	on	ADP
ma-234	512	14	lp	lp	PROPN
ma-234	512	15	,	,	PUNCT
ma-234	512	16	for	for	ADP
ma-234	512	17	1	1	NUM
ma-234	512	18	≤	≤	NUM
ma-234	513	1	j	j	PROPN
ma-234	513	2	≤	≤	PROPN
ma-234	513	3	d	d	NOUN
ma-234	513	4	.	.	PUNCT
ma-234	514	1	therefore	therefore	ADV
ma-234	514	2	,	,	PUNCT
ma-234	514	3	we	we	PRON
ma-234	514	4	deduce	deduce	VERB
ma-234	514	5	that	that	SCONJ
ma-234	514	6	fj	fj	PROPN
ma-234	514	7	=	=	SYM
ma-234	514	8	rj	rj	PROPN
ma-234	514	9	(	(	PUNCT
ma-234	514	10	i	i	NOUN
ma-234	514	11	1	1	NUM
ma-234	514	12	d	d	PROPN
ma-234	514	13	f	f	PROPN
ma-234	514	14	)	)	PUNCT
ma-234	514	15	belongs	belong	VERB
ma-234	514	16	to	to	ADP
ma-234	514	17	lp	lp	NOUN
ma-234	514	18	.2	.2	NUM
ma-234	514	19	)	)	PUNCT
ma-234	514	20	a	a	X
ma-234	514	21	)	)	PUNCT
ma-234	514	22	let	let	VERB
ma-234	514	23	ϕ	ϕ	NOUN
ma-234	514	24	be	be	AUX
ma-234	514	25	any	any	DET
ma-234	514	26	element	element	NOUN
ma-234	514	27	of	of	ADP
ma-234	514	28	c∞c	c∞c	ADJ
ma-234	514	29	.	.	PUNCT
ma-234	515	1	for	for	ADP
ma-234	515	2	1	1	NUM
ma-234	515	3	≤	≤	NUM
ma-234	515	4	j	j	PROPN
ma-234	515	5	≤	≤	PROPN
ma-234	515	6	d	d	PROPN
ma-234	515	7	,	,	PUNCT
ma-234	515	8	the	the	DET
ma-234	515	9	boundedness	boundedness	NOUN
ma-234	515	10	properties	property	NOUN
ma-234	515	11	of	of	ADP
ma-234	515	12	rj	rj	PROPN
ma-234	515	13	and	and	CCONJ
ma-234	515	14	i	i	PRON
ma-234	515	15	1	1	NUM
ma-234	515	16	d	d	NOUN
ma-234	515	17	showthat	showthat	NOUN
ma-234	515	18	ψj	ψj	ADP
ma-234	515	19	=	=	SYM
ma-234	515	20	rj	rj	PROPN
ma-234	515	21	(	(	PUNCT
ma-234	515	22	i	i	NOUN
ma-234	515	23	1	1	NUM
ma-234	515	24	d	d	PROPN
ma-234	515	25	ϕ	ϕ	PROPN
ma-234	515	26	)	)	PUNCT
ma-234	515	27	belongs	belong	VERB
ma-234	515	28	to	to	ADP
ma-234	515	29	⋂	⋂	PROPN
ma-234	515	30	r	r	X
ma-234	515	31	>	>	X
ma-234	515	32	d	d	PROPN
ma-234	515	33	d−1	d−1	PROPN
ma-234	515	34	lr	lr	PROPN
ma-234	515	35	.	.	PUNCT
ma-234	516	1	since	since	SCONJ
ma-234	516	2	d	d	PROPN
ma-234	516	3	d−1	d−1	PROPN
ma-234	516	4	<	<	X
ma-234	516	5	2	2	NUM
ma-234	516	6	,	,	PUNCT
ma-234	516	7	there	there	PRON
ma-234	516	8	exists	exist	VERB
ma-234	516	9	a	a	DET
ma-234	516	10	real	real	ADJ
ma-234	516	11	number	number	NOUN
ma-234	516	12	r	r	NOUN
ma-234	516	13	such	such	ADJ
ma-234	516	14	that	that	SCONJ
ma-234	516	15	d	d	PROPN
ma-234	516	16	d−1	d−1	PROPN
ma-234	516	17	<	<	X
ma-234	516	18	r	r	NOUN
ma-234	516	19	<	<	X
ma-234	516	20	2	2	NUM
ma-234	516	21	and	and	CCONJ
ma-234	516	22	ψj	ψj	ADV
ma-234	516	23	∈	∈	PROPN
ma-234	516	24	lr	lr	NOUN
ma-234	516	25	.	.	PUNCT
ma-234	517	1	therefore	therefore	ADV
ma-234	517	2	,	,	PUNCT
ma-234	517	3	we	we	PRON
ma-234	517	4	can	can	AUX
ma-234	517	5	use	use	VERB
ma-234	517	6	the	the	DET
ma-234	517	7	fourier	fourier	NOUN
ma-234	517	8	transform	transform	NOUN
ma-234	517	9	to	to	PART
ma-234	517	10	obtain	obtain	VERB
ma-234	517	11	cd	cd	PROPN
ma-234	517	12	d∑	d∑	PROPN
ma-234	518	1	j=1	j=1	NOUN
ma-234	518	2	∂jψj	∂jψj	PROPN
ma-234	518	3	=	=	SYM
ma-234	518	4	ϕ	ϕ	PROPN
ma-234	518	5	,	,	PUNCT
ma-234	518	6	where	where	SCONJ
ma-234	518	7	cd	cd	PROPN
ma-234	518	8	is	be	AUX
ma-234	518	9	a	a	DET
ma-234	518	10	real	real	ADV
ma-234	518	11	constant	constant	ADJ
ma-234	518	12	depending	depend	VERB
ma-234	518	13	only	only	ADV
ma-234	518	14	on	on	ADP
ma-234	518	15	d	d	PROPN
ma-234	518	16	(	(	PUNCT
ma-234	518	17	see	see	VERB
ma-234	518	18	[	[	X
ma-234	518	19	14	14	NUM
ma-234	518	20	,	,	PUNCT
ma-234	518	21	formula	formula	NOUN
ma-234	518	22	(	(	PUNCT
ma-234	518	23	17	17	NUM
ma-234	518	24	)	)	PUNCT
ma-234	518	25	,	,	PUNCT
ma-234	518	26	p.125]).b	p.125]).b	NOUN
ma-234	518	27	)	)	PUNCT
ma-234	518	28	fix	fix	VERB
ma-234	518	29	an	an	DET
ma-234	518	30	integer	integer	NOUN
ma-234	518	31	n	n	PRON
ma-234	518	32	≥	≥	NOUN
ma-234	518	33	1	1	NUM
ma-234	518	34	and	and	CCONJ
ma-234	518	35	set	set	VERB
ma-234	518	36	fn	fn	NOUN
ma-234	518	37	=	=	PUNCT
ma-234	518	38	(	(	PUNCT
ma-234	518	39	f	f	NOUN
ma-234	518	40	ωn	ωn	PROPN
ma-234	518	41	)	)	PUNCT
ma-234	518	42	∗	∗	NOUN
ma-234	518	43	φn	φn	INTJ
ma-234	518	44	.	.	PUNCT
ma-234	519	1	since	since	SCONJ
ma-234	519	2	fn	fn	PROPN
ma-234	519	3	∈	∈	PROPN
ma-234	519	4	c∞c	c∞c	NOUN
ma-234	519	5	,	,	PUNCT
ma-234	519	6	the	the	DET
ma-234	519	7	result	result	NOUN
ma-234	519	8	of	of	ADP
ma-234	519	9	point	point	NOUN
ma-234	519	10	a	a	PRON
ma-234	519	11	)	)	PUNCT
ma-234	519	12	implies	imply	VERB
ma-234	519	13	thatdiv	thatdiv	NOUN
ma-234	519	14	fn	fn	NOUN
ma-234	519	15	=	=	SYM
ma-234	519	16	fn	fn	NOUN
ma-234	519	17	,	,	PUNCT
ma-234	519	18	where	where	SCONJ
ma-234	519	19	fn	fn	NOUN
ma-234	519	20	=	=	SYM
ma-234	519	21	(	(	PUNCT
ma-234	519	22	fnj	fnj	NOUN
ma-234	519	23	)	)	PUNCT
ma-234	519	24	1≤j≤d	1≤j≤d	NUM
ma-234	519	25	with	with	ADP
ma-234	519	26	fnj	fnj	NOUN
ma-234	519	27	=	=	PUNCT
ma-234	519	28	cdrj	cdrj	PROPN
ma-234	519	29	(	(	PUNCT
ma-234	519	30	i	i	NOUN
ma-234	519	31	1	1	NUM
ma-234	519	32	d	d	NOUN
ma-234	519	33	fn	fn	NOUN
ma-234	519	34	)	)	PUNCT
ma-234	519	35	∈	∈	PROPN
ma-234	519	36	⋂	⋂	PROPN
ma-234	519	37	r	r	X
ma-234	519	38	>	>	X
ma-234	519	39	d	d	PROPN
ma-234	519	40	d−1	d−1	PROPN
ma-234	519	41	lr	lr	X
ma-234	519	42	,	,	PUNCT
ma-234	519	43	1	1	NUM
ma-234	519	44	≤	≤	NUM
ma-234	519	45	j	j	PROPN
ma-234	519	46	≤	≤	PROPN
ma-234	519	47	d.	d.	PROPN
ma-234	519	48	•	•	NUM
ma-234	519	49	according	accord	VERB
ma-234	519	50	to	to	ADP
ma-234	519	51	theorem	theorem	VERB
ma-234	519	52	2.5	2.5	NUM
ma-234	519	53	,	,	PUNCT
ma-234	519	54	(	(	PUNCT
ma-234	519	55	fn)n≥1	fn)n≥1	NOUN
ma-234	519	56	converges	converge	VERB
ma-234	519	57	to	to	ADP
ma-234	519	58	f	f	PROPN
ma-234	519	59	in	in	ADP
ma-234	519	60	mα	mα	PROPN
ma-234	519	61	1,p	1,p	PROPN
ma-234	519	62	.	.	PUNCT
ma-234	520	1	•	•	NUM
ma-234	520	2	for	for	ADP
ma-234	520	3	1	1	NUM
ma-234	520	4	≤	≤	NUM
ma-234	521	1	j	j	PROPN
ma-234	521	2	≤	≤	PROPN
ma-234	521	3	d	d	PROPN
ma-234	521	4	,	,	PUNCT
ma-234	521	5	the	the	DET
ma-234	521	6	boundedness	boundedness	NOUN
ma-234	521	7	properties	property	NOUN
ma-234	521	8	of	of	ADP
ma-234	521	9	rj	rj	PROPN
ma-234	521	10	and	and	CCONJ
ma-234	521	11	i	i	PRON
ma-234	521	12	1	1	NUM
ma-234	521	13	d	d	NOUN
ma-234	521	14	imply	imply	VERB
ma-234	521	15	that	that	SCONJ
ma-234	521	16	(	(	PUNCT
ma-234	521	17	fnj	fnj	NOUN
ma-234	521	18	)	)	PUNCT
ma-234	521	19	n≥1	n≥1	NOUN
ma-234	521	20	converges	converge	NOUN
ma-234	521	21	to	to	ADP
ma-234	521	22	cdfj	cdfj	ADJ
ma-234	521	23	=	=	PUNCT
ma-234	521	24	cdrj	cdrj	PROPN
ma-234	522	1	(	(	PUNCT
ma-234	522	2	i	i	NOUN
ma-234	522	3	1	1	NUM
ma-234	522	4	d	d	NOUN
ma-234	522	5	f	f	PROPN
ma-234	522	6	)	)	PUNCT
ma-234	522	7	in	in	ADP
ma-234	522	8	lp	lp	PROPN
ma-234	522	9	.	.	PUNCT
ma-234	523	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	523	2	eur	eur	PROPN
ma-234	523	3	.	.	PUNCT
ma-234	524	1	j.	j.	PROPN
ma-234	524	2	math	math	PROPN
ma-234	524	3	.	.	PUNCT
ma-234	525	1	anal	anal	PROPN
ma-234	525	2	.	.	PUNCT
ma-234	526	1	10.28924	10.28924	NUM
ma-234	526	2	/	/	SYM
ma-234	526	3	ada	ada	PROPN
ma-234	526	4	/	/	SYM
ma-234	526	5	ma.4.16	ma.4.16	PROPN
ma-234	526	6	20therefore	20therefore	NUM
ma-234	526	7	,	,	PUNCT
ma-234	526	8	for	for	ADP
ma-234	526	9	any	any	DET
ma-234	526	10	element	element	NOUN
ma-234	526	11	ϕ	ϕ	NOUN
ma-234	526	12	of	of	ADP
ma-234	526	13	c∞c	c∞c	ADJ
ma-234	526	14	,	,	PUNCT
ma-234	526	15	we	we	PRON
ma-234	526	16	have∫	have∫	VERB
ma-234	526	17	rd	rd	NOUN
ma-234	527	1	div	div	PROPN
ma-234	527	2	f	f	PROPN
ma-234	527	3	(	(	PUNCT
ma-234	527	4	x)ϕ(x)dx	x)ϕ(x)dx	PROPN
ma-234	527	5	=	=	NOUN
ma-234	527	6	−	−	PROPN
ma-234	528	1	d∑	d∑	INTJ
ma-234	529	1	j=1	j=1	PROPN
ma-234	529	2	∫	∫	PROPN
ma-234	529	3	rd	rd	PROPN
ma-234	529	4	cdfj(x	cdfj(x	PROPN
ma-234	529	5	)	)	PUNCT
ma-234	529	6	∂jϕ(x)dx	∂jϕ(x)dx	VERB
ma-234	530	1	=	=	SYM
ma-234	530	2	lim	lim	PROPN
ma-234	530	3	n→∞	n→∞	X
ma-234	530	4	−	−	ADV
ma-234	531	1	d∑	d∑	PROPN
ma-234	531	2	j=1	j=1	ADJ
ma-234	531	3	∫	∫	PROPN
ma-234	531	4	rd	rd	PROPN
ma-234	531	5	fnj	fnj	PROPN
ma-234	531	6	(	(	PUNCT
ma-234	531	7	x	x	NOUN
ma-234	531	8	)	)	PUNCT
ma-234	531	9	∂jϕ(x)dx	∂jϕ(x)dx	VERB
ma-234	531	10			NOUN
ma-234	531	11	=	=	SYM
ma-234	531	12	lim	lim	PROPN
ma-234	531	13	n→∞	n→∞	NUM
ma-234	531	14	∫	∫	PROPN
ma-234	531	15	rd	rd	PROPN
ma-234	532	1			PROPN
ma-234	532	2	d∑	d∑	PROPN
ma-234	532	3	j=1	j=1	PROPN
ma-234	532	4	∂jfnj	∂jfnj	PROPN
ma-234	532	5	(	(	PUNCT
ma-234	532	6	x	x	NOUN
ma-234	532	7	)	)	PUNCT
ma-234	532	8	ϕ(x)dx	ϕ(x)dx	ADV
ma-234	532	9	=	=	PUNCT
ma-234	532	10	lim	lim	PROPN
ma-234	532	11	n→∞	n→∞	NUM
ma-234	532	12	∫	∫	PROPN
ma-234	532	13	rd	rd	PROPN
ma-234	532	14	div	div	PROPN
ma-234	532	15	fn(x)ϕ(x)dx	fn(x)ϕ(x)dx	NOUN
ma-234	532	16	=	=	PROPN
ma-234	532	17	lim	lim	PROPN
ma-234	532	18	n→∞	n→∞	NUM
ma-234	532	19	∫	∫	PROPN
ma-234	532	20	rd	rd	PROPN
ma-234	532	21	fn(x)ϕ(x)dx	fn(x)ϕ(x)dx	PROPN
ma-234	532	22	=	=	SYM
ma-234	532	23	∫	∫	PROPN
ma-234	532	24	rd	rd	PROPN
ma-234	532	25	f	f	PROPN
ma-234	532	26	(	(	PUNCT
ma-234	532	27	x)ϕ(x)dx	x)ϕ(x)dx	PROPN
ma-234	532	28	.	.	PUNCT
ma-234	533	1	hence	hence	ADV
ma-234	533	2	,	,	PUNCT
ma-234	533	3	div	div	X
ma-234	533	4	f	f	PROPN
ma-234	533	5	=	=	SYM
ma-234	533	6	f	f	PROPN
ma-234	533	7	.	.	PUNCT
ma-234	534	1	thus	thus	ADV
ma-234	534	2	,	,	PUNCT
ma-234	534	3	we	we	PRON
ma-234	534	4	obtain	obtain	VERB
ma-234	534	5	the	the	DET
ma-234	534	6	desired	desire	VERB
ma-234	534	7	result	result	NOUN
ma-234	534	8	.	.	PUNCT
ma-234	535	1	�	�	PROPN
ma-234	535	2	acknowledgement	acknowledgement	PROPN
ma-234	535	3	.	.	PUNCT
ma-234	536	1	the	the	DET
ma-234	536	2	author	author	NOUN
ma-234	536	3	would	would	AUX
ma-234	536	4	like	like	VERB
ma-234	536	5	to	to	PART
ma-234	536	6	express	express	VERB
ma-234	536	7	his	his	PRON
ma-234	536	8	deep	deep	ADJ
ma-234	536	9	thanks	thank	NOUN
ma-234	536	10	to	to	ADP
ma-234	536	11	professor	professor	PROPN
ma-234	536	12	ibrahim	ibrahim	PROPN
ma-234	536	13	fofanafor	fofanafor	ADP
ma-234	536	14	his	his	PRON
ma-234	536	15	helpful	helpful	ADJ
ma-234	536	16	assistance	assistance	NOUN
ma-234	536	17	with	with	ADP
ma-234	536	18	this	this	DET
ma-234	536	19	note	note	NOUN
ma-234	536	20	.	.	PUNCT
ma-234	537	1	references	reference	NOUN
ma-234	537	2	[	[	X
ma-234	537	3	1	1	NUM
ma-234	537	4	]	]	PUNCT
ma-234	537	5	c.	c.	PROPN
ma-234	537	6	bennett	bennett	PROPN
ma-234	537	7	,	,	PUNCT
ma-234	537	8	r.c	r.c	PROPN
ma-234	537	9	.	.	PROPN
ma-234	537	10	sharpley	sharpley	PROPN
ma-234	537	11	,	,	PUNCT
ma-234	537	12	interpolation	interpolation	NOUN
ma-234	537	13	of	of	ADP
ma-234	537	14	operators	operator	NOUN
ma-234	537	15	,	,	PUNCT
ma-234	537	16	academic	academic	ADJ
ma-234	537	17	press	press	NOUN
ma-234	537	18	,	,	PUNCT
ma-234	537	19	new	new	PROPN
ma-234	537	20	york	york	PROPN
ma-234	537	21	,	,	PUNCT
ma-234	537	22	(	(	PUNCT
ma-234	537	23	1988).[2	1988).[2	X
ma-234	537	24	]	]	X
ma-234	537	25	j.	j.	PROPN
ma-234	537	26	bourgain	bourgain	PROPN
ma-234	537	27	,	,	PUNCT
ma-234	537	28	on	on	ADP
ma-234	537	29	the	the	DET
ma-234	537	30	restriction	restriction	NOUN
ma-234	537	31	and	and	CCONJ
ma-234	537	32	multiplier	multipli	ADJ
ma-234	537	33	problems	problem	NOUN
ma-234	537	34	in	in	ADP
ma-234	537	35	r3	r3	PROPN
ma-234	537	36	,	,	PUNCT
ma-234	537	37	in	in	ADP
ma-234	537	38	:	:	PUNCT
ma-234	537	39	geometric	geometric	ADJ
ma-234	537	40	aspects	aspect	NOUN
ma-234	537	41	of	of	ADP
ma-234	537	42	functional	functional	ADJ
ma-234	537	43	analysis	analysis	NOUN
ma-234	537	44	(	(	PUNCT
ma-234	537	45	1989	1989	NUM
ma-234	537	46	-	-	SYM
ma-234	537	47	90	90	NUM
ma-234	537	48	)	)	PUNCT
ma-234	537	49	,	,	PUNCT
ma-234	537	50	in	in	ADP
ma-234	537	51	:	:	PUNCT
ma-234	537	52	lecture	lecture	NOUN
ma-234	537	53	notes	note	NOUN
ma-234	537	54	in	in	ADP
ma-234	537	55	math	math	NOUN
ma-234	537	56	.	.	PUNCT
ma-234	537	57	,	,	PUNCT
ma-234	537	58	springer	springer	NOUN
ma-234	537	59	,	,	PUNCT
ma-234	537	60	berlin	berlin	PROPN
ma-234	537	61	,	,	PUNCT
ma-234	537	62	1469	1469	NUM
ma-234	537	63	,	,	PUNCT
ma-234	537	64	(	(	PUNCT
ma-234	537	65	1991	1991	NUM
ma-234	537	66	)	)	PUNCT
ma-234	537	67	,	,	PUNCT
ma-234	537	68	179	179	NUM
ma-234	537	69	-	-	SYM
ma-234	537	70	191.[3	191.[3	NUM
ma-234	537	71	]	]	PUNCT
ma-234	537	72	d.	d.	PROPN
ma-234	537	73	cruz	cruz	PROPN
ma-234	537	74	-	-	PUNCT
ma-234	537	75	uribe	uribe	PROPN
ma-234	537	76	,	,	PUNCT
ma-234	537	77	two	two	NUM
ma-234	537	78	weight	weight	NOUN
ma-234	537	79	inequalities	inequality	NOUN
ma-234	537	80	for	for	ADP
ma-234	537	81	fractional	fractional	ADJ
ma-234	537	82	integral	integral	ADJ
ma-234	537	83	operators	operator	NOUN
ma-234	537	84	and	and	CCONJ
ma-234	537	85	commutators	commutator	NOUN
ma-234	537	86	,	,	PUNCT
ma-234	537	87	in	in	ADP
ma-234	537	88	:	:	PUNCT
ma-234	537	89	advanced	advanced	ADJ
ma-234	537	90	courses	course	NOUN
ma-234	537	91	ofmathematical	ofmathematical	ADJ
ma-234	537	92	analysis	analysis	NOUN
ma-234	537	93	vi	vi	NOUN
ma-234	537	94	,	,	PUNCT
ma-234	537	95	world	world	NOUN
ma-234	537	96	scientific	scientific	ADJ
ma-234	537	97	,	,	PUNCT
ma-234	537	98	universidad	universidad	PROPN
ma-234	537	99	de	de	PROPN
ma-234	537	100	málaga	málaga	PROPN
ma-234	537	101	,	,	PUNCT
ma-234	537	102	spain	spain	PROPN
ma-234	537	103	,	,	PUNCT
ma-234	537	104	2017	2017	NUM
ma-234	537	105	:	:	PUNCT
ma-234	537	106	pp	pp	ADP
ma-234	537	107	.	.	PUNCT
ma-234	538	1	25–85	25–85	NUM
ma-234	538	2	.	.	PUNCT
ma-234	539	1	https://doi.org/	https://doi.org/	VERB
ma-234	539	2	10.1142/9789813147645_0002.[4	10.1142/9789813147645_0002.[4	NUM
ma-234	539	3	]	]	X
ma-234	539	4	n.	n.	PROPN
ma-234	539	5	diarra	diarra	PROPN
ma-234	539	6	,	,	PUNCT
ma-234	539	7	i.	i.	PROPN
ma-234	539	8	fofana	fofana	PROPN
ma-234	539	9	,	,	PUNCT
ma-234	539	10	characterization	characterization	NOUN
ma-234	539	11	of	of	ADP
ma-234	539	12	some	some	DET
ma-234	539	13	closed	close	VERB
ma-234	539	14	linear	linear	ADJ
ma-234	539	15	subspaces	subspace	NOUN
ma-234	539	16	of	of	ADP
ma-234	539	17	morrey	morrey	NOUN
ma-234	539	18	spaces	space	NOUN
ma-234	539	19	and	and	CCONJ
ma-234	539	20	approximation	approximation	NOUN
ma-234	539	21	,	,	PUNCT
ma-234	539	22	adv.pure	adv.pure	NOUN
ma-234	539	23	appl	appl	PROPN
ma-234	539	24	.	.	PROPN
ma-234	539	25	math	math	NOUN
ma-234	539	26	.	.	PUNCT
ma-234	540	1	14	14	NUM
ma-234	540	2	(	(	PUNCT
ma-234	540	3	2023	2023	NUM
ma-234	540	4	)	)	PUNCT
ma-234	540	5	41–72	41–72	NUM
ma-234	540	6	.	.	PUNCT
ma-234	541	1	https://doi.org/10.21494/iste.op.2023.0980.[5	https://doi.org/10.21494/iste.op.2023.0980.[5	NOUN
ma-234	541	2	]	]	X
ma-234	541	3	n.	n.	PROPN
ma-234	541	4	diarra	diarra	PROPN
ma-234	541	5	,	,	PUNCT
ma-234	541	6	i.	i.	PROPN
ma-234	541	7	fofana	fofana	PROPN
ma-234	541	8	,	,	PUNCT
ma-234	541	9	complex	complex	ADJ
ma-234	541	10	interpolation	interpolation	NOUN
ma-234	541	11	of	of	ADP
ma-234	541	12	some	some	DET
ma-234	541	13	banach	banach	NOUN
ma-234	541	14	spaces	space	NOUN
ma-234	541	15	including	include	VERB
ma-234	541	16	morrey	morrey	PROPN
ma-234	541	17	spaces	space	NOUN
ma-234	541	18	,	,	PUNCT
ma-234	541	19	khayyam	khayyam	PROPN
ma-234	541	20	j.	j.	PROPN
ma-234	541	21	math.(2024).[6	math.(2024).[6	PROPN
ma-234	541	22	]	]	PUNCT
ma-234	541	23	m.	m.	NOUN
ma-234	541	24	dosso	dosso	PROPN
ma-234	541	25	,	,	PUNCT
ma-234	541	26	i.	i.	PROPN
ma-234	541	27	fofana	fofana	PROPN
ma-234	541	28	,	,	PUNCT
ma-234	541	29	m.	m.	NOUN
ma-234	541	30	sanogo	sanogo	PROPN
ma-234	541	31	,	,	PUNCT
ma-234	541	32	on	on	ADP
ma-234	541	33	some	some	DET
ma-234	541	34	subspaces	subspace	NOUN
ma-234	541	35	of	of	ADP
ma-234	541	36	morrey	morrey	NOUN
ma-234	541	37	–	–	PUNCT
ma-234	541	38	sobolev	sobolev	NOUN
ma-234	541	39	spaces	space	NOUN
ma-234	541	40	and	and	CCONJ
ma-234	541	41	boundedness	boundedness	NOUN
ma-234	541	42	of	of	ADP
ma-234	541	43	riesz	riesz	NOUN
ma-234	541	44	integrals	integral	NOUN
ma-234	541	45	,	,	PUNCT
ma-234	541	46	ann	ann	PROPN
ma-234	541	47	.	.	PUNCT
ma-234	541	48	polon	polon	PROPN
ma-234	541	49	.	.	PUNCT
ma-234	542	1	math	math	NOUN
ma-234	542	2	.	.	PUNCT
ma-234	543	1	108	108	NUM
ma-234	543	2	(	(	PUNCT
ma-234	543	3	2013	2013	NUM
ma-234	543	4	)	)	PUNCT
ma-234	543	5	133–153	133–153	NUM
ma-234	543	6	.	.	PUNCT
ma-234	544	1	https://doi.org/10.4064/ap108-2-2.[7	https://doi.org/10.4064/ap108-2-2.[7	PROPN
ma-234	544	2	]	]	PUNCT
ma-234	544	3	i.	i.	PROPN
ma-234	544	4	fofana	fofana	PROPN
ma-234	544	5	,	,	PUNCT
ma-234	544	6	f.r	f.r	PROPN
ma-234	544	7	.	.	PROPN
ma-234	544	8	faléa	faléa	PROPN
ma-234	544	9	,	,	PUNCT
ma-234	544	10	b.a	b.a	PROPN
ma-234	544	11	.	.	PROPN
ma-234	544	12	kpata	kpata	PROPN
ma-234	544	13	,	,	PUNCT
ma-234	544	14	a	a	DET
ma-234	544	15	class	class	NOUN
ma-234	544	16	of	of	ADP
ma-234	544	17	subspaces	subspace	NOUN
ma-234	544	18	of	of	ADP
ma-234	544	19	morrey	morrey	NOUN
ma-234	544	20	spaces	space	NOUN
ma-234	544	21	and	and	CCONJ
ma-234	544	22	norm	norm	NOUN
ma-234	544	23	inequalities	inequality	NOUN
ma-234	544	24	on	on	ADP
ma-234	544	25	riesz	riesz	NOUN
ma-234	544	26	potentialoperators	potentialoperator	NOUN
ma-234	544	27	,	,	PUNCT
ma-234	544	28	afr	afr	PROPN
ma-234	544	29	.	.	PUNCT
ma-234	545	1	mat	mat	PROPN
ma-234	545	2	.	.	PROPN
ma-234	545	3	26	26	NUM
ma-234	545	4	(	(	PUNCT
ma-234	545	5	2015	2015	NUM
ma-234	545	6	)	)	PUNCT
ma-234	546	1	717–739	717–739	NUM
ma-234	546	2	.	.	PUNCT
ma-234	547	1	https://doi.org/10.1007/s13370-014-0241-3.[8	https://doi.org/10.1007/s13370-014-0241-3.[8	PROPN
ma-234	547	2	]	]	PUNCT
ma-234	547	3	n.	n.	PROPN
ma-234	547	4	hatano	hatano	PROPN
ma-234	547	5	,	,	PUNCT
ma-234	547	6	t.	t.	NOUN
ma-234	547	7	nogayama	nogayama	PROPN
ma-234	547	8	,	,	PUNCT
ma-234	547	9	y.	y.	PROPN
ma-234	547	10	sawano	sawano	PROPN
ma-234	547	11	,	,	PUNCT
ma-234	547	12	d.i	d.i	PROPN
ma-234	547	13	.	.	PROPN
ma-234	547	14	hakim	hakim	PROPN
ma-234	547	15	,	,	PUNCT
ma-234	547	16	bourgain	bourgain	PROPN
ma-234	547	17	–	–	PUNCT
ma-234	547	18	morrey	morrey	NOUN
ma-234	547	19	spaces	space	NOUN
ma-234	547	20	and	and	CCONJ
ma-234	547	21	their	their	PRON
ma-234	547	22	applications	application	NOUN
ma-234	547	23	to	to	ADP
ma-234	547	24	boundednessof	boundednessof	NOUN
ma-234	547	25	operators	operator	NOUN
ma-234	547	26	,	,	PUNCT
ma-234	547	27	j.	j.	PROPN
ma-234	547	28	funct	funct	PROPN
ma-234	547	29	.	.	PUNCT
ma-234	548	1	anal	anal	PROPN
ma-234	548	2	.	.	PUNCT
ma-234	549	1	284	284	NUM
ma-234	549	2	(	(	PUNCT
ma-234	549	3	2023	2023	NUM
ma-234	549	4	)	)	PUNCT
ma-234	549	5	,	,	PUNCT
ma-234	549	6	109720	109720	NUM
ma-234	549	7	.	.	PUNCT
ma-234	550	1	https://doi.org/10.1016/j.jfa.2022.109720.[9	https://doi.org/10.1016/j.jfa.2022.109720.[9	CCONJ
ma-234	550	2	]	]	X
ma-234	550	3	s.	s.	PROPN
ma-234	550	4	masaki	masaki	PROPN
ma-234	550	5	,	,	PUNCT
ma-234	550	6	j.	j.	PROPN
ma-234	550	7	segata	segata	PROPN
ma-234	550	8	,	,	PUNCT
ma-234	550	9	existence	existence	NOUN
ma-234	550	10	of	of	ADP
ma-234	550	11	a	a	DET
ma-234	550	12	minimal	minimal	ADJ
ma-234	550	13	non	non	ADJ
ma-234	550	14	-	-	ADJ
ma-234	550	15	scattering	scattering	ADJ
ma-234	550	16	solution	solution	NOUN
ma-234	550	17	to	to	ADP
ma-234	550	18	the	the	DET
ma-234	550	19	mass	mass	ADJ
ma-234	550	20	-	-	PUNCT
ma-234	550	21	subcritical	subcritical	ADJ
ma-234	550	22	generalized	generalize	VERB
ma-234	550	23	korteweg	korteweg	NOUN
ma-234	550	24	–	–	PUNCT
ma-234	550	25	de	de	PROPN
ma-234	550	26	vries	vries	PROPN
ma-234	550	27	equation	equation	NOUN
ma-234	550	28	,	,	PUNCT
ma-234	550	29	ann	ann	PROPN
ma-234	550	30	.	.	PROPN
ma-234	550	31	inst	inst	PROPN
ma-234	550	32	.	.	PUNCT
ma-234	551	1	h.	h.	PROPN
ma-234	551	2	poincaré	poincaré	PROPN
ma-234	551	3	anal	anal	PROPN
ma-234	551	4	.	.	PUNCT
ma-234	552	1	non	non	PROPN
ma-234	552	2	linéaire	linéaire	PROPN
ma-234	552	3	35	35	NUM
ma-234	552	4	(	(	PUNCT
ma-234	552	5	2018	2018	NUM
ma-234	552	6	)	)	PUNCT
ma-234	552	7	283–326	283–326	NUM
ma-234	552	8	.	.	PUNCT
ma-234	553	1	https://doi.org/10.1016/j	https://doi.org/10.1016/j	NOUN
ma-234	553	2	.	.	PUNCT
ma-234	554	1	anihpc.2017.04.003.[10	anihpc.2017.04.003.[10	VERB
ma-234	554	2	]	]	PUNCT
ma-234	554	3	s.	s.	PROPN
ma-234	554	4	masaki	masaki	PROPN
ma-234	554	5	,	,	PUNCT
ma-234	554	6	j.	j.	PROPN
ma-234	554	7	segata	segata	PROPN
ma-234	554	8	,	,	PUNCT
ma-234	554	9	refinement	refinement	NOUN
ma-234	554	10	of	of	ADP
ma-234	554	11	strichartz	strichartz	ADJ
ma-234	554	12	estimates	estimate	NOUN
ma-234	554	13	for	for	ADP
ma-234	554	14	airy	airy	ADJ
ma-234	554	15	equation	equation	NOUN
ma-234	554	16	and	and	CCONJ
ma-234	554	17	application	application	NOUN
ma-234	554	18	,	,	PUNCT
ma-234	554	19	rims	rims	PROPN
ma-234	554	20	kôkyûrokubessatsu	kôkyûrokubessatsu	PROPN
ma-234	554	21	,	,	PUNCT
ma-234	554	22	b80	b80	NOUN
ma-234	554	23	(	(	PUNCT
ma-234	554	24	2020	2020	NUM
ma-234	554	25	)	)	PUNCT
ma-234	554	26	11–25	11–25	NUM
ma-234	554	27	.	.	PUNCT
ma-234	554	28	http://hdl.handle.net/2433/260656.[11	http://hdl.handle.net/2433/260656.[11	X
ma-234	554	29	]	]	X
ma-234	555	1	c.b	c.b	PROPN
ma-234	555	2	.	.	PROPN
ma-234	555	3	morrey	morrey	PROPN
ma-234	555	4	,	,	PUNCT
ma-234	555	5	on	on	ADP
ma-234	555	6	the	the	DET
ma-234	555	7	solutions	solution	NOUN
ma-234	555	8	of	of	ADP
ma-234	555	9	quasi	quasi	ADJ
ma-234	555	10	-	-	ADJ
ma-234	555	11	linear	linear	ADJ
ma-234	555	12	elliptic	elliptic	ADJ
ma-234	555	13	partial	partial	ADJ
ma-234	555	14	differential	differential	NOUN
ma-234	555	15	equations	equation	NOUN
ma-234	555	16	,	,	PUNCT
ma-234	555	17	trans	trans	PROPN
ma-234	555	18	.	.	PROPN
ma-234	556	1	amer	amer	PROPN
ma-234	556	2	.	.	PUNCT
ma-234	556	3	math	math	PROPN
ma-234	556	4	.	.	PUNCT
ma-234	557	1	soc	soc	PROPN
ma-234	557	2	.	.	PUNCT
ma-234	558	1	43(1938	43(1938	X
ma-234	558	2	)	)	PUNCT
ma-234	559	1	126–166	126–166	NUM
ma-234	559	2	.	.	PUNCT
ma-234	560	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
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ma-234	560	3	https://doi.org/10.1142/9789813147645_0002	https://doi.org/10.1142/9789813147645_0002	PROPN
ma-234	560	4	https://doi.org/10.21494/iste.op.2023.0980	https://doi.org/10.21494/iste.op.2023.0980	X
ma-234	560	5	https://doi.org/10.4064/ap108-2-2	https://doi.org/10.4064/ap108-2-2	X
ma-234	560	6	https://doi.org/10.1007/s13370-014-0241-3	https://doi.org/10.1007/s13370-014-0241-3	NUM
ma-234	561	1	https://doi.org/10.1016/j.jfa.2022.109720	https://doi.org/10.1016/j.jfa.2022.109720	PROPN
ma-234	561	2	https://doi.org/10.1016/j.anihpc.2017.04.003	https://doi.org/10.1016/j.anihpc.2017.04.003	NUM
ma-234	561	3	https://doi.org/10.1016/j.anihpc.2017.04.003	https://doi.org/10.1016/j.anihpc.2017.04.003	NUM
ma-234	561	4	http://hdl.handle.net/2433/260656	http://hdl.handle.net/2433/260656	ADJ
ma-234	561	5	eur	eur	NOUN
ma-234	561	6	.	.	PUNCT
ma-234	562	1	j.	j.	PROPN
ma-234	562	2	math	math	PROPN
ma-234	562	3	.	.	PUNCT
ma-234	563	1	anal	anal	PROPN
ma-234	563	2	.	.	PUNCT
ma-234	564	1	10.28924	10.28924	NUM
ma-234	564	2	/	/	SYM
ma-234	564	3	ada	ada	PROPN
ma-234	564	4	/	/	SYM
ma-234	564	5	ma.4.16	ma.4.16	PROPN
ma-234	564	6	21	21	NUM
ma-234	565	1	[	[	X
ma-234	565	2	12	12	NUM
ma-234	565	3	]	]	X
ma-234	565	4	b.	b.	PROPN
ma-234	565	5	muckenhoupt	muckenhoupt	PROPN
ma-234	565	6	,	,	PUNCT
ma-234	565	7	r.	r.	PROPN
ma-234	565	8	wheeden	wheeden	PROPN
ma-234	565	9	,	,	PUNCT
ma-234	565	10	weighted	weight	VERB
ma-234	565	11	norm	norm	NOUN
ma-234	565	12	inequalities	inequality	NOUN
ma-234	565	13	for	for	ADP
ma-234	565	14	fractional	fractional	ADJ
ma-234	565	15	integrals	integral	NOUN
ma-234	565	16	,	,	PUNCT
ma-234	565	17	trans	trans	PROPN
ma-234	565	18	.	.	PROPN
ma-234	566	1	amer	amer	PROPN
ma-234	566	2	.	.	PUNCT
ma-234	566	3	math	math	PROPN
ma-234	566	4	.	.	PUNCT
ma-234	567	1	soc	soc	PROPN
ma-234	567	2	.	.	PUNCT
ma-234	568	1	192(1974	192(1974	NUM
ma-234	568	2	)	)	PUNCT
ma-234	568	3	261–274	261–274	NUM
ma-234	568	4	.	.	PUNCT
ma-234	569	1	https://doi.org/10.1090/s0002-9947-1974-0340523-6.[13	https://doi.org/10.1090/s0002-9947-1974-0340523-6.[13	PROPN
ma-234	569	2	]	]	PUNCT
ma-234	569	3	n.c	n.c	PROPN
ma-234	569	4	.	.	PROPN
ma-234	569	5	phuc	phuc	PROPN
ma-234	569	6	,	,	PUNCT
ma-234	569	7	m.	m.	NOUN
ma-234	569	8	torrès	torrès	PROPN
ma-234	569	9	,	,	PUNCT
ma-234	569	10	characterizations	characterization	NOUN
ma-234	569	11	of	of	ADP
ma-234	569	12	the	the	DET
ma-234	569	13	existence	existence	NOUN
ma-234	569	14	and	and	CCONJ
ma-234	569	15	removable	removable	ADJ
ma-234	569	16	singularities	singularity	NOUN
ma-234	569	17	of	of	ADP
ma-234	569	18	divergence	divergence	ADJ
ma-234	569	19	-	-	PUNCT
ma-234	569	20	measure	measure	NOUN
ma-234	569	21	vectorfields	vectorfield	NOUN
ma-234	569	22	,	,	PUNCT
ma-234	569	23	indiana	indiana	PROPN
ma-234	569	24	univ	univ	PROPN
ma-234	569	25	.	.	PUNCT
ma-234	570	1	math	math	PROPN
ma-234	570	2	.	.	PUNCT
ma-234	571	1	j.	j.	PROPN
ma-234	571	2	,	,	PUNCT
ma-234	571	3	57	57	NUM
ma-234	571	4	(	(	PUNCT
ma-234	571	5	2008	2008	NUM
ma-234	571	6	)	)	PUNCT
ma-234	571	7	1573–1597	1573–1597	NUM
ma-234	571	8	.	.	PUNCT
ma-234	572	1	https://www.jstor.org/stable/24902998.[14	https://www.jstor.org/stable/24902998.[14	PROPN
ma-234	572	2	]	]	X
ma-234	572	3	e.m	e.m	PROPN
ma-234	572	4	.	.	PROPN
ma-234	572	5	stein	stein	PROPN
ma-234	572	6	,	,	PUNCT
ma-234	572	7	singular	singular	PROPN
ma-234	572	8	integrals	integral	NOUN
ma-234	572	9	and	and	CCONJ
ma-234	572	10	differentiability	differentiability	NOUN
ma-234	572	11	properties	property	NOUN
ma-234	572	12	of	of	ADP
ma-234	572	13	functions	function	NOUN
ma-234	572	14	,	,	PUNCT
ma-234	572	15	princeton	princeton	PROPN
ma-234	572	16	university	university	PROPN
ma-234	572	17	press	press	PROPN
ma-234	572	18	,	,	PUNCT
ma-234	572	19	princeton	princeton	PROPN
ma-234	572	20	,	,	PUNCT
ma-234	572	21	new	new	PROPN
ma-234	572	22	jersey	jersey	PROPN
ma-234	572	23	(	(	PUNCT
ma-234	572	24	1970).[15	1970).[15	PROPN
ma-234	572	25	]	]	X
ma-234	572	26	j.	j.	PROPN
ma-234	572	27	tao	tao	PROPN
ma-234	572	28	,	,	PUNCT
ma-234	572	29	d.	d.	PROPN
ma-234	572	30	yang	yang	PROPN
ma-234	572	31	,	,	PUNCT
ma-234	572	32	w.	w.	PROPN
ma-234	572	33	yuan	yuan	PROPN
ma-234	572	34	,	,	PUNCT
ma-234	572	35	a	a	DET
ma-234	572	36	bridge	bridge	NOUN
ma-234	572	37	connecting	connect	VERB
ma-234	572	38	lebesgue	lebesgue	NOUN
ma-234	572	39	and	and	CCONJ
ma-234	572	40	morrey	morrey	PROPN
ma-234	572	41	spaces	space	NOUN
ma-234	572	42	via	via	ADP
ma-234	572	43	riesz	riesz	NOUN
ma-234	572	44	norms	norm	NOUN
ma-234	572	45	,	,	PUNCT
ma-234	572	46	banach	banach	NOUN
ma-234	572	47	j.	j.	PROPN
ma-234	572	48	math	math	PROPN
ma-234	572	49	.	.	PUNCT
ma-234	573	1	anal.15	anal.15	NOUN
ma-234	573	2	(	(	PUNCT
ma-234	573	3	2021	2021	NUM
ma-234	573	4	)	)	PUNCT
ma-234	573	5	20	20	NUM
ma-234	573	6	.	.	PUNCT
ma-234	574	1	https://doi.org/10.1007/s43037-020-00106-6	https://doi.org/10.1007/s43037-020-00106-6	PROPN
ma-234	574	2	.	.	PUNCT
ma-234	575	1	https://doi.org/10.28924/ada/ma.4.16	https://doi.org/10.28924/ada/ma.4.16	PROPN
ma-234	576	1	https://doi.org/10.1090/s0002-9947-1974-0340523-6	https://doi.org/10.1090/s0002-9947-1974-0340523-6	PROPN
ma-234	576	2	https://www.jstor.org/stable/24902998	https://www.jstor.org/stable/24902998	PROPN
ma-234	576	3	https://doi.org/10.1007/s43037-020-00106-6	https://doi.org/10.1007/s43037-020-00106-6	PROPN
ma-234	576	4	1	1	NUM
ma-234	576	5	.	.	PUNCT
ma-234	577	1	introduction	introduction	NOUN
ma-234	577	2	2	2	NUM
ma-234	577	3	.	.	PUNCT
ma-234	577	4	statement	statement	NOUN
ma-234	577	5	of	of	ADP
ma-234	577	6	the	the	DET
ma-234	577	7	main	main	ADJ
ma-234	577	8	results	result	NOUN
ma-234	577	9	3	3	NUM
ma-234	577	10	.	.	PUNCT
ma-234	577	11	preliminaries	preliminary	NOUN
ma-234	577	12	3.1	3.1	NUM
ma-234	577	13	.	.	PUNCT
ma-234	578	1	equivalent	equivalent	ADJ
ma-234	578	2	norms	norm	NOUN
ma-234	578	3	on	on	ADP
ma-234	578	4	mq	mq	PROPN
ma-234	578	5	,	,	PUNCT
ma-234	578	6	p	p	NOUN
ma-234	578	7	3.2	3.2	NUM
ma-234	578	8	.	.	PUNCT
ma-234	579	1	continuity	continuity	NOUN
ma-234	579	2	of	of	ADP
ma-234	579	3	the	the	DET
ma-234	579	4	translation	translation	NOUN
ma-234	579	5	operator	operator	NOUN
ma-234	579	6	in	in	ADP
ma-234	579	7	mq	mq	PROPN
ma-234	579	8	,	,	PUNCT
ma-234	579	9	p	p	PROPN
ma-234	579	10	4	4	NUM
ma-234	579	11	.	.	PUNCT
ma-234	579	12	inclusion	inclusion	NOUN
ma-234	579	13	and	and	CCONJ
ma-234	579	14	approximation	approximation	NOUN
ma-234	579	15	results	result	VERB
ma-234	579	16	4.1	4.1	NUM
ma-234	579	17	.	.	PUNCT
ma-234	580	1	inclusion	inclusion	NOUN
ma-234	580	2	of	of	ADP
ma-234	580	3	mq	mq	PROPN
ma-234	580	4	,	,	PUNCT
ma-234	580	5	p	p	NOUN
ma-234	580	6	in	in	ADP
ma-234	580	7	f(q	f(q	PROPN
ma-234	580	8	,	,	PUNCT
ma-234	580	9	p	p	X
ma-234	580	10	,	,	PUNCT
ma-234	580	11	)	)	PUNCT
ma-234	580	12	4.2	4.2	NUM
ma-234	580	13	.	.	PUNCT
ma-234	581	1	approximation	approximation	NOUN
ma-234	581	2	in	in	ADP
ma-234	581	3	mq	mq	PROPN
ma-234	581	4	,	,	PUNCT
ma-234	581	5	p	p	PROPN
ma-234	581	6	5	5	NUM
ma-234	581	7	.	.	PUNCT
ma-234	581	8	fractional	fractional	ADJ
ma-234	581	9	operators	operator	NOUN
ma-234	581	10	in	in	ADP
ma-234	581	11	mq	mq	PROPN
ma-234	581	12	,	,	PUNCT
ma-234	581	13	p	p	PROPN
ma-234	581	14	6	6	NUM
ma-234	581	15	.	.	PUNCT
ma-234	581	16	application	application	NOUN
ma-234	581	17	references	reference	NOUN