id	sid	tid	token	lemma	pos
ma-381	1	1	2025	2025	NUM
ma-381	1	2	ada	ada	PROPN
ma-381	1	3	academica	academica	PROPN
ma-381	1	4	https://adac.eeeur	https://adac.eeeur	PROPN
ma-381	1	5	.	.	PUNCT
ma-381	2	1	j.	j.	PROPN
ma-381	2	2	math	math	PROPN
ma-381	2	3	.	.	PUNCT
ma-381	3	1	anal	anal	ADJ
ma-381	3	2	.	.	PUNCT
ma-381	4	1	5	5	NUM
ma-381	4	2	(	(	PUNCT
ma-381	4	3	2025	2025	NUM
ma-381	4	4	)	)	PUNCT
ma-381	4	5	21doi	21doi	NOUN
ma-381	4	6	:	:	PUNCT
ma-381	4	7	10.28924	10.28924	NUM
ma-381	4	8	/	/	SYM
ma-381	4	9	ada	ada	PROPN
ma-381	4	10	/	/	SYM
ma-381	4	11	ma.5.21	ma.5.21	PROPN
ma-381	4	12	new	new	ADJ
ma-381	4	13	characterization	characterization	NOUN
ma-381	4	14	of	of	ADP
ma-381	4	15	hardy	hardy	ADJ
ma-381	4	16	-	-	PUNCT
ma-381	4	17	fofana	fofana	NOUN
ma-381	4	18	spaces	space	NOUN
ma-381	4	19	and	and	CCONJ
ma-381	4	20	temperature	temperature	NOUN
ma-381	4	21	equation	equation	NOUN
ma-381	4	22	martial	martial	ADJ
ma-381	4	23	agbly	agbly	ADV
ma-381	4	24	dakoury1	dakoury1	PROPN
ma-381	4	25	,	,	PUNCT
ma-381	4	26	justin	justin	PROPN
ma-381	4	27	feuto2,∗	feuto2,∗	PROPN
ma-381	4	28	1laboratoire	1laboratoire	PROPN
ma-381	4	29	de	de	ADP
ma-381	4	30	mathématiques	mathématiques	PROPN
ma-381	4	31	et	et	NOUN
ma-381	4	32	applications	application	NOUN
ma-381	4	33	,	,	PUNCT
ma-381	4	34	ufr	ufr	NOUN
ma-381	4	35	mathématiques	mathématique	NOUN
ma-381	4	36	et	et	PROPN
ma-381	4	37	informatique	informatique	PROPN
ma-381	4	38	,	,	PUNCT
ma-381	4	39	université	université	PROPN
ma-381	4	40	félix	félix	ADJ
ma-381	4	41	houphouët	houphouët	PROPN
ma-381	4	42	-	-	PUNCT
ma-381	4	43	boigny	boigny	PROPN
ma-381	4	44	abidjan	abidjan	PROPN
ma-381	4	45	-	-	NOUN
ma-381	4	46	cocody	cocody	NOUN
ma-381	4	47	,	,	PUNCT
ma-381	4	48	22	22	NUM
ma-381	4	49	b.p	b.p	PROPN
ma-381	4	50	582	582	NUM
ma-381	4	51	abidjan	abidjan	PROPN
ma-381	4	52	22	22	NUM
ma-381	4	53	.	.	PUNCT
ma-381	5	1	côte	côte	PROPN
ma-381	5	2	d’ivoire	d’ivoire	PROPN
ma-381	5	3	dakourymartial@gmail.com	dakourymartial@gmail.com	X
ma-381	6	1	2laboratoire	2laboratoire	NUM
ma-381	6	2	de	de	X
ma-381	6	3	mathématiques	mathématiques	X
ma-381	6	4	et	et	NOUN
ma-381	6	5	applications	application	NOUN
ma-381	6	6	,	,	PUNCT
ma-381	6	7	ufr	ufr	NOUN
ma-381	6	8	mathématiques	mathématique	NOUN
ma-381	6	9	et	et	PROPN
ma-381	6	10	informatique	informatique	PROPN
ma-381	6	11	,	,	PUNCT
ma-381	6	12	université	université	PROPN
ma-381	6	13	félix	félix	ADJ
ma-381	6	14	houphouët	houphouët	PROPN
ma-381	6	15	-	-	PUNCT
ma-381	6	16	boigny	boigny	PROPN
ma-381	6	17	abidjan	abidjan	PROPN
ma-381	6	18	-	-	NOUN
ma-381	6	19	cocody	cocody	NOUN
ma-381	6	20	,	,	PUNCT
ma-381	6	21	22	22	NUM
ma-381	6	22	b.p	b.p	PROPN
ma-381	6	23	1194	1194	NUM
ma-381	6	24	abidjan	abidjan	PROPN
ma-381	6	25	22	22	NUM
ma-381	6	26	.	.	PUNCT
ma-381	7	1	côte	côte	PROPN
ma-381	7	2	d’ivoire	d’ivoire	PROPN
ma-381	7	3	justfeuto@yahoo.fr	justfeuto@yahoo.fr	NOUN
ma-381	7	4	∗correspondence	∗correspondence	NOUN
ma-381	7	5	:	:	PUNCT
ma-381	7	6	justfeuto@yahoo.fr	justfeuto@yahoo.fr	PROPN
ma-381	7	7	abstract	abstract	NOUN
ma-381	7	8	.	.	PUNCT
ma-381	8	1	the	the	DET
ma-381	8	2	aim	aim	NOUN
ma-381	8	3	of	of	ADP
ma-381	8	4	this	this	DET
ma-381	8	5	paper	paper	NOUN
ma-381	8	6	is	be	AUX
ma-381	8	7	to	to	PART
ma-381	8	8	give	give	VERB
ma-381	8	9	a	a	DET
ma-381	8	10	characterization	characterization	NOUN
ma-381	8	11	of	of	ADP
ma-381	8	12	hardy	hardy	ADJ
ma-381	8	13	-	-	PUNCT
ma-381	8	14	fofana	fofana	NOUN
ma-381	8	15	spaces	space	NOUN
ma-381	8	16	via	via	ADP
ma-381	8	17	riesztransforms	riesztransform	NOUN
ma-381	8	18	.	.	PUNCT
ma-381	9	1	this	this	DET
ma-381	9	2	characterization	characterization	NOUN
ma-381	9	3	allows	allow	VERB
ma-381	9	4	us	we	PRON
ma-381	9	5	to	to	PART
ma-381	9	6	describe	describe	VERB
ma-381	9	7	the	the	DET
ma-381	9	8	distributions	distribution	NOUN
ma-381	9	9	belonging	belong	VERB
ma-381	9	10	to	to	ADP
ma-381	9	11	these	these	DET
ma-381	9	12	spacesas	spacesa	NOUN
ma-381	9	13	a	a	DET
ma-381	9	14	bounded	bound	VERB
ma-381	9	15	solutions	solution	NOUN
ma-381	9	16	of	of	ADP
ma-381	9	17	cauchy	cauchy	PROPN
ma-381	9	18	-	-	PUNCT
ma-381	9	19	riemann	riemann	PROPN
ma-381	9	20	’s	’s	PART
ma-381	9	21	general	general	ADJ
ma-381	9	22	temperature	temperature	NOUN
ma-381	9	23	equations	equation	NOUN
ma-381	9	24	.	.	PUNCT
ma-381	10	1	1	1	X
ma-381	10	2	.	.	X
ma-381	10	3	introduction	introduction	NOUN
ma-381	10	4	let	let	VERB
ma-381	10	5	rd	rd	PROPN
ma-381	10	6	(	(	PUNCT
ma-381	10	7	d	d	NOUN
ma-381	10	8	is	be	AUX
ma-381	10	9	a	a	DET
ma-381	10	10	positive	positive	ADJ
ma-381	10	11	integer	integer	NOUN
ma-381	10	12	)	)	PUNCT
ma-381	10	13	be	be	AUX
ma-381	10	14	the	the	DET
ma-381	10	15	euclidean	euclidean	ADJ
ma-381	10	16	space	space	NOUN
ma-381	10	17	of	of	ADP
ma-381	10	18	dimension	dimension	NOUN
ma-381	10	19	d	d	PROPN
ma-381	10	20	equipped	equip	VERB
ma-381	10	21	with	with	ADP
ma-381	10	22	thelebesgue	thelebesgue	NOUN
ma-381	10	23	measure	measure	NOUN
ma-381	10	24	dx	dx	PROPN
ma-381	10	25	and	and	CCONJ
ma-381	10	26	the	the	DET
ma-381	10	27	euclidean	euclidean	ADJ
ma-381	10	28	norm	norm	NOUN
ma-381	10	29	.	.	PUNCT
ma-381	11	1	the	the	DET
ma-381	11	2	classical	classical	ADJ
ma-381	11	3	hardy	hardy	ADJ
ma-381	11	4	space	space	NOUN
ma-381	11	5	hp(rd	hp(rd	NOUN
ma-381	11	6	)	)	PUNCT
ma-381	11	7	(	(	PUNCT
ma-381	11	8	0	0	PUNCT
ma-381	11	9	<	<	X
ma-381	11	10	p	p	X
ma-381	11	11	<	<	X
ma-381	11	12	∞)is	∞)is	ADJ
ma-381	11	13	defined	define	VERB
ma-381	11	14	as	as	ADP
ma-381	11	15	the	the	DET
ma-381	11	16	space	space	NOUN
ma-381	11	17	of	of	ADP
ma-381	11	18	tempered	temper	VERB
ma-381	11	19	distributions	distribution	NOUN
ma-381	11	20	f	f	X
ma-381	11	21	satisfying	satisfy	VERB
ma-381	11	22	‖mf	‖mf	PROPN
ma-381	11	23	‖p	‖p	PROPN
ma-381	11	24	<	<	X
ma-381	11	25	∞	∞	PROPN
ma-381	11	26	,	,	PUNCT
ma-381	11	27	where	where	SCONJ
ma-381	11	28	the	the	DET
ma-381	11	29	maximalfunction	maximalfunction	NOUN
ma-381	11	30	mf	mf	NOUN
ma-381	11	31	is	be	AUX
ma-381	11	32	defined	define	VERB
ma-381	11	33	by	by	ADP
ma-381	11	34	mf	mf	X
ma-381	11	35	(	(	PUNCT
ma-381	11	36	x	x	NOUN
ma-381	11	37	)	)	PUNCT
ma-381	11	38	=	=	SYM
ma-381	11	39	sup	sup	NOUN
ma-381	11	40	t>0	t>0	NOUN
ma-381	11	41	|(f	|(f	PROPN
ma-381	11	42	∗	∗	NOUN
ma-381	11	43	ϕt)(x)|	ϕt)(x)|	NOUN
ma-381	11	44	,	,	PUNCT
ma-381	11	45	(	(	PUNCT
ma-381	11	46	1.1	1.1	NUM
ma-381	11	47	)	)	PUNCT
ma-381	11	48	with	with	ADP
ma-381	11	49	ϕ	ϕ	NOUN
ma-381	11	50	in	in	ADP
ma-381	11	51	the	the	DET
ma-381	11	52	schwartz	schwartz	PROPN
ma-381	11	53	class	class	PROPN
ma-381	11	54	s(rd	s(rd	PROPN
ma-381	11	55	)	)	PUNCT
ma-381	11	56	having	have	VERB
ma-381	11	57	non	non	ADJ
ma-381	11	58	vanish	vanish	VERB
ma-381	11	59	integral	integral	ADJ
ma-381	11	60	,	,	PUNCT
ma-381	11	61	and	and	CCONJ
ma-381	11	62	ϕt(x	ϕt(x	PUNCT
ma-381	11	63	)	)	PUNCT
ma-381	12	1	=	=	PRON
ma-381	12	2	t−dϕ(t−1x).it	t−dϕ(t−1x).it	NOUN
ma-381	12	3	is	be	AUX
ma-381	12	4	well	well	ADV
ma-381	12	5	known	know	VERB
ma-381	12	6	that	that	SCONJ
ma-381	12	7	not	not	PART
ma-381	12	8	only	only	ADV
ma-381	12	9	this	this	DET
ma-381	12	10	space	space	NOUN
ma-381	12	11	does	do	AUX
ma-381	12	12	not	not	PART
ma-381	12	13	depends	depend	VERB
ma-381	12	14	on	on	ADP
ma-381	12	15	ϕ	ϕ	NOUN
ma-381	12	16	,	,	PUNCT
ma-381	12	17	but	but	CCONJ
ma-381	12	18	one	one	PRON
ma-381	12	19	can	can	AUX
ma-381	12	20	replaced	replace	VERB
ma-381	12	21	schwartzfunction	schwartzfunction	NOUN
ma-381	12	22	by	by	ADP
ma-381	12	23	poisson	poisson	PROPN
ma-381	12	24	kernel	kernel	PROPN
ma-381	12	25	in	in	ADP
ma-381	12	26	the	the	DET
ma-381	12	27	definition	definition	NOUN
ma-381	12	28	of	of	ADP
ma-381	12	29	the	the	DET
ma-381	12	30	maximal	maximal	ADJ
ma-381	12	31	function	function	NOUN
ma-381	12	32	(	(	PUNCT
ma-381	12	33	1.1).in	1.1).in	NUM
ma-381	12	34	[	[	X
ma-381	12	35	1	1	NUM
ma-381	12	36	]	]	PUNCT
ma-381	12	37	,	,	PUNCT
ma-381	12	38	ablé	ablé	PROPN
ma-381	13	1	and	and	CCONJ
ma-381	13	2	the	the	DET
ma-381	13	3	second	second	ADJ
ma-381	13	4	author	author	NOUN
ma-381	13	5	studied	study	VERB
ma-381	13	6	hardy	hardy	ADJ
ma-381	13	7	-	-	PUNCT
ma-381	13	8	amalgam	amalgam	NOUN
ma-381	13	9	spaces	space	NOUN
ma-381	13	10	h(p	h(p	NOUN
ma-381	13	11	,	,	PUNCT
ma-381	13	12	q)(rd	q)(rd	NUM
ma-381	13	13	)	)	PUNCT
ma-381	13	14	(	(	PUNCT
ma-381	13	15	0	0	PUNCT
ma-381	13	16	<	<	X
ma-381	13	17	p	p	X
ma-381	13	18	,	,	PUNCT
ma-381	13	19	q	q	ADJ
ma-381	13	20	<	<	X
ma-381	13	21	∞)by	∞)by	NOUN
ma-381	13	22	taking	take	VERB
ma-381	13	23	in	in	ADP
ma-381	13	24	the	the	DET
ma-381	13	25	above	above	ADJ
ma-381	13	26	maximal	maximal	ADJ
ma-381	13	27	characterization	characterization	NOUN
ma-381	13	28	of	of	ADP
ma-381	13	29	classical	classical	ADJ
ma-381	13	30	hardy	hardy	ADJ
ma-381	13	31	space	space	NOUN
ma-381	13	32	the	the	DET
ma-381	13	33	wiener	wiener	NOUN
ma-381	13	34	amalgamquasi	amalgamquasi	NOUN
ma-381	13	35	-	-	PUNCT
ma-381	13	36	norm	norm	NOUN
ma-381	13	37	‖·‖p	‖·‖p	ADJ
ma-381	13	38	,	,	PUNCT
ma-381	13	39	q	q	X
ma-381	13	40	instead	instead	ADV
ma-381	13	41	of	of	ADP
ma-381	13	42	lebesgue	lebesgue	NOUN
ma-381	13	43	’s	’s	PART
ma-381	13	44	.	.	PUNCT
ma-381	14	1	received	receive	VERB
ma-381	14	2	:	:	PUNCT
ma-381	14	3	14	14	NUM
ma-381	14	4	may	may	PROPN
ma-381	14	5	2025	2025	NUM
ma-381	14	6	.	.	PUNCT
ma-381	15	1	key	key	ADJ
ma-381	15	2	words	word	NOUN
ma-381	15	3	and	and	CCONJ
ma-381	15	4	phrases	phrase	NOUN
ma-381	15	5	.	.	PUNCT
ma-381	16	1	amalgam	amalgam	NOUN
ma-381	16	2	spaces	space	NOUN
ma-381	16	3	,	,	PUNCT
ma-381	16	4	hardy	hardy	ADJ
ma-381	16	5	-	-	PUNCT
ma-381	16	6	amalgam	amalgam	NOUN
ma-381	16	7	spaces	space	NOUN
ma-381	16	8	,	,	PUNCT
ma-381	16	9	generalized	generalize	VERB
ma-381	16	10	hardy	hardy	ADJ
ma-381	16	11	-	-	PUNCT
ma-381	16	12	morrey	morrey	NOUN
ma-381	16	13	spaces	space	NOUN
ma-381	16	14	,	,	PUNCT
ma-381	16	15	calderón	calderón	NOUN
ma-381	16	16	-	-	PUNCT
ma-381	16	17	zygmund	zygmund	ADJ
ma-381	16	18	operators	operator	NOUN
ma-381	16	19	,	,	PUNCT
ma-381	16	20	molecular	molecular	ADJ
ma-381	16	21	decomposition	decomposition	NOUN
ma-381	16	22	.	.	PUNCT
ma-381	17	1	1	1	NUM
ma-381	17	2	https://adac.ee	https://adac.ee	PROPN
ma-381	17	3	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	17	4	eur	eur	PROPN
ma-381	17	5	.	.	PUNCT
ma-381	18	1	j.	j.	PROPN
ma-381	18	2	math	math	PROPN
ma-381	18	3	.	.	PUNCT
ma-381	19	1	anal	anal	PROPN
ma-381	19	2	.	.	PUNCT
ma-381	20	1	10.28924	10.28924	NUM
ma-381	20	2	/	/	SYM
ma-381	20	3	ada	ada	PROPN
ma-381	20	4	/	/	SYM
ma-381	20	5	ma.5.21	ma.5.21	PROPN
ma-381	20	6	2a	2a	NUM
ma-381	20	7	locally	locally	ADV
ma-381	20	8	integrable	integrable	ADJ
ma-381	20	9	function	function	NOUN
ma-381	20	10	u	u	PROPN
ma-381	20	11	belongs	belong	VERB
ma-381	20	12	to	to	ADP
ma-381	20	13	the	the	DET
ma-381	20	14	amalgam	amalgam	NOUN
ma-381	20	15	space	space	NOUN
ma-381	20	16	(	(	PUNCT
ma-381	20	17	lp	lp	ADJ
ma-381	20	18	,	,	PUNCT
ma-381	20	19	`	`	PUNCT
ma-381	20	20	q)(rd	q)(rd	X
ma-381	20	21	)	)	PUNCT
ma-381	20	22	if	if	SCONJ
ma-381	20	23	‖u‖p	‖u‖p	NOUN
ma-381	20	24	,	,	PUNCT
ma-381	20	25	q	q	NOUN
ma-381	20	26	:	:	PUNCT
ma-381	20	27	=	=	SYM
ma-381	20	28	∑	∑	NOUN
ma-381	20	29	k∈zd	k∈zd	VERB
ma-381	20	30	‖uχqk‖	‖uχqk‖	PRON
ma-381	20	31	q	q	NOUN
ma-381	20	32	p	p	NOUN
ma-381	20	33			NOUN
ma-381	20	34	1	1	NUM
ma-381	20	35	q	q	NOUN
ma-381	20	36	<	<	X
ma-381	20	37	∞	∞	PROPN
ma-381	20	38	,	,	PUNCT
ma-381	20	39	where	where	SCONJ
ma-381	20	40	for	for	ADP
ma-381	20	41	k	k	PROPN
ma-381	20	42	∈	∈	PROPN
ma-381	20	43	zd	zd	PROPN
ma-381	20	44	,	,	PUNCT
ma-381	20	45	qk	qk	ADP
ma-381	20	46	=	=	SYM
ma-381	20	47	k	k	PROPN
ma-381	21	1	+	+	PUNCT
ma-381	21	2	[	[	X
ma-381	21	3	0	0	NUM
ma-381	21	4	,	,	PUNCT
ma-381	21	5	1)d	1)d	NUM
ma-381	21	6	and	and	CCONJ
ma-381	21	7	χqk	χqk	NOUN
ma-381	21	8	stands	stand	VERB
ma-381	21	9	for	for	ADP
ma-381	21	10	the	the	DET
ma-381	21	11	characteristic	characteristic	ADJ
ma-381	21	12	function	function	NOUN
ma-381	21	13	of	of	ADP
ma-381	21	14	qk	qk	NOUN
ma-381	21	15	.multiple	.multiple	PUNCT
ma-381	21	16	characterizations	characterization	NOUN
ma-381	21	17	of	of	ADP
ma-381	21	18	h(p	h(p	NOUN
ma-381	21	19	,	,	PUNCT
ma-381	21	20	q)(rd	q)(rd	NUM
ma-381	21	21	)	)	PUNCT
ma-381	21	22	spaces	space	NOUN
ma-381	21	23	including	include	VERB
ma-381	21	24	atomic	atomic	ADJ
ma-381	21	25	and	and	CCONJ
ma-381	21	26	poisson	poisson	PROPN
ma-381	21	27	kernel	kernel	PROPN
ma-381	21	28	character	character	PROPN
ma-381	21	29	-	-	PUNCT
ma-381	21	30	ization	ization	PROPN
ma-381	21	31	,	,	PUNCT
ma-381	21	32	were	be	AUX
ma-381	21	33	given	give	VERB
ma-381	21	34	in	in	ADP
ma-381	21	35	[	[	X
ma-381	21	36	1	1	NUM
ma-381	21	37	]	]	PUNCT
ma-381	21	38	.	.	PUNCT
ma-381	22	1	we	we	PRON
ma-381	22	2	notice	notice	VERB
ma-381	22	3	that	that	SCONJ
ma-381	22	4	the	the	DET
ma-381	22	5	atoms	atom	NOUN
ma-381	22	6	in	in	ADP
ma-381	22	7	this	this	DET
ma-381	22	8	context	context	NOUN
ma-381	22	9	are	be	AUX
ma-381	22	10	exactely	exactely	ADV
ma-381	22	11	the	the	DET
ma-381	22	12	one	one	NOUN
ma-381	22	13	used	use	VERB
ma-381	22	14	inclassical	inclassical	ADJ
ma-381	22	15	hardy	hardy	ADV
ma-381	22	16	space.recently	space.recently	ADV
ma-381	22	17	,	,	PUNCT
ma-381	22	18	assaubay	assaubay	VERB
ma-381	22	19	et	et	PROPN
ma-381	22	20	al	al	PROPN
ma-381	22	21	in	in	ADP
ma-381	22	22	[	[	X
ma-381	22	23	3	3	NUM
ma-381	22	24	]	]	PUNCT
ma-381	22	25	characterized	characterize	VERB
ma-381	22	26	this	this	DET
ma-381	22	27	spaces	space	NOUN
ma-381	22	28	by	by	ADP
ma-381	22	29	using	use	VERB
ma-381	22	30	first	first	ADJ
ma-381	22	31	-	-	PUNCT
ma-381	22	32	order	order	NOUN
ma-381	22	33	classical	classical	ADJ
ma-381	22	34	riesztransforms	riesztransform	NOUN
ma-381	22	35	and	and	CCONJ
ma-381	22	36	composition	composition	NOUN
ma-381	22	37	of	of	ADP
ma-381	22	38	first	first	ADJ
ma-381	22	39	-	-	PUNCT
ma-381	22	40	order	order	NOUN
ma-381	22	41	riesz	riesz	NOUN
ma-381	22	42	transformations	transformation	NOUN
ma-381	22	43	.	.	PUNCT
ma-381	23	1	they	they	PRON
ma-381	23	2	also	also	ADV
ma-381	23	3	describe	describe	VERB
ma-381	23	4	the	the	DET
ma-381	23	5	distributionsinh(p	distributionsinh(p	PROPN
ma-381	23	6	,	,	PUNCT
ma-381	23	7	q)(rd	q)(rd	NUM
ma-381	23	8	)	)	PUNCT
ma-381	23	9	as	as	ADP
ma-381	23	10	the	the	DET
ma-381	23	11	boundary	boundary	ADJ
ma-381	23	12	values	value	NOUN
ma-381	23	13	of	of	ADP
ma-381	23	14	solutions	solution	NOUN
ma-381	23	15	of	of	ADP
ma-381	23	16	harmonic	harmonic	ADJ
ma-381	23	17	and	and	CCONJ
ma-381	23	18	caloric	caloric	PROPN
ma-381	23	19	cauchy	cauchy	PROPN
ma-381	23	20	-	-	PUNCT
ma-381	23	21	riemann	riemann	PROPN
ma-381	23	22	systems.here	systems.here	NOUN
ma-381	23	23	we	we	PRON
ma-381	23	24	intend	intend	VERB
ma-381	23	25	to	to	PART
ma-381	23	26	prove	prove	VERB
ma-381	23	27	that	that	SCONJ
ma-381	23	28	similar	similar	ADJ
ma-381	23	29	characterizations	characterization	NOUN
ma-381	23	30	are	be	AUX
ma-381	23	31	possible	possible	ADJ
ma-381	23	32	in	in	ADP
ma-381	23	33	the	the	DET
ma-381	23	34	context	context	NOUN
ma-381	23	35	of	of	ADP
ma-381	23	36	hardy-fofanaspaces.it	hardy-fofanaspaces.it	NOUN
ma-381	23	37	is	be	AUX
ma-381	23	38	well	well	ADV
ma-381	23	39	known	know	VERB
ma-381	23	40	that	that	SCONJ
ma-381	23	41	for	for	ADP
ma-381	23	42	0	0	NUM
ma-381	23	43	<	<	X
ma-381	23	44	p	p	X
ma-381	23	45	,	,	PUNCT
ma-381	23	46	α	α	NOUN
ma-381	23	47	,	,	PUNCT
ma-381	23	48	q	q	X
ma-381	23	49	<	<	X
ma-381	23	50	∞	∞	NOUN
ma-381	23	51	and	and	CCONJ
ma-381	23	52	r	r	NOUN
ma-381	23	53	>	>	X
ma-381	23	54	0	0	NUM
ma-381	23	55	,	,	PUNCT
ma-381	23	56	there	there	PRON
ma-381	23	57	exists	exist	VERB
ma-381	23	58	a	a	DET
ma-381	23	59	constant	constant	ADJ
ma-381	23	60	cr	cr	NOUN
ma-381	23	61	;	;	PUNCT
ma-381	23	62	α	α	X
ma-381	23	63	>	>	X
ma-381	23	64	0	0	NUM
ma-381	23	65	dependingon	dependingon	PROPN
ma-381	23	66	r	r	NOUN
ma-381	23	67	and	and	CCONJ
ma-381	23	68	α	α	NOUN
ma-381	23	69	such	such	ADJ
ma-381	23	70	that	that	SCONJ
ma-381	23	71	c−1	c−1	PROPN
ma-381	23	72	r	r	NOUN
ma-381	23	73	;	;	PUNCT
ma-381	23	74	α‖u‖p	α‖u‖p	NOUN
ma-381	23	75	,	,	PUNCT
ma-381	23	76	q	q	NOUN
ma-381	23	77	≤	≤	X
ma-381	23	78	‖stαr	‖stαr	PUNCT
ma-381	23	79	u‖q	u‖q	NOUN
ma-381	23	80	,	,	PUNCT
ma-381	23	81	p	p	NOUN
ma-381	23	82	≤	≤	NUM
ma-381	23	83	cr	cr	NOUN
ma-381	23	84	;	;	PUNCT
ma-381	23	85	α‖u‖p	α‖u‖p	NOUN
ma-381	23	86	,	,	PUNCT
ma-381	23	87	q	q	NOUN
ma-381	23	88	,	,	PUNCT
ma-381	23	89	u	u	PROPN
ma-381	23	90	∈	∈	PROPN
ma-381	23	91	(	(	PUNCT
ma-381	23	92	lp	lp	NOUN
ma-381	23	93	,	,	PUNCT
ma-381	23	94	`	`	PUNCT
ma-381	23	95	q)(rd	q)(rd	NUM
ma-381	23	96	)	)	PUNCT
ma-381	23	97	,	,	PUNCT
ma-381	23	98	(	(	PUNCT
ma-381	23	99	1.2	1.2	NUM
ma-381	23	100	)	)	PUNCT
ma-381	23	101	where	where	SCONJ
ma-381	23	102	(	(	PUNCT
ma-381	23	103	stαr	stαr	VERB
ma-381	23	104	u)(x	u)(x	NUM
ma-381	23	105	)	)	PUNCT
ma-381	24	1	=	=	SYM
ma-381	24	2	r−	r−	PROPN
ma-381	24	3	d	d	PROPN
ma-381	24	4	α	α	NOUN
ma-381	24	5	u(r−1x	u(r−1x	NOUN
ma-381	24	6	)	)	PUNCT
ma-381	24	7	.	.	PUNCT
ma-381	25	1	it	it	PRON
ma-381	25	2	follows	follow	VERB
ma-381	25	3	from	from	ADP
ma-381	25	4	the	the	DET
ma-381	25	5	above	above	ADJ
ma-381	25	6	relation	relation	NOUN
ma-381	25	7	that	that	PRON
ma-381	25	8	for	for	ADP
ma-381	25	9	u	u	PROPN
ma-381	25	10	∈	∈	PROPN
ma-381	25	11	(	(	PUNCT
ma-381	25	12	lp	lp	NOUN
ma-381	25	13	,	,	PUNCT
ma-381	25	14	`	`	PUNCT
ma-381	25	15	q)(rd),we	q)(rd),we	INTJ
ma-381	25	16	have	have	AUX
ma-381	25	17	stαr	stαr	NOUN
ma-381	25	18	u	u	X
ma-381	25	19	∈	∈	PROPN
ma-381	25	20	(	(	PUNCT
ma-381	25	21	lp	lp	PROPN
ma-381	25	22	,	,	PUNCT
ma-381	25	23	`	`	PUNCT
ma-381	25	24	q)(rd	q)(rd	NUM
ma-381	25	25	)	)	PUNCT
ma-381	25	26	for	for	ADP
ma-381	25	27	α	α	PROPN
ma-381	25	28	>	>	X
ma-381	25	29	0	0	PUNCT
ma-381	26	1	and	and	CCONJ
ma-381	26	2	r	r	X
ma-381	26	3	>	>	X
ma-381	26	4	0	0	NUM
ma-381	26	5	.	.	PUNCT
ma-381	27	1	unfortunately	unfortunately	ADV
ma-381	27	2	,	,	PUNCT
ma-381	27	3	the	the	DET
ma-381	27	4	family	family	NOUN
ma-381	27	5	{	{	PUNCT
ma-381	27	6	stαr	stαr	X
ma-381	27	7	u}r>0	u}r>0	VERB
ma-381	27	8	is	be	AUX
ma-381	27	9	notbounded	notbounde	VERB
ma-381	27	10	in	in	ADP
ma-381	27	11	(	(	PUNCT
ma-381	27	12	lp	lp	ADJ
ma-381	27	13	,	,	PUNCT
ma-381	27	14	`	`	PUNCT
ma-381	27	15	q)(rd	q)(rd	NUM
ma-381	27	16	)	)	PUNCT
ma-381	27	17	.	.	PUNCT
ma-381	28	1	ibrahim	ibrahim	PROPN
ma-381	28	2	fofana	fofana	PROPN
ma-381	28	3	considered	consider	VERB
ma-381	28	4	in	in	ADP
ma-381	28	5	[	[	X
ma-381	28	6	7	7	NUM
ma-381	28	7	]	]	PUNCT
ma-381	28	8	,	,	PUNCT
ma-381	28	9	the	the	DET
ma-381	28	10	spaces	space	NOUN
ma-381	28	11	(	(	PUNCT
ma-381	28	12	lp	lp	ADJ
ma-381	28	13	,	,	PUNCT
ma-381	28	14	`	`	PUNCT
ma-381	28	15	q)α(rd	q)α(rd	X
ma-381	28	16	)	)	PUNCT
ma-381	28	17	defined	define	VERB
ma-381	28	18	for	for	ADP
ma-381	28	19	0	0	NUM
ma-381	28	20	<	<	X
ma-381	28	21	p	p	X
ma-381	28	22	,	,	PUNCT
ma-381	28	23	q	q	INTJ
ma-381	28	24	,	,	PUNCT
ma-381	28	25	α	α	NOUN
ma-381	28	26	≤	≤	NOUN
ma-381	28	27	∞	∞	NUM
ma-381	28	28	by	by	ADP
ma-381	28	29	(	(	PUNCT
ma-381	28	30	lp	lp	ADJ
ma-381	28	31	,	,	PUNCT
ma-381	28	32	`	`	PUNCT
ma-381	28	33	q)α(rd	q)α(rd	X
ma-381	28	34	)	)	PUNCT
ma-381	28	35	=	=	PRON
ma-381	28	36	{	{	PUNCT
ma-381	28	37	f	f	PROPN
ma-381	28	38	∈	∈	PROPN
ma-381	28	39	(	(	PUNCT
ma-381	28	40	lp	lp	NOUN
ma-381	28	41	,	,	PUNCT
ma-381	28	42	`	`	PUNCT
ma-381	28	43	q)(rd)/	q)(rd)/	X
ma-381	28	44	‖f	‖f	ADJ
ma-381	28	45	‖p	‖p	NOUN
ma-381	28	46	,	,	PUNCT
ma-381	28	47	q	q	X
ma-381	28	48	,	,	PUNCT
ma-381	28	49	α	α	X
ma-381	28	50	<	<	X
ma-381	28	51	∞	∞	NUM
ma-381	28	52	}	}	PUNCT
ma-381	28	53	where	where	SCONJ
ma-381	28	54	‖f	‖f	ADJ
ma-381	28	55	‖p	‖p	NOUN
ma-381	28	56	,	,	PUNCT
ma-381	28	57	q	q	X
ma-381	28	58	,	,	PUNCT
ma-381	28	59	α	α	NOUN
ma-381	28	60	:	:	PUNCT
ma-381	28	61	=	=	NUM
ma-381	28	62	sup	sup	NOUN
ma-381	28	63	r>0	r>0	PROPN
ma-381	28	64	‖stαr	‖stαr	PROPN
ma-381	28	65	f	f	PROPN
ma-381	28	66	‖p	‖p	PROPN
ma-381	28	67	,	,	PUNCT
ma-381	28	68	q	q	X
ma-381	28	69	.	.	PUNCT
ma-381	29	1	(	(	PUNCT
ma-381	29	2	1.3	1.3	NUM
ma-381	29	3	)	)	PUNCT
ma-381	29	4	these	these	DET
ma-381	29	5	spaces	space	NOUN
ma-381	29	6	known	know	VERB
ma-381	29	7	as	as	ADP
ma-381	29	8	fofana	fofana	PROPN
ma-381	29	9	’s	’s	PART
ma-381	29	10	spaces	space	NOUN
ma-381	29	11	are	be	AUX
ma-381	29	12	non	non	ADJ
ma-381	29	13	trivial	trivial	ADJ
ma-381	29	14	if	if	SCONJ
ma-381	29	15	and	and	CCONJ
ma-381	29	16	only	only	ADV
ma-381	29	17	if	if	SCONJ
ma-381	29	18	p	p	ADP
ma-381	29	19	≤	≤	X
ma-381	29	20	α	α	NOUN
ma-381	29	21	≤	≤	ADJ
ma-381	29	22	q	q	PUNCT
ma-381	29	23	(	(	PUNCT
ma-381	29	24	see	see	VERB
ma-381	29	25	[	[	X
ma-381	29	26	7	7	NUM
ma-381	29	27	]	]	NUM
ma-381	29	28	)	)	PUNCT
ma-381	29	29	.	.	PUNCT
ma-381	30	1	in	in	ADP
ma-381	30	2	therest	therest	NOUN
ma-381	30	3	of	of	ADP
ma-381	30	4	the	the	DET
ma-381	30	5	paper	paper	NOUN
ma-381	30	6	we	we	PRON
ma-381	30	7	will	will	AUX
ma-381	30	8	always	always	ADV
ma-381	30	9	assume	assume	VERB
ma-381	30	10	that	that	SCONJ
ma-381	30	11	this	this	DET
ma-381	30	12	condition	condition	NOUN
ma-381	30	13	is	be	AUX
ma-381	30	14	fulfilled	fulfil	VERB
ma-381	30	15	.	.	PUNCT
ma-381	31	1	it	it	PRON
ma-381	31	2	is	be	AUX
ma-381	31	3	proved	prove	VERB
ma-381	31	4	in	in	ADP
ma-381	31	5	[	[	X
ma-381	31	6	6	6	NUM
ma-381	31	7	]	]	PUNCT
ma-381	31	8	that	that	SCONJ
ma-381	31	9	for	for	ADP
ma-381	31	10	u	u	PROPN
ma-381	31	11	∈	∈	PROPN
ma-381	31	12	(	(	PUNCT
ma-381	31	13	lp	lp	ADJ
ma-381	31	14	,	,	PUNCT
ma-381	31	15	`	`	PUNCT
ma-381	31	16	q)α(rd	q)α(rd	X
ma-381	31	17	)	)	PUNCT
ma-381	31	18	,	,	PUNCT
ma-381	31	19	we	we	PRON
ma-381	31	20	have	have	VERB
ma-381	31	21	‖stαr	‖stαr	NOUN
ma-381	31	22	u‖p	u‖p	NOUN
ma-381	31	23	,	,	PUNCT
ma-381	31	24	q	q	NOUN
ma-381	31	25	,	,	PUNCT
ma-381	31	26	α	α	NOUN
ma-381	31	27	=	=	PUNCT
ma-381	31	28	‖u‖p	‖u‖p	NOUN
ma-381	31	29	,	,	PUNCT
ma-381	31	30	q	q	NOUN
ma-381	31	31	,	,	PUNCT
ma-381	31	32	α	α	NOUN
ma-381	31	33	and	and	CCONJ
ma-381	31	34	that	that	SCONJ
ma-381	31	35	(	(	PUNCT
ma-381	31	36	lp	lp	ADJ
ma-381	31	37	,	,	PUNCT
ma-381	31	38	`	`	PUNCT
ma-381	31	39	q)α(rd	q)α(rd	X
ma-381	31	40	)	)	PUNCT
ma-381	31	41	(	(	PUNCT
ma-381	31	42	1	1	NUM
ma-381	31	43	≤	≤	NOUN
ma-381	31	44	p	p	NOUN
ma-381	31	45	≤	≤	NUM
ma-381	31	46	α	α	PRON
ma-381	31	47	≤	≤	NUM
ma-381	32	1	q	q	X
ma-381	32	2	)	)	PUNCT
ma-381	32	3	is	be	AUX
ma-381	32	4	thebiggest	thebiggest	ADJ
ma-381	32	5	norm	norm	NOUN
ma-381	32	6	space	space	NOUN
ma-381	32	7	which	which	PRON
ma-381	32	8	is	be	AUX
ma-381	32	9	continuously	continuously	ADV
ma-381	32	10	embedded	embed	VERB
ma-381	32	11	in	in	ADP
ma-381	32	12	(	(	PUNCT
ma-381	32	13	lp	lp	ADJ
ma-381	32	14	,	,	PUNCT
ma-381	32	15	`	`	PUNCT
ma-381	32	16	q)(rd	q)(rd	NUM
ma-381	32	17	)	)	PUNCT
ma-381	32	18	and	and	CCONJ
ma-381	32	19	for	for	ADP
ma-381	32	20	which	which	PRON
ma-381	32	21	the	the	DET
ma-381	32	22	translation	translation	NOUN
ma-381	32	23	stαr	stαr	VERB
ma-381	32	24	is	be	AUX
ma-381	32	25	an	an	DET
ma-381	32	26	isometry	isometry	NOUN
ma-381	32	27	.	.	PUNCT
ma-381	33	1	these	these	DET
ma-381	33	2	spaces	space	NOUN
ma-381	33	3	can	can	AUX
ma-381	33	4	also	also	ADV
ma-381	33	5	be	be	AUX
ma-381	33	6	viewed	view	VERB
ma-381	33	7	as	as	ADP
ma-381	33	8	some	some	DET
ma-381	33	9	generalized	generalize	VERB
ma-381	33	10	morrey	morrey	NOUN
ma-381	33	11	spaces	space	NOUN
ma-381	33	12	since	since	SCONJ
ma-381	33	13	for	for	ADP
ma-381	33	14	p	p	NOUN
ma-381	33	15	<	<	X
ma-381	33	16	α	α	PROPN
ma-381	33	17	,	,	PUNCT
ma-381	33	18	the	the	DET
ma-381	33	19	space	space	NOUN
ma-381	33	20	(	(	PUNCT
ma-381	33	21	lp	lp	ADJ
ma-381	33	22	,	,	PUNCT
ma-381	33	23	`	`	PUNCT
ma-381	33	24	∞)α(rd	∞)α(rd	ADJ
ma-381	33	25	)	)	PUNCT
ma-381	33	26	is	be	AUX
ma-381	33	27	exactly	exactly	ADV
ma-381	33	28	the	the	DET
ma-381	33	29	classical	classical	ADJ
ma-381	33	30	morrey	morrey	PROPN
ma-381	33	31	space	space	NOUN
ma-381	33	32	lp	lp	PROPN
ma-381	33	33	,	,	PUNCT
ma-381	33	34	d	d	NOUN
ma-381	33	35	pα	pα	INTJ
ma-381	33	36	(	(	PUNCT
ma-381	33	37	rd).for	rd).for	ADP
ma-381	33	38	0	0	NUM
ma-381	33	39	<	<	X
ma-381	33	40	p	p	X
ma-381	33	41	≤	≤	NUM
ma-381	33	42	α	α	NOUN
ma-381	33	43	≤	≤	NUM
ma-381	33	44	q	q	X
ma-381	33	45	<	<	X
ma-381	33	46	∞	∞	ADJ
ma-381	33	47	,	,	PUNCT
ma-381	33	48	hardy	hardy	ADJ
ma-381	33	49	-	-	PUNCT
ma-381	33	50	fofana	fofana	NOUN
ma-381	33	51	space	space	NOUN
ma-381	33	52	h(p	h(p	NOUN
ma-381	33	53	,	,	PUNCT
ma-381	33	54	q	q	NOUN
ma-381	33	55	,	,	PUNCT
ma-381	33	56	α)(rd	α)(rd	NOUN
ma-381	33	57	)	)	PUNCT
ma-381	33	58	,	,	PUNCT
ma-381	33	59	introduced	introduce	VERB
ma-381	33	60	by	by	ADP
ma-381	33	61	the	the	DET
ma-381	33	62	authors	author	NOUN
ma-381	33	63	in	in	ADP
ma-381	33	64	[	[	X
ma-381	33	65	4	4	NUM
ma-381	33	66	]	]	PUNCT
ma-381	33	67	,	,	PUNCT
ma-381	33	68	isa	isa	NOUN
ma-381	33	69	subspace	subspace	NOUN
ma-381	33	70	of	of	ADP
ma-381	33	71	hardy	hardy	ADJ
ma-381	33	72	-	-	PUNCT
ma-381	33	73	amalgam	amalgam	NOUN
ma-381	33	74	spaces	space	NOUN
ma-381	33	75	consists	consist	VERB
ma-381	33	76	of	of	ADP
ma-381	33	77	tempered	temper	VERB
ma-381	33	78	distributions	distribution	NOUN
ma-381	33	79	f	f	X
ma-381	33	80	satisfying	satisfy	VERB
ma-381	33	81	‖f	‖f	ADP
ma-381	33	82	‖h(p	‖h(p	NOUN
ma-381	33	83	,	,	PUNCT
ma-381	33	84	q	q	NOUN
ma-381	33	85	,	,	PUNCT
ma-381	33	86	α	α	NOUN
ma-381	33	87	)	)	PUNCT
ma-381	33	88	:	:	PUNCT
ma-381	34	1	=	=	SYM
ma-381	34	2	‖mf	‖mf	NUM
ma-381	34	3	‖p	‖p	PROPN
ma-381	34	4	,	,	PUNCT
ma-381	34	5	q	q	X
ma-381	34	6	,	,	PUNCT
ma-381	34	7	α	α	PRON
ma-381	34	8	<	<	X
ma-381	34	9	∞.	∞.	PROPN
ma-381	34	10	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	34	11	eur	eur	PROPN
ma-381	34	12	.	.	PUNCT
ma-381	35	1	j.	j.	PROPN
ma-381	35	2	math	math	PROPN
ma-381	35	3	.	.	PUNCT
ma-381	36	1	anal	anal	PROPN
ma-381	36	2	.	.	PUNCT
ma-381	37	1	10.28924	10.28924	NUM
ma-381	37	2	/	/	SYM
ma-381	37	3	ada	ada	PROPN
ma-381	37	4	/	/	SYM
ma-381	37	5	ma.5.21	ma.5.21	NOUN
ma-381	37	6	3the	3the	NUM
ma-381	37	7	purpose	purpose	NOUN
ma-381	37	8	of	of	ADP
ma-381	37	9	this	this	DET
ma-381	37	10	article	article	NOUN
ma-381	37	11	is	be	AUX
ma-381	37	12	twofold	twofold	ADV
ma-381	37	13	.	.	PUNCT
ma-381	38	1	we	we	PRON
ma-381	38	2	first	first	ADV
ma-381	38	3	characterize	characterize	VERB
ma-381	38	4	these	these	DET
ma-381	38	5	spaces	space	NOUN
ma-381	38	6	via	via	ADP
ma-381	38	7	riesz	riesz	PROPN
ma-381	38	8	transforms	transform	VERB
ma-381	38	9	andsecondly	andsecondly	ADV
ma-381	38	10	,	,	PUNCT
ma-381	38	11	we	we	PRON
ma-381	38	12	describe	describe	VERB
ma-381	38	13	the	the	DET
ma-381	38	14	distributions	distribution	NOUN
ma-381	38	15	belonging	belong	VERB
ma-381	38	16	to	to	ADP
ma-381	38	17	these	these	DET
ma-381	38	18	spaces	space	NOUN
ma-381	38	19	as	as	SCONJ
ma-381	38	20	bounded	bounded	ADJ
ma-381	38	21	solutions	solution	NOUN
ma-381	38	22	of	of	ADP
ma-381	38	23	certaingeneral	certaingeneral	ADJ
ma-381	38	24	temperature	temperature	NOUN
ma-381	38	25	equations	equation	NOUN
ma-381	38	26	of	of	ADP
ma-381	38	27	cauchy-riemann.this	cauchy-riemann.this	PRON
ma-381	38	28	paper	paper	NOUN
ma-381	38	29	is	be	AUX
ma-381	38	30	organized	organize	VERB
ma-381	38	31	as	as	ADP
ma-381	38	32	follow	follow	NOUN
ma-381	38	33	:	:	PUNCT
ma-381	38	34	the	the	DET
ma-381	38	35	next	next	ADJ
ma-381	38	36	section	section	NOUN
ma-381	38	37	is	be	AUX
ma-381	38	38	devoted	devote	VERB
ma-381	38	39	to	to	ADP
ma-381	38	40	the	the	DET
ma-381	38	41	prerequisites	prerequisite	NOUN
ma-381	38	42	on	on	ADP
ma-381	38	43	hardy	hardy	ADJ
ma-381	38	44	-	-	PUNCT
ma-381	38	45	fofana	fofana	NOUN
ma-381	38	46	spaces	space	NOUN
ma-381	38	47	.	.	PUNCT
ma-381	39	1	in	in	ADP
ma-381	39	2	section	section	NOUN
ma-381	39	3	3	3	NUM
ma-381	39	4	,	,	PUNCT
ma-381	39	5	wegive	wegive	ADJ
ma-381	39	6	the	the	DET
ma-381	39	7	characterizations	characterization	NOUN
ma-381	39	8	of	of	ADP
ma-381	39	9	hardy	hardy	ADJ
ma-381	39	10	-	-	PUNCT
ma-381	39	11	fofana	fofana	NOUN
ma-381	39	12	spaces	space	NOUN
ma-381	39	13	with	with	ADP
ma-381	39	14	riesz	riesz	NOUN
ma-381	39	15	transforms	transform	VERB
ma-381	39	16	.	.	PUNCT
ma-381	40	1	in	in	ADP
ma-381	40	2	the	the	DET
ma-381	40	3	last	last	ADJ
ma-381	40	4	section	section	NOUN
ma-381	40	5	,	,	PUNCT
ma-381	40	6	we	we	PRON
ma-381	40	7	characterize	characterize	VERB
ma-381	40	8	distributions	distribution	NOUN
ma-381	40	9	belonging	belong	VERB
ma-381	40	10	to	to	ADP
ma-381	40	11	our	our	PRON
ma-381	40	12	spaces	space	NOUN
ma-381	40	13	as	as	ADP
ma-381	40	14	bounded	bounded	ADJ
ma-381	40	15	solutions	solution	NOUN
ma-381	40	16	of	of	ADP
ma-381	40	17	certain	certain	ADJ
ma-381	40	18	generaltemperature	generaltemperature	NOUN
ma-381	40	19	equations	equation	NOUN
ma-381	40	20	of	of	ADP
ma-381	40	21	cauchy-riemann.in	cauchy-riemann.in	ADJ
ma-381	40	22	this	this	DET
ma-381	40	23	work	work	NOUN
ma-381	40	24	,	,	PUNCT
ma-381	40	25	s	s	PART
ma-381	40	26	:	:	PUNCT
ma-381	40	27	=	=	SYM
ma-381	40	28	s(rd	s(rd	X
ma-381	40	29	)	)	PUNCT
ma-381	40	30	will	will	AUX
ma-381	40	31	denote	denote	VERB
ma-381	40	32	the	the	DET
ma-381	40	33	schwartz	schwartz	PROPN
ma-381	40	34	class	class	NOUN
ma-381	40	35	of	of	ADP
ma-381	40	36	rapidly	rapidly	ADV
ma-381	40	37	decreasing	decrease	VERB
ma-381	40	38	smooth	smooth	ADJ
ma-381	40	39	functionsequipped	functionsequippe	VERB
ma-381	40	40	with	with	ADP
ma-381	40	41	its	its	PRON
ma-381	40	42	usual	usual	ADJ
ma-381	40	43	topology	topology	NOUN
ma-381	40	44	.	.	PUNCT
ma-381	41	1	the	the	DET
ma-381	41	2	dual	dual	ADJ
ma-381	41	3	space	space	NOUN
ma-381	41	4	of	of	ADP
ma-381	41	5	s	s	PROPN
ma-381	41	6	is	be	AUX
ma-381	41	7	the	the	DET
ma-381	41	8	space	space	NOUN
ma-381	41	9	of	of	ADP
ma-381	41	10	tempered	temper	VERB
ma-381	41	11	distributionsdenoted	distributionsdenote	VERB
ma-381	41	12	by	by	ADP
ma-381	41	13	s	s	NOUN
ma-381	41	14	′	′	NUM
ma-381	41	15	:	:	PUNCT
ma-381	41	16	=	=	SYM
ma-381	41	17	s	s	VERB
ma-381	41	18	′(rd	′(rd	NOUN
ma-381	41	19	)	)	PUNCT
ma-381	41	20	.	.	PUNCT
ma-381	42	1	the	the	DET
ma-381	42	2	pairing	pairing	NOUN
ma-381	42	3	between	between	ADP
ma-381	42	4	s	s	PRON
ma-381	42	5	′	′	NUM
ma-381	42	6	and	and	CCONJ
ma-381	42	7	s	s	VERB
ma-381	42	8	is	be	AUX
ma-381	42	9	denoted	denote	VERB
ma-381	42	10	by	by	ADP
ma-381	42	11	〈	〈	PROPN
ma-381	42	12	·	·	PROPN
ma-381	42	13	,	,	PUNCT
ma-381	42	14	·	·	SYM
ma-381	42	15	〉	〉	PROPN
ma-381	42	16	.we	.we	PUNCT
ma-381	42	17	denote	denote	VERB
ma-381	42	18	by	by	ADP
ma-381	42	19	|e|	|e|	PROPN
ma-381	42	20	,	,	PUNCT
ma-381	42	21	the	the	DET
ma-381	42	22	lebesgue	lebesgue	ADJ
ma-381	42	23	measure	measure	NOUN
ma-381	42	24	of	of	ADP
ma-381	42	25	a	a	DET
ma-381	42	26	measurable	measurable	ADJ
ma-381	42	27	subset	subset	NOUN
ma-381	42	28	e	e	PROPN
ma-381	42	29	of	of	ADP
ma-381	42	30	rd	rd	PROPN
ma-381	42	31	.	.	PUNCT
ma-381	43	1	the	the	DET
ma-381	43	2	notation	notation	NOUN
ma-381	43	3	a	a	DET
ma-381	43	4	≈	≈	PROPN
ma-381	43	5	bmeans	bmean	NOUN
ma-381	43	6	that	that	SCONJ
ma-381	43	7	there	there	PRON
ma-381	43	8	exist	exist	VERB
ma-381	43	9	two	two	NUM
ma-381	43	10	constants	constant	NOUN
ma-381	43	11	0	0	NUM
ma-381	43	12	<	<	X
ma-381	43	13	c1	c1	PROPN
ma-381	43	14	and	and	CCONJ
ma-381	43	15	0	0	NUM
ma-381	43	16	<	<	X
ma-381	43	17	c2	c2	PROPN
ma-381	43	18	such	such	ADJ
ma-381	43	19	that	that	SCONJ
ma-381	43	20	a	a	DET
ma-381	43	21	≤	≤	NUM
ma-381	43	22	c1b	c1b	PROPN
ma-381	43	23	and	and	CCONJ
ma-381	43	24	b	b	NOUN
ma-381	43	25	≤	≤	NUM
ma-381	43	26	c2a	c2a	NOUN
ma-381	43	27	,	,	PUNCT
ma-381	43	28	while	while	SCONJ
ma-381	43	29	a	a	DET
ma-381	43	30	:	:	PUNCT
ma-381	43	31	=	=	SYM
ma-381	43	32	b	b	NOUN
ma-381	43	33	means	mean	VERB
ma-381	43	34	that	that	SCONJ
ma-381	43	35	b	b	NOUN
ma-381	43	36	is	be	AUX
ma-381	43	37	the	the	DET
ma-381	43	38	definition	definition	NOUN
ma-381	43	39	of	of	ADP
ma-381	43	40	a.	a.	NOUN
ma-381	43	41	2	2	NUM
ma-381	43	42	.	.	PUNCT
ma-381	43	43	prerequisites	prerequisite	NOUN
ma-381	43	44	for	for	ADP
ma-381	43	45	hardy	hardy	ADJ
ma-381	43	46	-	-	PUNCT
ma-381	43	47	fofana	fofana	ADJ
ma-381	43	48	spaces	space	NOUN
ma-381	43	49	fofana	fofana	PROPN
ma-381	43	50	’s	’s	PART
ma-381	43	51	spaces	space	NOUN
ma-381	43	52	have	have	VERB
ma-381	43	53	among	among	ADP
ma-381	43	54	others	other	NOUN
ma-381	43	55	,	,	PUNCT
ma-381	43	56	the	the	DET
ma-381	43	57	following	follow	VERB
ma-381	43	58	properties	property	NOUN
ma-381	43	59	(	(	PUNCT
ma-381	43	60	see	see	VERB
ma-381	43	61	for	for	ADP
ma-381	43	62	example	example	NOUN
ma-381	44	1	[	[	X
ma-381	44	2	6	6	NUM
ma-381	44	3	]	]	PUNCT
ma-381	44	4	and	and	CCONJ
ma-381	44	5	[	[	X
ma-381	44	6	7]):(1	7]):(1	X
ma-381	44	7	)	)	PUNCT
ma-381	44	8	let	let	VERB
ma-381	44	9	0	0	NUM
ma-381	44	10	<	<	X
ma-381	44	11	p	p	X
ma-381	44	12	,	,	PUNCT
ma-381	44	13	α	α	PROPN
ma-381	44	14	,	,	PUNCT
ma-381	44	15	q	q	PROPN
ma-381	44	16	≤	≤	PUNCT
ma-381	44	17	∞.	∞.	PROPN
ma-381	44	18	the	the	DET
ma-381	44	19	space	space	NOUN
ma-381	44	20	(	(	PUNCT
ma-381	44	21	(	(	PUNCT
ma-381	44	22	lp	lp	ADJ
ma-381	44	23	,	,	PUNCT
ma-381	44	24	`	`	PUNCT
ma-381	44	25	q)α(rd	q)α(rd	X
ma-381	44	26	)	)	PUNCT
ma-381	44	27	,	,	PUNCT
ma-381	44	28	‖·‖p	‖·‖p	ADJ
ma-381	44	29	,	,	PUNCT
ma-381	44	30	q	q	NOUN
ma-381	44	31	,	,	PUNCT
ma-381	44	32	α	α	PROPN
ma-381	44	33	)	)	PUNCT
ma-381	44	34	is	be	AUX
ma-381	44	35	a	a	DET
ma-381	44	36	banach	banach	NOUN
ma-381	44	37	space	space	NOUN
ma-381	44	38	if	if	SCONJ
ma-381	44	39	1	1	NUM
ma-381	44	40	≤	≤	NOUN
ma-381	44	41	p	p	NOUN
ma-381	44	42	≤	≤	NUM
ma-381	44	43	α	α	PRON
ma-381	44	44	≤	≤	ADJ
ma-381	44	45	q	q	NOUN
ma-381	44	46	and	and	CCONJ
ma-381	44	47	a	a	DET
ma-381	44	48	quasi	quasi	ADJ
ma-381	44	49	-	-	ADJ
ma-381	44	50	banach	banach	ADJ
ma-381	44	51	space	space	NOUN
ma-381	44	52	if	if	SCONJ
ma-381	44	53	0	0	NUM
ma-381	44	54	<	<	X
ma-381	44	55	p	p	X
ma-381	44	56	<	<	X
ma-381	44	57	1;(2	1;(2	NUM
ma-381	44	58	)	)	PUNCT
ma-381	44	59	if	if	SCONJ
ma-381	44	60	α	α	PRON
ma-381	44	61	∈	∈	PROPN
ma-381	44	62	{	{	PUNCT
ma-381	44	63	p	p	X
ma-381	44	64	,	,	PUNCT
ma-381	44	65	q	q	ADJ
ma-381	44	66	}	}	PUNCT
ma-381	44	67	then	then	ADV
ma-381	44	68	(	(	PUNCT
ma-381	44	69	lp	lp	ADJ
ma-381	44	70	,	,	PUNCT
ma-381	44	71	`	`	PUNCT
ma-381	44	72	q)α(rd	q)α(rd	X
ma-381	44	73	)	)	PUNCT
ma-381	44	74	=	=	SYM
ma-381	44	75	lα(rd	lα(rd	ADJ
ma-381	44	76	)	)	PUNCT
ma-381	44	77	with	with	ADP
ma-381	44	78	equivalent	equivalent	ADJ
ma-381	44	79	norms;(3	norms;(3	PROPN
ma-381	44	80	)	)	PUNCT
ma-381	44	81	if	if	SCONJ
ma-381	44	82	p	p	PROPN
ma-381	44	83	<	<	X
ma-381	44	84	α	α	X
ma-381	44	85	<	<	X
ma-381	44	86	q	q	X
ma-381	44	87	then	then	ADV
ma-381	44	88	lα(rd	lα(rd	ADJ
ma-381	44	89	)	)	PUNCT
ma-381	44	90	(	(	PUNCT
ma-381	44	91	(	(	PUNCT
ma-381	44	92	lp	lp	ADJ
ma-381	44	93	,	,	PUNCT
ma-381	44	94	`	`	PUNCT
ma-381	44	95	q)α(rd	q)α(rd	X
ma-381	44	96	)	)	PUNCT
ma-381	44	97	(	(	PUNCT
ma-381	44	98	(	(	PUNCT
ma-381	44	99	lp	lp	ADJ
ma-381	44	100	,	,	PUNCT
ma-381	44	101	`	`	PUNCT
ma-381	44	102	q)(rd);(4	q)(rd);(4	VERB
ma-381	44	103	)	)	PUNCT
ma-381	44	104	let	let	VERB
ma-381	44	105	f	f	PROPN
ma-381	44	106	and	and	CCONJ
ma-381	44	107	g	g	PROPN
ma-381	44	108	be	be	AUX
ma-381	44	109	two	two	NUM
ma-381	44	110	measurable	measurable	ADJ
ma-381	44	111	functions	function	NOUN
ma-381	44	112	on	on	ADP
ma-381	44	113	rd	rd	NOUN
ma-381	44	114	.	.	PUNCT
ma-381	45	1	if	if	SCONJ
ma-381	45	2	|f	|f	PROPN
ma-381	45	3	|	|	ADV
ma-381	45	4	≤	≤	NUM
ma-381	45	5	|g|	|g|	ADJ
ma-381	45	6	,	,	PUNCT
ma-381	45	7	then	then	ADV
ma-381	45	8	‖f	‖f	ADP
ma-381	45	9	‖p	‖p	PROPN
ma-381	45	10	,	,	PUNCT
ma-381	45	11	q	q	X
ma-381	45	12	,	,	PUNCT
ma-381	45	13	α	α	PROPN
ma-381	45	14	≤	≤	NUM
ma-381	45	15	‖g‖p	‖g‖p	NOUN
ma-381	45	16	,	,	PUNCT
ma-381	45	17	q	q	X
ma-381	45	18	,	,	PUNCT
ma-381	45	19	α.for	α.for	ADP
ma-381	45	20	many	many	ADJ
ma-381	45	21	operators	operator	NOUN
ma-381	45	22	including	include	VERB
ma-381	45	23	the	the	DET
ma-381	45	24	maximal	maximal	ADJ
ma-381	45	25	hardy	hardy	ADJ
ma-381	45	26	-	-	PUNCT
ma-381	45	27	littlewood	littlewood	NOUN
ma-381	45	28	operator	operator	NOUN
ma-381	45	29	,	,	PUNCT
ma-381	45	30	norm	norm	NOUN
ma-381	45	31	inequalities	inequality	NOUN
ma-381	45	32	aregiven	aregiven	VERB
ma-381	45	33	in	in	ADP
ma-381	45	34	these	these	DET
ma-381	45	35	spaces	space	NOUN
ma-381	45	36	for	for	ADP
ma-381	45	37	1	1	NUM
ma-381	45	38	≤	≤	NOUN
ma-381	45	39	p	p	NOUN
ma-381	45	40	≤	≤	NUM
ma-381	45	41	α	α	NOUN
ma-381	45	42	≤	≤	NOUN
ma-381	46	1	q.let	q.let	PROPN
ma-381	46	2	f	f	PROPN
ma-381	46	3	be	be	AUX
ma-381	46	4	a	a	DET
ma-381	46	5	locally	locally	ADV
ma-381	46	6	integrable	integrable	ADJ
ma-381	46	7	function	function	NOUN
ma-381	46	8	and	and	CCONJ
ma-381	46	9	m(f	m(f	PROPN
ma-381	46	10	)	)	PUNCT
ma-381	46	11	be	be	AUX
ma-381	46	12	the	the	DET
ma-381	46	13	centered	center	VERB
ma-381	46	14	hardy	hardy	ADJ
ma-381	46	15	-	-	PUNCT
ma-381	46	16	littlewood	littlewood	NOUN
ma-381	46	17	maximalfunction	maximalfunction	NOUN
ma-381	46	18	defined	define	VERB
ma-381	46	19	by	by	ADP
ma-381	46	20	m(f	m(f	PROPN
ma-381	46	21	)	)	PUNCT
ma-381	46	22	(	(	PUNCT
ma-381	46	23	x	x	X
ma-381	46	24	)	)	PUNCT
ma-381	46	25	:	:	PUNCT
ma-381	47	1	=	=	SYM
ma-381	47	2	sup	sup	NUM
ma-381	47	3	r>0	r>0	PROPN
ma-381	47	4	|b(x	|b(x	PROPN
ma-381	47	5	,	,	PUNCT
ma-381	47	6	r)|−1	r)|−1	VERB
ma-381	47	7	∫	∫	PROPN
ma-381	47	8	b(x	b(x	PROPN
ma-381	47	9	,	,	PUNCT
ma-381	47	10	r	r	NOUN
ma-381	47	11	)	)	PUNCT
ma-381	47	12	|f	|f	PROPN
ma-381	48	1	(	(	PUNCT
ma-381	48	2	y)|dy	y)|dy	NOUN
ma-381	48	3	,	,	PUNCT
ma-381	48	4	∀	∀	X
ma-381	48	5	x	x	SYM
ma-381	48	6	∈	∈	PROPN
ma-381	48	7	rd	rd	NOUN
ma-381	48	8	.	.	PUNCT
ma-381	49	1	it	it	PRON
ma-381	49	2	is	be	AUX
ma-381	49	3	proved	prove	VERB
ma-381	49	4	in	in	ADP
ma-381	49	5	[	[	X
ma-381	49	6	6	6	NUM
ma-381	49	7	,	,	PUNCT
ma-381	49	8	proposition	proposition	NOUN
ma-381	49	9	4.2	4.2	NUM
ma-381	49	10	]	]	PUNCT
ma-381	49	11	that	that	PRON
ma-381	49	12	m	m	VERB
ma-381	49	13	is	be	AUX
ma-381	49	14	bounded	bound	VERB
ma-381	49	15	on	on	ADP
ma-381	49	16	(	(	PUNCT
ma-381	49	17	lp	lp	ADJ
ma-381	49	18	,	,	PUNCT
ma-381	49	19	`	`	PUNCT
ma-381	49	20	q)α(rd	q)α(rd	X
ma-381	49	21	)	)	PUNCT
ma-381	49	22	,	,	PUNCT
ma-381	49	23	whenever	whenever	SCONJ
ma-381	49	24	1	1	NUM
ma-381	49	25	<	<	X
ma-381	49	26	p	p	X
ma-381	49	27	≤	≤	NUM
ma-381	49	28	α	α	PRON
ma-381	49	29	≤	≤	NUM
ma-381	49	30	q	q	PROPN
ma-381	49	31	≤	≤	NUM
ma-381	49	32	∞.	∞.	PROPN
ma-381	49	33	using	use	VERB
ma-381	49	34	[	[	PUNCT
ma-381	49	35	8	8	NUM
ma-381	49	36	,	,	PUNCT
ma-381	49	37	proposition	proposition	NOUN
ma-381	49	38	11.12	11.12	NUM
ma-381	49	39	]	]	PUNCT
ma-381	49	40	,	,	PUNCT
ma-381	49	41	it	it	PRON
ma-381	49	42	is	be	AUX
ma-381	49	43	easy	easy	ADJ
ma-381	49	44	to	to	PART
ma-381	49	45	extablish	extablish	VERB
ma-381	49	46	the	the	DET
ma-381	49	47	following	following	ADJ
ma-381	49	48	result	result	NOUN
ma-381	49	49	whose	whose	DET
ma-381	49	50	proof	proof	NOUN
ma-381	49	51	is	be	AUX
ma-381	49	52	omitted	omit	VERB
ma-381	49	53	.	.	PUNCT
ma-381	50	1	proposition	proposition	NOUN
ma-381	50	2	2.1	2.1	NUM
ma-381	50	3	.	.	PUNCT
ma-381	51	1	let	let	VERB
ma-381	51	2	1	1	NUM
ma-381	51	3	<	<	X
ma-381	51	4	p	p	X
ma-381	51	5	≤	≤	NUM
ma-381	51	6	α	α	NOUN
ma-381	51	7	≤	≤	NUM
ma-381	51	8	q	q	X
ma-381	51	9	<	<	X
ma-381	52	1	+	+	NOUN
ma-381	52	2	∞	∞	NUM
ma-381	52	3	and	and	CCONJ
ma-381	52	4	1	1	NUM
ma-381	52	5	<	<	X
ma-381	52	6	u	u	X
ma-381	52	7	≤	≤	X
ma-381	53	1	+	+	CCONJ
ma-381	53	2	∞.	∞.	PROPN
ma-381	53	3	for	for	ADP
ma-381	53	4	all	all	DET
ma-381	53	5	sequences	sequence	NOUN
ma-381	53	6	{	{	PUNCT
ma-381	53	7	fn}n≥0	fn}n≥0	NOUN
ma-381	53	8	of	of	ADP
ma-381	53	9	measurable	measurable	ADJ
ma-381	53	10	functions	function	NOUN
ma-381	53	11	,	,	PUNCT
ma-381	53	12	we	we	PRON
ma-381	53	13	have∥∥∥∥∥∥∥	have∥∥∥∥∥∥∥	VERB
ma-381	53	14	∑	∑	VERB
ma-381	53	15	n≥0	n≥0	ADJ
ma-381	53	16	|m(fn)|u	|m(fn)|u	VERB
ma-381	53	17			PROPN
ma-381	53	18	1	1	NUM
ma-381	53	19	u	u	NOUN
ma-381	53	20	∥∥∥∥∥∥∥	∥∥∥∥∥∥∥	PROPN
ma-381	53	21	p	p	PROPN
ma-381	53	22	,	,	PUNCT
ma-381	53	23	q	q	X
ma-381	53	24	,	,	PUNCT
ma-381	53	25	α	α	PROPN
ma-381	53	26	≈	≈	PROPN
ma-381	53	27	∥∥∥∥∥∥∥	∥∥∥∥∥∥∥	PROPN
ma-381	53	28	∑	∑	PROPN
ma-381	53	29	n≥0	n≥0	ADJ
ma-381	53	30	|fn|u	|fn|u	X
ma-381	53	31			PROPN
ma-381	53	32	1	1	NUM
ma-381	53	33	u	u	NOUN
ma-381	53	34	∥∥∥∥∥∥∥	∥∥∥∥∥∥∥	PROPN
ma-381	53	35	p	p	PROPN
ma-381	53	36	,	,	PUNCT
ma-381	53	37	q	q	X
ma-381	53	38	,	,	PUNCT
ma-381	53	39	α	α	NOUN
ma-381	53	40	,	,	PUNCT
ma-381	53	41	with	with	ADP
ma-381	53	42	the	the	DET
ma-381	53	43	equivalence	equivalence	NOUN
ma-381	53	44	constants	constant	NOUN
ma-381	53	45	not	not	PART
ma-381	53	46	depending	depend	VERB
ma-381	53	47	on	on	ADP
ma-381	53	48	the	the	DET
ma-381	53	49	sequence	sequence	NOUN
ma-381	53	50	{	{	PUNCT
ma-381	53	51	fn}n≥0	fn}n≥0	NOUN
ma-381	53	52	.	.	PUNCT
ma-381	54	1	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	54	2	eur	eur	PROPN
ma-381	54	3	.	.	PUNCT
ma-381	55	1	j.	j.	PROPN
ma-381	55	2	math	math	PROPN
ma-381	55	3	.	.	PUNCT
ma-381	56	1	anal	anal	PROPN
ma-381	56	2	.	.	PUNCT
ma-381	57	1	10.28924	10.28924	NUM
ma-381	57	2	/	/	SYM
ma-381	57	3	ada	ada	PROPN
ma-381	57	4	/	/	SYM
ma-381	57	5	ma.5.21	ma.5.21	PROPN
ma-381	57	6	4as	4as	ADJ
ma-381	57	7	hardy	hardy	ADJ
ma-381	57	8	-	-	PUNCT
ma-381	57	9	fofana	fofana	NOUN
ma-381	57	10	spaces	space	NOUN
ma-381	57	11	are	be	AUX
ma-381	57	12	concerned	concern	VERB
ma-381	57	13	,	,	PUNCT
ma-381	57	14	we	we	PRON
ma-381	57	15	have	have	VERB
ma-381	57	16	among	among	ADP
ma-381	57	17	others	other	NOUN
ma-381	57	18	,	,	PUNCT
ma-381	57	19	the	the	DET
ma-381	57	20	following	follow	VERB
ma-381	57	21	properties	property	NOUN
ma-381	57	22	whichcan	whichcan	AUX
ma-381	57	23	be	be	AUX
ma-381	57	24	found	find	VERB
ma-381	57	25	in	in	ADP
ma-381	57	26	[	[	X
ma-381	57	27	4	4	NUM
ma-381	57	28	]	]	PUNCT
ma-381	57	29	.	.	PUNCT
ma-381	58	1	proposition	proposition	NOUN
ma-381	58	2	2.2	2.2	NUM
ma-381	58	3	.	.	PUNCT
ma-381	59	1	let	let	VERB
ma-381	59	2	1	1	NUM
ma-381	59	3	≤	≤	NOUN
ma-381	59	4	p	p	NOUN
ma-381	59	5	≤	≤	NUM
ma-381	59	6	α	α	PRON
ma-381	59	7	≤	≤	NUM
ma-381	59	8	q	q	X
ma-381	59	9	<	<	X
ma-381	59	10	∞.(1	∞.(1	X
ma-381	59	11	)	)	PUNCT
ma-381	59	12	if	if	SCONJ
ma-381	59	13	1	1	NUM
ma-381	59	14	<	<	X
ma-381	59	15	p	p	X
ma-381	59	16	then	then	ADV
ma-381	59	17	the	the	DET
ma-381	59	18	space	space	NOUN
ma-381	59	19	h(p	h(p	NOUN
ma-381	59	20	,	,	PUNCT
ma-381	59	21	q	q	NOUN
ma-381	59	22	,	,	PUNCT
ma-381	59	23	α)(rd	α)(rd	NOUN
ma-381	59	24	)	)	PUNCT
ma-381	59	25	and	and	CCONJ
ma-381	59	26	(	(	PUNCT
ma-381	59	27	lp	lp	ADJ
ma-381	59	28	,	,	PUNCT
ma-381	59	29	lq)α(rd	lq)α(rd	ADJ
ma-381	59	30	)	)	PUNCT
ma-381	59	31	are	be	AUX
ma-381	59	32	equal	equal	ADJ
ma-381	59	33	with	with	ADP
ma-381	59	34	equivalence	equivalence	NOUN
ma-381	59	35	norms.(2	norms.(2	PROPN
ma-381	59	36	)	)	PUNCT
ma-381	59	37	the	the	DET
ma-381	59	38	space	space	NOUN
ma-381	59	39	h(1,q	h(1,q	NOUN
ma-381	59	40	,	,	PUNCT
ma-381	59	41	α)(rd	α)(rd	NUM
ma-381	59	42	)	)	PUNCT
ma-381	59	43	is	be	AUX
ma-381	59	44	continuously	continuously	ADV
ma-381	59	45	embedded	embed	VERB
ma-381	59	46	in	in	ADP
ma-381	59	47	(	(	PUNCT
ma-381	59	48	l1	l1	PROPN
ma-381	59	49	,	,	PUNCT
ma-381	59	50	`	`	PUNCT
ma-381	59	51	q)α(rd	q)α(rd	X
ma-381	59	52	)	)	PUNCT
ma-381	59	53	.	.	PUNCT
ma-381	60	1	notice	notice	VERB
ma-381	60	2	that	that	SCONJ
ma-381	60	3	for	for	ADP
ma-381	60	4	p	p	PRON
ma-381	60	5	<	<	X
ma-381	60	6	1	1	NUM
ma-381	60	7	,	,	PUNCT
ma-381	60	8	we	we	PRON
ma-381	60	9	have	have	VERB
ma-381	60	10	as	as	ADP
ma-381	60	11	in	in	ADP
ma-381	60	12	the	the	DET
ma-381	60	13	classical	classical	ADJ
ma-381	60	14	hardy	hardy	ADJ
ma-381	60	15	and	and	CCONJ
ma-381	60	16	hardy	hardy	ADJ
ma-381	60	17	-	-	PUNCT
ma-381	60	18	amalgam	amalgam	NOUN
ma-381	60	19	spaces	space	NOUN
ma-381	60	20	,	,	PUNCT
ma-381	60	21	that	that	SCONJ
ma-381	60	22	thespaces	thespace	NOUN
ma-381	60	23	(	(	PUNCT
ma-381	60	24	h(p	h(p	NOUN
ma-381	60	25	,	,	PUNCT
ma-381	60	26	q	q	NOUN
ma-381	60	27	,	,	PUNCT
ma-381	60	28	α)(rd	α)(rd	NOUN
ma-381	60	29	)	)	PUNCT
ma-381	60	30	,	,	PUNCT
ma-381	60	31	‖	‖	PROPN
ma-381	60	32	·	·	PUNCT
ma-381	60	33	‖h(p	‖h(p	NOUN
ma-381	60	34	,	,	PUNCT
ma-381	60	35	q	q	NOUN
ma-381	60	36	,	,	PUNCT
ma-381	60	37	α	α	NOUN
ma-381	60	38	)	)	PUNCT
ma-381	60	39	)	)	PUNCT
ma-381	60	40	are	be	AUX
ma-381	60	41	quasi	quasi	ADJ
ma-381	60	42	-	-	ADJ
ma-381	60	43	banach	banach	NOUN
ma-381	60	44	and	and	CCONJ
ma-381	60	45	for	for	ADP
ma-381	60	46	f	f	PROPN
ma-381	60	47	,	,	PUNCT
ma-381	60	48	g	g	PROPN
ma-381	60	49	∈	∈	PROPN
ma-381	60	50	h(p	h(p	PROPN
ma-381	60	51	,	,	PUNCT
ma-381	60	52	q	q	NOUN
ma-381	60	53	,	,	PUNCT
ma-381	60	54	α)(rd	α)(rd	NOUN
ma-381	60	55	)	)	PUNCT
ma-381	60	56	,	,	PUNCT
ma-381	61	1	‖f	‖f	ADP
ma-381	61	2	+	+	X
ma-381	61	3	g‖ph(p	g‖ph(p	PROPN
ma-381	61	4	,	,	PUNCT
ma-381	61	5	q	q	NOUN
ma-381	61	6	,	,	PUNCT
ma-381	61	7	α	α	NOUN
ma-381	61	8	)	)	PUNCT
ma-381	61	9	≤	≤	NOUN
ma-381	61	10	‖f	‖f	PUNCT
ma-381	61	11	‖	‖	PROPN
ma-381	61	12	p	p	ADJ
ma-381	61	13	h(p	h(p	PROPN
ma-381	61	14	,	,	PUNCT
ma-381	61	15	q	q	NOUN
ma-381	61	16	,	,	PUNCT
ma-381	61	17	α	α	NOUN
ma-381	61	18	)	)	PUNCT
ma-381	61	19	+	+	CCONJ
ma-381	61	20	‖g‖ph(p	‖g‖ph(p	NOUN
ma-381	61	21	,	,	PUNCT
ma-381	61	22	q	q	NOUN
ma-381	61	23	,	,	PUNCT
ma-381	61	24	α	α	NOUN
ma-381	61	25	)	)	PUNCT
ma-381	61	26	.	.	PUNCT
ma-381	62	1	we	we	PRON
ma-381	62	2	can	can	AUX
ma-381	62	3	also	also	ADV
ma-381	62	4	define	define	VERB
ma-381	62	5	(	(	PUNCT
ma-381	62	6	see	see	VERB
ma-381	62	7	[	[	X
ma-381	62	8	5	5	NUM
ma-381	62	9	]	]	PUNCT
ma-381	62	10	)	)	PUNCT
ma-381	62	11	these	these	DET
ma-381	62	12	spaces	space	NOUN
ma-381	62	13	as	as	ADP
ma-381	62	14	subspaces	subspace	NOUN
ma-381	62	15	of	of	ADP
ma-381	62	16	hardy	hardy	ADJ
ma-381	62	17	-	-	PUNCT
ma-381	62	18	amalgam	amalgam	NOUN
ma-381	62	19	spaces	space	NOUN
ma-381	62	20	for	for	ADP
ma-381	62	21	which	which	PRON
ma-381	62	22	thefamilly	thefamilly	NOUN
ma-381	62	23	of	of	ADP
ma-381	62	24	dilations	dilation	NOUN
ma-381	62	25	{	{	PUNCT
ma-381	62	26	stαρ	stαρ	NOUN
ma-381	62	27	}	}	PUNCT
ma-381	62	28	ρ>0	ρ>0	PROPN
ma-381	62	29	is	be	AUX
ma-381	62	30	locally	locally	ADV
ma-381	62	31	bounded.more	bounded.more	ADV
ma-381	62	32	precisely	precisely	ADV
ma-381	62	33	,	,	PUNCT
ma-381	62	34	for	for	ADP
ma-381	62	35	a	a	DET
ma-381	62	36	tempered	temper	VERB
ma-381	62	37	distribution	distribution	NOUN
ma-381	62	38	f	f	PROPN
ma-381	62	39	,	,	PUNCT
ma-381	62	40	ρ	ρ	PROPN
ma-381	62	41	>	>	X
ma-381	62	42	0	0	PROPN
ma-381	62	43	and	and	CCONJ
ma-381	62	44	α	α	PRON
ma-381	62	45	two	two	NUM
ma-381	62	46	real	real	ADJ
ma-381	62	47	numbers	number	NOUN
ma-381	62	48	we	we	PRON
ma-381	62	49	put	put	VERB
ma-381	62	50	〈	〈	PROPN
ma-381	62	51	stαρ	stαρ	NOUN
ma-381	62	52	f	f	PROPN
ma-381	62	53	,	,	PUNCT
ma-381	62	54	ϕ	ϕ	X
ma-381	62	55	〉	〉	NOUN
ma-381	62	56	:	:	PUNCT
ma-381	62	57	=	=	PUNCT
ma-381	62	58	〈	〈	PROPN
ma-381	62	59	f	f	NOUN
ma-381	62	60	,	,	PUNCT
ma-381	62	61	stα	stα	ADJ
ma-381	62	62	′	′	NUM
ma-381	62	63	ρ−1ϕ	ρ−1ϕ	NOUN
ma-381	62	64	〉	〉	NOUN
ma-381	62	65	,	,	PUNCT
ma-381	62	66	where	where	SCONJ
ma-381	62	67	1	1	NUM
ma-381	62	68	α′	α′	NUM
ma-381	62	69	+	+	CCONJ
ma-381	62	70	1	1	NUM
ma-381	62	71	α	α	NOUN
ma-381	62	72	=	=	SYM
ma-381	62	73	1	1	X
ma-381	62	74	.	.	X
ma-381	63	1	we	we	PRON
ma-381	63	2	have	have	AUX
ma-381	63	3	(	(	PUNCT
ma-381	63	4	see	see	VERB
ma-381	63	5	[	[	X
ma-381	63	6	4	4	NUM
ma-381	63	7	]	]	PUNCT
ma-381	63	8	)	)	PUNCT
ma-381	64	1	that	that	SCONJ
ma-381	64	2	for	for	ADP
ma-381	64	3	0	0	NUM
ma-381	64	4	<	<	X
ma-381	64	5	p	p	X
ma-381	64	6	≤	≤	NUM
ma-381	64	7	α	α	PRON
ma-381	64	8	≤	≤	NUM
ma-381	64	9	q	q	PROPN
ma-381	64	10	≤	≤	NUM
ma-381	64	11	∞	∞	PROPN
ma-381	64	12	,	,	PUNCT
ma-381	64	13	‖f	‖f	ADP
ma-381	64	14	‖h(p	‖h(p	SYM
ma-381	64	15	,	,	PUNCT
ma-381	64	16	q	q	NOUN
ma-381	64	17	,	,	PUNCT
ma-381	64	18	α	α	NOUN
ma-381	64	19	)	)	PUNCT
ma-381	64	20	=	=	SYM
ma-381	64	21	sup	sup	NOUN
ma-381	64	22	ρ>0	ρ>0	NOUN
ma-381	64	23	‖stαρ	‖stαρ	VERB
ma-381	64	24	f	f	NOUN
ma-381	64	25	‖h(p	‖h(p	PROPN
ma-381	64	26	,	,	PUNCT
ma-381	64	27	q	q	NOUN
ma-381	64	28	)	)	PUNCT
ma-381	64	29	.	.	PUNCT
ma-381	65	1	(	(	PUNCT
ma-381	65	2	2.1	2.1	NUM
ma-381	65	3	)	)	PUNCT
ma-381	65	4	just	just	ADV
ma-381	65	5	as	as	SCONJ
ma-381	65	6	hardy	hardy	ADJ
ma-381	65	7	-	-	PUNCT
ma-381	65	8	amalgam	amalgam	NOUN
ma-381	65	9	spaces	space	NOUN
ma-381	65	10	was	be	AUX
ma-381	65	11	characterized	characterize	VERB
ma-381	65	12	in	in	ADP
ma-381	65	13	[	[	X
ma-381	65	14	1	1	NUM
ma-381	65	15	]	]	PUNCT
ma-381	65	16	with	with	ADP
ma-381	65	17	poisson	poisson	PROPN
ma-381	65	18	kernel	kernel	PROPN
ma-381	65	19	,	,	PUNCT
ma-381	65	20	so	so	ADV
ma-381	65	21	are	be	AUX
ma-381	65	22	hardy	hardy	ADJ
ma-381	65	23	-	-	PUNCT
ma-381	65	24	fofana’sspaces	fofana’sspace	NOUN
ma-381	65	25	.	.	PUNCT
ma-381	66	1	in	in	ADP
ma-381	66	2	fact	fact	NOUN
ma-381	66	3	,	,	PUNCT
ma-381	66	4	a	a	DET
ma-381	66	5	tempered	temper	VERB
ma-381	66	6	distribution	distribution	NOUN
ma-381	66	7	f	f	PROPN
ma-381	66	8	belonging	belong	VERB
ma-381	66	9	to	to	ADP
ma-381	66	10	hardy	hardy	ADJ
ma-381	66	11	-	-	PUNCT
ma-381	66	12	amalgam	amalgam	NOUN
ma-381	66	13	spaces	space	NOUN
ma-381	66	14	is	be	AUX
ma-381	66	15	bounded	bound	VERB
ma-381	66	16	;	;	PUNCT
ma-381	66	17	i.e	i.e	PROPN
ma-381	66	18	f	f	PROPN
ma-381	66	19	∗	∗	X
ma-381	66	20	ψ	ψ	X
ma-381	66	21	∈	∈	PROPN
ma-381	66	22	l∞(rd	l∞(rd	NOUN
ma-381	66	23	)	)	PUNCT
ma-381	66	24	for	for	ADP
ma-381	66	25	all	all	DET
ma-381	66	26	ψ	ψ	PRON
ma-381	66	27	∈	∈	PROPN
ma-381	66	28	s(rd	s(rd	NUM
ma-381	66	29	)	)	PUNCT
ma-381	66	30	.	.	PUNCT
ma-381	67	1	a	a	DET
ma-381	67	2	convolution	convolution	NOUN
ma-381	67	3	of	of	ADP
ma-381	67	4	such	such	ADJ
ma-381	67	5	distribution	distribution	NOUN
ma-381	67	6	with	with	ADP
ma-381	67	7	integrable	integrable	ADJ
ma-381	67	8	functionscan	functionscan	PROPN
ma-381	67	9	be	be	AUX
ma-381	67	10	defined	define	VERB
ma-381	67	11	in	in	ADP
ma-381	67	12	term	term	NOUN
ma-381	67	13	of	of	ADP
ma-381	67	14	distribution	distribution	NOUN
ma-381	67	15	.	.	PUNCT
ma-381	68	1	more	more	ADV
ma-381	68	2	precisely	precisely	ADV
ma-381	68	3	,	,	PUNCT
ma-381	68	4	if	if	SCONJ
ma-381	68	5	f	f	PROPN
ma-381	68	6	∈	∈	PROPN
ma-381	68	7	s	s	VERB
ma-381	68	8	′(rd	′(rd	NOUN
ma-381	68	9	)	)	PUNCT
ma-381	68	10	is	be	AUX
ma-381	68	11	bounded	bound	VERB
ma-381	68	12	and	and	CCONJ
ma-381	68	13	u	u	PROPN
ma-381	68	14	∈	∈	PROPN
ma-381	68	15	l1(rd),then	l1(rd),then	PUNCT
ma-381	69	1	the	the	DET
ma-381	69	2	convolution	convolution	NOUN
ma-381	69	3	f	f	PROPN
ma-381	69	4	∗	∗	NOUN
ma-381	69	5	u	u	PROPN
ma-381	69	6	is	be	AUX
ma-381	69	7	defined	define	VERB
ma-381	69	8	as	as	ADP
ma-381	69	9	a	a	DET
ma-381	69	10	tempered	temper	VERB
ma-381	69	11	distribution	distribution	NOUN
ma-381	69	12	acting	act	VERB
ma-381	69	13	on	on	ADP
ma-381	69	14	s(rd	s(rd	NUM
ma-381	69	15	)	)	PUNCT
ma-381	69	16	by	by	ADP
ma-381	69	17	the	the	DET
ma-381	69	18	pairing	pair	VERB
ma-381	69	19	〈	〈	PROPN
ma-381	69	20	f	f	PROPN
ma-381	69	21	∗	∗	X
ma-381	69	22	u	u	PROPN
ma-381	69	23	,	,	PUNCT
ma-381	69	24	ϕ	ϕ	PROPN
ma-381	69	25	〉	〉	NOUN
ma-381	69	26	:	:	PUNCT
ma-381	69	27	=	=	SYM
ma-381	69	28	〈	〈	PROPN
ma-381	69	29	f	f	PROPN
ma-381	69	30	∗	∗	PROPN
ma-381	69	31	ϕ̃	ϕ̃	PROPN
ma-381	69	32	,	,	PUNCT
ma-381	69	33	ũ〉(l∞,l1	ũ〉(l∞,l1	PROPN
ma-381	69	34	)	)	PUNCT
ma-381	69	35	ϕ	ϕ	PROPN
ma-381	69	36	∈	∈	PROPN
ma-381	69	37	s(rd	s(rd	PROPN
ma-381	69	38	)	)	PUNCT
ma-381	69	39	where	where	SCONJ
ma-381	69	40	ũ(x	ũ(x	ADJ
ma-381	69	41	)	)	PUNCT
ma-381	69	42	=	=	PUNCT
ma-381	69	43	u(−x	u(−x	X
ma-381	69	44	)	)	PUNCT
ma-381	69	45	and	and	CCONJ
ma-381	69	46	〈	〈	PROPN
ma-381	69	47	f	f	PROPN
ma-381	69	48	∗	∗	PROPN
ma-381	69	49	ϕ̃	ϕ̃	PROPN
ma-381	69	50	,	,	PUNCT
ma-381	69	51	ũ〉(l∞,l1	ũ〉(l∞,l1	PROPN
ma-381	69	52	)	)	PUNCT
ma-381	69	53	is	be	AUX
ma-381	69	54	the	the	DET
ma-381	69	55	pairing	pairing	NOUN
ma-381	69	56	between	between	ADP
ma-381	69	57	l∞(rd	l∞(rd	NOUN
ma-381	69	58	)	)	PUNCT
ma-381	69	59	and	and	CCONJ
ma-381	69	60	l1(rd	l1(rd	PROPN
ma-381	69	61	)	)	PUNCT
ma-381	69	62	.	.	PUNCT
ma-381	70	1	but	but	CCONJ
ma-381	70	2	if	if	SCONJ
ma-381	70	3	wetake	wetake	NOUN
ma-381	70	4	as	as	SCONJ
ma-381	70	5	u	u	PROPN
ma-381	70	6	the	the	DET
ma-381	70	7	poisson	poisson	PROPN
ma-381	70	8	kernel	kernel	PROPN
ma-381	70	9	p	p	PROPN
ma-381	70	10	defined	define	VERB
ma-381	70	11	by	by	ADP
ma-381	70	12	p	p	X
ma-381	70	13	(	(	PUNCT
ma-381	70	14	x	x	NOUN
ma-381	70	15	)	)	PUNCT
ma-381	70	16	:	:	PUNCT
ma-381	71	1	=	=	SYM
ma-381	71	2	γ(d+1	γ(d+1	X
ma-381	71	3	2	2	X
ma-381	71	4	)	)	PUNCT
ma-381	71	5	π	π	PROPN
ma-381	71	6	d+1	d+1	NUM
ma-381	71	7	2	2	NUM
ma-381	71	8	1	1	NUM
ma-381	71	9	(	(	PUNCT
ma-381	71	10	1	1	NUM
ma-381	71	11	+	+	NUM
ma-381	71	12	|x	|x	NOUN
ma-381	71	13	|2	|2	NUM
ma-381	71	14	)	)	PUNCT
ma-381	71	15	d+1	d+1	PROPN
ma-381	71	16	2	2	NUM
ma-381	71	17	x	x	SYM
ma-381	71	18	∈	∈	PROPN
ma-381	71	19	rd	rd	NOUN
ma-381	71	20	,	,	PUNCT
ma-381	71	21	then	then	ADV
ma-381	71	22	f	f	PROPN
ma-381	71	23	∗	∗	NOUN
ma-381	71	24	pt	pt	NOUN
ma-381	71	25	can	can	AUX
ma-381	71	26	be	be	AUX
ma-381	71	27	identified	identify	VERB
ma-381	71	28	for	for	ADP
ma-381	71	29	all	all	DET
ma-381	71	30	t	t	PROPN
ma-381	71	31	>	>	X
ma-381	71	32	0	0	NUM
ma-381	71	33	,	,	PUNCT
ma-381	71	34	to	to	ADP
ma-381	71	35	a	a	DET
ma-381	71	36	well	well	ADV
ma-381	71	37	defined	define	VERB
ma-381	71	38	bounded	bounded	ADJ
ma-381	71	39	function	function	NOUN
ma-381	71	40	.	.	PUNCT
ma-381	72	1	as	as	SCONJ
ma-381	72	2	we	we	PRON
ma-381	72	3	can	can	AUX
ma-381	72	4	see	see	VERB
ma-381	72	5	forexample	forexample	NOUN
ma-381	72	6	in	in	ADP
ma-381	72	7	[	[	X
ma-381	72	8	9	9	NUM
ma-381	72	9	]	]	PUNCT
ma-381	72	10	,	,	PUNCT
ma-381	72	11	there	there	PRON
ma-381	72	12	exist	exist	VERB
ma-381	72	13	ϕ,ψ	ϕ,ψ	PROPN
ma-381	72	14	∈	∈	PROPN
ma-381	72	15	s(rd	s(rd	NUM
ma-381	72	16	)	)	PUNCT
ma-381	72	17	such	such	ADJ
ma-381	72	18	that	that	SCONJ
ma-381	72	19	f	f	PROPN
ma-381	72	20	∗	∗	NOUN
ma-381	72	21	pt	pt	X
ma-381	73	1	=	=	SYM
ma-381	74	1	(	(	PUNCT
ma-381	74	2	f	f	PROPN
ma-381	74	3	∗	∗	X
ma-381	74	4	ϕt	ϕt	PROPN
ma-381	74	5	)	)	PUNCT
ma-381	74	6	∗	∗	NOUN
ma-381	74	7	pt	pt	NOUN
ma-381	75	1	+	+	CCONJ
ma-381	75	2	f	f	PROPN
ma-381	75	3	∗	∗	NOUN
ma-381	75	4	ψt	ψt	VERB
ma-381	75	5	for	for	ADP
ma-381	75	6	t	t	PROPN
ma-381	75	7	>	>	X
ma-381	75	8	0	0	X
ma-381	75	9	.	.	PUNCT
ma-381	76	1	it	it	PRON
ma-381	76	2	is	be	AUX
ma-381	76	3	proved	prove	VERB
ma-381	76	4	in	in	ADP
ma-381	76	5	[	[	X
ma-381	76	6	1	1	X
ma-381	76	7	]	]	PUNCT
ma-381	76	8	that	that	SCONJ
ma-381	76	9	for	for	ADP
ma-381	76	10	an	an	DET
ma-381	76	11	element	element	NOUN
ma-381	76	12	f	f	PROPN
ma-381	76	13	∈	∈	PROPN
ma-381	76	14	h(p	h(p	PROPN
ma-381	76	15	,	,	PUNCT
ma-381	76	16	q)(rd	q)(rd	NUM
ma-381	76	17	)	)	PUNCT
ma-381	76	18	,	,	PUNCT
ma-381	76	19	we	we	PRON
ma-381	76	20	have	have	VERB
ma-381	76	21	‖x	‖x	NOUN
ma-381	76	22	7→	7→	NUM
ma-381	76	23	sup	sup	NOUN
ma-381	76	24	t>0	t>0	NOUN
ma-381	76	25	sup	sup	NOUN
ma-381	76	26	|x−y	|x−y	ADJ
ma-381	76	27	|<t	|<t	NOUN
ma-381	76	28	|f	|f	ADP
ma-381	76	29	∗	∗	NOUN
ma-381	76	30	pt(y)|‖p	pt(y)|‖p	NOUN
ma-381	76	31	,	,	PUNCT
ma-381	76	32	q	q	PROPN
ma-381	77	1	≈	≈	PROPN
ma-381	77	2	‖mf	‖mf	PROPN
ma-381	77	3	‖p	‖p	PROPN
ma-381	77	4	,	,	PUNCT
ma-381	77	5	q	q	X
ma-381	77	6	(	(	PUNCT
ma-381	77	7	2.2	2.2	NUM
ma-381	77	8	)	)	PUNCT
ma-381	77	9	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	77	10	eur	eur	PROPN
ma-381	77	11	.	.	PUNCT
ma-381	78	1	j.	j.	PROPN
ma-381	78	2	math	math	PROPN
ma-381	78	3	.	.	PUNCT
ma-381	79	1	anal	anal	PROPN
ma-381	79	2	.	.	PUNCT
ma-381	80	1	10.28924	10.28924	NUM
ma-381	80	2	/	/	SYM
ma-381	80	3	ada	ada	PROPN
ma-381	80	4	/	/	SYM
ma-381	80	5	ma.5.21	ma.5.21	PROPN
ma-381	80	6	5where	5where	NUM
ma-381	80	7	mf	mf	NOUN
ma-381	80	8	is	be	AUX
ma-381	80	9	the	the	DET
ma-381	80	10	maximal	maximal	ADJ
ma-381	80	11	function	function	NOUN
ma-381	80	12	defined	define	VERB
ma-381	80	13	in	in	ADP
ma-381	80	14	relation	relation	NOUN
ma-381	80	15	(	(	PUNCT
ma-381	80	16	1.1	1.1	NUM
ma-381	80	17	)	)	PUNCT
ma-381	80	18	.	.	PUNCT
ma-381	81	1	it	it	PRON
ma-381	81	2	follows	follow	VERB
ma-381	81	3	that	that	SCONJ
ma-381	81	4	‖x	‖x	PROPN
ma-381	81	5	7→	7→	NUM
ma-381	81	6	sup	sup	NOUN
ma-381	81	7	t>0	t>0	NOUN
ma-381	81	8	sup	sup	NOUN
ma-381	81	9	|x−y	|x−y	ADJ
ma-381	81	10	|<t	|<t	NOUN
ma-381	81	11	|f	|f	ADP
ma-381	81	12	∗	∗	NOUN
ma-381	81	13	pt(y)|‖p	pt(y)|‖p	NOUN
ma-381	81	14	,	,	PUNCT
ma-381	81	15	q	q	X
ma-381	81	16	,	,	PUNCT
ma-381	81	17	α	α	PROPN
ma-381	81	18	≈	≈	PROPN
ma-381	81	19	‖mf	‖mf	PROPN
ma-381	81	20	‖p	‖p	PROPN
ma-381	81	21	,	,	PUNCT
ma-381	81	22	q	q	X
ma-381	81	23	,	,	PUNCT
ma-381	81	24	α	α	NOUN
ma-381	81	25	,	,	PUNCT
ma-381	81	26	(	(	PUNCT
ma-381	81	27	2.3	2.3	NUM
ma-381	81	28	)	)	PUNCT
ma-381	81	29	thanks	thank	NOUN
ma-381	81	30	to	to	ADP
ma-381	81	31	relations	relation	NOUN
ma-381	81	32	(	(	PUNCT
ma-381	81	33	2.1	2.1	NUM
ma-381	81	34	)	)	PUNCT
ma-381	81	35	and	and	CCONJ
ma-381	81	36	(	(	PUNCT
ma-381	81	37	2.2	2.2	NUM
ma-381	81	38	)	)	PUNCT
ma-381	81	39	and	and	CCONJ
ma-381	81	40	the	the	DET
ma-381	81	41	fact	fact	NOUN
ma-381	81	42	that	that	SCONJ
ma-381	81	43	stαρ	stαρ	NOUN
ma-381	81	44	commute	commute	VERB
ma-381	81	45	with	with	ADP
ma-381	81	46	the	the	DET
ma-381	81	47	maximal	maximal	ADJ
ma-381	81	48	function	function	NOUN
ma-381	81	49	m.	m.	NOUN
ma-381	81	50	lemma	lemma	PROPN
ma-381	81	51	2.3	2.3	NUM
ma-381	81	52	.	.	PUNCT
ma-381	82	1	let	let	VERB
ma-381	82	2	f	f	PROPN
ma-381	82	3	∈	∈	PROPN
ma-381	82	4	s	s	VERB
ma-381	82	5	′(rd	′(rd	NOUN
ma-381	82	6	)	)	PUNCT
ma-381	82	7	,	,	PUNCT
ma-381	82	8	ϕ	ϕ	PROPN
ma-381	82	9	∈	∈	PROPN
ma-381	82	10	s(rd	s(rd	PROPN
ma-381	82	11	)	)	PUNCT
ma-381	82	12	,	,	PUNCT
ma-381	82	13	ρ	ρ	PROPN
ma-381	82	14	and	and	CCONJ
ma-381	82	15	α	α	PRON
ma-381	82	16	positive	positive	ADJ
ma-381	82	17	real	real	ADJ
ma-381	82	18	numbers	number	NOUN
ma-381	82	19	.	.	PUNCT
ma-381	83	1	we	we	PRON
ma-381	83	2	have	have	VERB
ma-381	83	3	stαρ	stαρ	NOUN
ma-381	83	4	(	(	PUNCT
ma-381	83	5	f	f	PROPN
ma-381	83	6	∗	∗	X
ma-381	83	7	ϕt	ϕt	PROPN
ma-381	83	8	)	)	PUNCT
ma-381	83	9	=	=	PRON
ma-381	84	1	(	(	PUNCT
ma-381	84	2	stαρ	stαρ	NOUN
ma-381	84	3	f	f	PROPN
ma-381	84	4	)	)	PUNCT
ma-381	84	5	∗	∗	NOUN
ma-381	84	6	ϕρt	ϕρt	NOUN
ma-381	84	7	,	,	PUNCT
ma-381	84	8	t	t	X
ma-381	84	9	>	>	X
ma-381	84	10	0	0	X
ma-381	84	11	.	.	PUNCT
ma-381	85	1	infact	infact	PROPN
ma-381	85	2	,	,	PUNCT
ma-381	85	3	ρ	ρ	PROPN
ma-381	85	4	−d	−d	PROPN
ma-381	85	5	α	α	PROPN
ma-381	85	6	(	(	PUNCT
ma-381	85	7	f	f	PROPN
ma-381	85	8	∗	∗	X
ma-381	85	9	ϕt	ϕt	PROPN
ma-381	85	10	)	)	PUNCT
ma-381	85	11	(	(	PUNCT
ma-381	85	12	ρ−1x	ρ−1x	NOUN
ma-381	85	13	)	)	PUNCT
ma-381	85	14	=	=	SYM
ma-381	86	1	ρ	ρ	PROPN
ma-381	86	2	−d	−d	PROPN
ma-381	87	1	α	α	PROPN
ma-381	87	2	〈	〈	PROPN
ma-381	87	3	f	f	X
ma-381	87	4	,	,	PUNCT
ma-381	87	5	ρdϕρt(x	ρdϕρt(x	NOUN
ma-381	87	6	−	−	PROPN
ma-381	87	7	ρ	ρ	PROPN
ma-381	87	8	·	·	PROPN
ma-381	87	9	)	)	PUNCT
ma-381	87	10	〉	〉	NOUN
ma-381	87	11	=	=	SYM
ma-381	87	12	ρ	ρ	PROPN
ma-381	87	13	d	d	PROPN
ma-381	87	14	α′	α′	NUM
ma-381	87	15	〈	〈	PROPN
ma-381	87	16	f	f	X
ma-381	87	17	,	,	PUNCT
ma-381	87	18	ϕρt(x	ϕρt(x	PROPN
ma-381	87	19	−	−	PROPN
ma-381	87	20	ρ	ρ	PROPN
ma-381	87	21	·	·	PROPN
ma-381	87	22	)	)	PUNCT
ma-381	87	23	〉	〉	NOUN
ma-381	87	24	=	=	SYM
ma-381	87	25	〈	〈	PROPN
ma-381	87	26	f	f	NOUN
ma-381	87	27	,	,	PUNCT
ma-381	87	28	stα	stα	PROPN
ma-381	87	29	′	′	NUM
ma-381	88	1	ρ−1	ρ−1	PROPN
ma-381	89	1	[	[	X
ma-381	89	2	ϕρt(x	ϕρt(x	X
ma-381	89	3	−	−	PROPN
ma-381	89	4	·	·	PUNCT
ma-381	89	5	)	)	PUNCT
ma-381	89	6	]	]	PUNCT
ma-381	89	7	〉	〉	NOUN
ma-381	89	8	=	=	PUNCT
ma-381	89	9	(	(	PUNCT
ma-381	89	10	stαρ	stαρ	NOUN
ma-381	89	11	f	f	PROPN
ma-381	89	12	∗	∗	PROPN
ma-381	89	13	ϕρt	ϕρt	PROPN
ma-381	89	14	)	)	PUNCT
ma-381	89	15	(	(	PUNCT
ma-381	89	16	x	x	NOUN
ma-381	89	17	)	)	PUNCT
ma-381	89	18	.	.	PUNCT
ma-381	90	1	lemma	lemma	PROPN
ma-381	90	2	2.4	2.4	NUM
ma-381	90	3	.	.	PUNCT
ma-381	91	1	let	let	VERB
ma-381	91	2	f	f	PROPN
ma-381	91	3	∈	∈	PROPN
ma-381	91	4	s	s	VERB
ma-381	91	5	′(rd	′(rd	NOUN
ma-381	91	6	)	)	PUNCT
ma-381	91	7	,	,	PUNCT
ma-381	91	8	ϕ	ϕ	PROPN
ma-381	91	9	∈	∈	PROPN
ma-381	91	10	s(rd	s(rd	PROPN
ma-381	91	11	)	)	PUNCT
ma-381	91	12	,	,	PUNCT
ma-381	91	13	ρ	ρ	PROPN
ma-381	91	14	and	and	CCONJ
ma-381	91	15	α	α	PRON
ma-381	91	16	positive	positive	ADJ
ma-381	91	17	real	real	ADJ
ma-381	91	18	numbers	number	NOUN
ma-381	91	19	.	.	PUNCT
ma-381	92	1	we	we	PRON
ma-381	92	2	have	have	VERB
ma-381	92	3	stαρ	stαρ	NOUN
ma-381	92	4	[	[	X
ma-381	92	5	(	(	PUNCT
ma-381	92	6	f	f	PROPN
ma-381	92	7	∗	∗	PROPN
ma-381	92	8	ϕt	ϕt	PROPN
ma-381	92	9	)	)	PUNCT
ma-381	92	10	∗	∗	NOUN
ma-381	92	11	pt	pt	X
ma-381	92	12	]	]	X
ma-381	93	1	=	=	PUNCT
ma-381	93	2	(	(	PUNCT
ma-381	93	3	stαρ	stαρ	NOUN
ma-381	93	4	f	f	PROPN
ma-381	93	5	∗	∗	PROPN
ma-381	93	6	ϕρt	ϕρt	PROPN
ma-381	93	7	)	)	PUNCT
ma-381	93	8	∗	∗	NOUN
ma-381	93	9	pρt	pρt	NOUN
ma-381	93	10	.	.	PUNCT
ma-381	94	1	(	(	PUNCT
ma-381	94	2	2.4	2.4	NUM
ma-381	94	3	)	)	PUNCT
ma-381	94	4	relation	relation	NOUN
ma-381	94	5	(	(	PUNCT
ma-381	94	6	2.4	2.4	NUM
ma-381	94	7	)	)	PUNCT
ma-381	94	8	follows	follow	VERB
ma-381	94	9	from	from	ADP
ma-381	94	10	the	the	DET
ma-381	94	11	fact	fact	NOUN
ma-381	94	12	that	that	SCONJ
ma-381	94	13	ρ	ρ	PROPN
ma-381	94	14	−d	−d	PROPN
ma-381	94	15	α	α	PROPN
ma-381	94	16	(	(	PUNCT
ma-381	94	17	f	f	PROPN
ma-381	94	18	∗	∗	VERB
ma-381	94	19	ϕρ−1	ϕρ−1	PROPN
ma-381	94	20	t	t	PROPN
ma-381	94	21	)	)	PUNCT
ma-381	94	22	∗	∗	X
ma-381	94	23	pρ−1t(ρ	pρ−1t(ρ	X
ma-381	94	24	−1x	−1x	PROPN
ma-381	94	25	)	)	PUNCT
ma-381	95	1	=	=	SYM
ma-381	96	1	ρ	ρ	PROPN
ma-381	96	2	−d	−d	PROPN
ma-381	96	3	α	α	PROPN
ma-381	96	4	∫	∫	PROPN
ma-381	96	5	rd	rd	PROPN
ma-381	96	6	(	(	PUNCT
ma-381	96	7	f	f	PROPN
ma-381	96	8	∗	∗	VERB
ma-381	96	9	ϕρ−1	ϕρ−1	PROPN
ma-381	96	10	t	t	PROPN
ma-381	96	11	)	)	PUNCT
ma-381	96	12	(	(	PUNCT
ma-381	96	13	ρ−1x	ρ−1x	NOUN
ma-381	96	14	−	−	PROPN
ma-381	96	15	y)pρ−1t(y)dy	y)pρ−1t(y)dy	PROPN
ma-381	96	16	=	=	SYM
ma-381	96	17	∫	∫	PROPN
ma-381	96	18	rd	rd	PROPN
ma-381	96	19	〈	〈	PROPN
ma-381	96	20	f	f	PROPN
ma-381	96	21	,	,	PUNCT
ma-381	96	22	ρ	ρ	PROPN
ma-381	96	23	d	d	PROPN
ma-381	96	24	α′ϕt(x	α′ϕt(x	PROPN
ma-381	96	25	−	−	PROPN
ma-381	96	26	z	z	NOUN
ma-381	96	27	−	−	PROPN
ma-381	96	28	ρ	ρ	PROPN
ma-381	96	29	·	·	NUM
ma-381	96	30	)	)	PUNCT
ma-381	96	31	〉	〉	NOUN
ma-381	96	32	pt(z)dz	pt(z)dz	PROPN
ma-381	96	33	=	=	SYM
ma-381	96	34	∫	∫	PROPN
ma-381	96	35	rd	rd	PROPN
ma-381	96	36	〈	〈	PROPN
ma-381	96	37	stαρ	stαρ	PROPN
ma-381	96	38	f	f	PROPN
ma-381	96	39	,	,	PUNCT
ma-381	96	40	ϕt(x	ϕt(x	PUNCT
ma-381	97	1	−	−	PROPN
ma-381	97	2	z	z	NOUN
ma-381	97	3	−	−	PROPN
ma-381	97	4	·	·	SYM
ma-381	97	5	)	)	PUNCT
ma-381	97	6	〉	〉	NOUN
ma-381	97	7	pt(z)dz	pt(z)dz	PROPN
ma-381	97	8	=	=	SYM
ma-381	97	9	∫	∫	PROPN
ma-381	97	10	rd	rd	PROPN
ma-381	97	11	(	(	PUNCT
ma-381	97	12	stαρ	stαρ	PROPN
ma-381	97	13	f	f	PROPN
ma-381	97	14	∗	∗	VERB
ma-381	97	15	ϕt	ϕt	PROPN
ma-381	97	16	)	)	PUNCT
ma-381	98	1	(	(	PUNCT
ma-381	98	2	x	x	X
ma-381	98	3	−	−	PROPN
ma-381	98	4	z)pt(z)dz	z)pt(z)dz	NUM
ma-381	98	5	=	=	SYM
ma-381	98	6	(	(	PUNCT
ma-381	98	7	stαρ	stαρ	NOUN
ma-381	98	8	f	f	PROPN
ma-381	98	9	∗	∗	X
ma-381	98	10	ϕt	ϕt	PROPN
ma-381	98	11	)	)	PUNCT
ma-381	98	12	∗	∗	NOUN
ma-381	98	13	pt(x	pt(x	NUM
ma-381	98	14	)	)	PUNCT
ma-381	98	15	.	.	PUNCT
ma-381	99	1	it	it	PRON
ma-381	99	2	comes	come	VERB
ma-381	99	3	from	from	ADP
ma-381	99	4	lemma	lemma	PROPN
ma-381	99	5	2.3	2.3	NUM
ma-381	99	6	and	and	CCONJ
ma-381	99	7	2.4	2.4	NUM
ma-381	99	8	that	that	PRON
ma-381	99	9	for	for	ADP
ma-381	99	10	a	a	DET
ma-381	99	11	bounded	bound	VERB
ma-381	99	12	tempered	temper	VERB
ma-381	99	13	distribution	distribution	NOUN
ma-381	99	14	f	f	PROPN
ma-381	99	15	and	and	CCONJ
ma-381	99	16	u(x	u(x	PROPN
ma-381	99	17	,	,	PUNCT
ma-381	99	18	t	t	PROPN
ma-381	99	19	)	)	PUNCT
ma-381	99	20	=	=	SYM
ma-381	99	21	f	f	PROPN
ma-381	99	22	∗pt(x	∗pt(x	PROPN
ma-381	99	23	)	)	PUNCT
ma-381	99	24	,	,	PUNCT
ma-381	99	25	(	(	PUNCT
ma-381	99	26	stαρ	stαρ	NOUN
ma-381	99	27	u	u	PROPN
ma-381	99	28	)	)	PUNCT
ma-381	99	29	(	(	PUNCT
ma-381	99	30	x	x	X
ma-381	99	31	,	,	PUNCT
ma-381	99	32	t	t	PROPN
ma-381	99	33	)	)	PUNCT
ma-381	99	34	=	=	PUNCT
ma-381	100	1	[	[	X
ma-381	100	2	(	(	PUNCT
ma-381	100	3	stαρ	stαρ	PROPN
ma-381	100	4	f	f	PROPN
ma-381	100	5	)	)	PUNCT
ma-381	100	6	∗	∗	NOUN
ma-381	100	7	pt	pt	X
ma-381	100	8	]	]	PUNCT
ma-381	100	9	(	(	PUNCT
ma-381	100	10	x	x	NOUN
ma-381	100	11	)	)	PUNCT
ma-381	100	12	,	,	PUNCT
ma-381	100	13	ρ	ρ	PROPN
ma-381	100	14	>	>	X
ma-381	100	15	0	0	PROPN
ma-381	100	16	and	and	CCONJ
ma-381	100	17	α	α	X
ma-381	100	18	>	>	X
ma-381	100	19	0	0	PUNCT
ma-381	100	20	(	(	PUNCT
ma-381	100	21	2.5	2.5	NUM
ma-381	100	22	)	)	PUNCT
ma-381	100	23	for	for	ADP
ma-381	100	24	all	all	DET
ma-381	100	25	t	t	PROPN
ma-381	100	26	>	>	X
ma-381	100	27	0	0	NUM
ma-381	100	28	.	.	NOUN
ma-381	101	1	3	3	X
ma-381	101	2	.	.	X
ma-381	101	3	cauchy	cauchy	PROPN
ma-381	101	4	-	-	PUNCT
ma-381	101	5	riemann	riemann	PROPN
ma-381	101	6	equations	equation	NOUN
ma-381	101	7	,	,	PUNCT
ma-381	101	8	riesz	riesz	NOUN
ma-381	101	9	transforms	transform	VERB
ma-381	101	10	and	and	CCONJ
ma-381	101	11	hardy	hardy	ADJ
ma-381	101	12	-	-	PUNCT
ma-381	101	13	fofana	fofana	NOUN
ma-381	101	14	spaces	space	NOUN
ma-381	101	15	let	let	VERB
ma-381	101	16	u	u	PRON
ma-381	101	17	be	be	AUX
ma-381	101	18	a	a	DET
ma-381	101	19	harmonic	harmonic	ADJ
ma-381	101	20	function	function	NOUN
ma-381	101	21	on	on	ADP
ma-381	101	22	rd+1	rd+1	ADV
ma-381	101	23	+	+	CCONJ
ma-381	101	24	;	;	PUNCT
ma-381	101	25	i.e	i.e	X
ma-381	101	26	,	,	PUNCT
ma-381	101	27	u	u	PROPN
ma-381	101	28	∈	∈	PROPN
ma-381	101	29	c2(rd+1	c2(rd+1	NOUN
ma-381	101	30	+	+	X
ma-381	101	31	)	)	PUNCT
ma-381	101	32	and	and	CCONJ
ma-381	101	33	∆u	∆u	ADV
ma-381	101	34	:	:	PUNCT
ma-381	102	1	=	=	SYM
ma-381	102	2	∑d+1	∑d+1	PROPN
ma-381	102	3	j=1	j=1	PROPN
ma-381	102	4	∂2u	∂2u	PROPN
ma-381	102	5	(	(	PUNCT
ma-381	102	6	∂xj	∂xj	NOUN
ma-381	102	7	)	)	PUNCT
ma-381	102	8	2	2	NUM
ma-381	102	9	=	=	SYM
ma-381	102	10	0	0	NUM
ma-381	102	11	,	,	PUNCT
ma-381	102	12	where	where	SCONJ
ma-381	102	13	xd+1	xd+1	PROPN
ma-381	102	14	=	=	SYM
ma-381	102	15	t	t	PROPN
ma-381	102	16	and	and	CCONJ
ma-381	102	17	rd+1	rd+1	INTJ
ma-381	102	18	+	+	CCONJ
ma-381	102	19	:	:	PUNCT
ma-381	102	20	=	=	PUNCT
ma-381	103	1	rd×]0,+∞	rd×]0,+∞	PROPN
ma-381	104	1	[	[	X
ma-381	104	2	.	.	PUNCT
ma-381	105	1	we	we	PRON
ma-381	105	2	define	define	VERB
ma-381	105	3	its	its	PRON
ma-381	105	4	non	non	ADJ
ma-381	105	5	tangential	tangential	ADJ
ma-381	105	6	maximal	maximal	ADJ
ma-381	105	7	function	function	NOUN
ma-381	105	8	u∗	u∗	VERB
ma-381	105	9	by	by	ADP
ma-381	105	10	u∗(x	u∗(x	PROPN
ma-381	105	11	)	)	PUNCT
ma-381	105	12	:	:	PUNCT
ma-381	106	1	=	=	SYM
ma-381	106	2	sup	sup	NOUN
ma-381	106	3	t>0	t>0	NOUN
ma-381	106	4	sup	sup	NOUN
ma-381	106	5	|x−y	|x−y	ADJ
ma-381	106	6	|<t	|<t	VERB
ma-381	106	7	|u(y	|u(y	NOUN
ma-381	106	8	,	,	PUNCT
ma-381	106	9	t)|	t)|	ADV
ma-381	106	10	∀x	∀x	PUNCT
ma-381	106	11	∈	∈	PROPN
ma-381	106	12	rd	rd	NOUN
ma-381	106	13	.	.	PUNCT
ma-381	107	1	(	(	PUNCT
ma-381	107	2	3.1	3.1	NUM
ma-381	107	3	)	)	PUNCT
ma-381	107	4	let	let	VERB
ma-381	107	5	f	f	PRON
ma-381	107	6	be	be	AUX
ma-381	107	7	a	a	DET
ma-381	107	8	bounded	bound	VERB
ma-381	107	9	tempered	temper	VERB
ma-381	107	10	distribution	distribution	NOUN
ma-381	107	11	,	,	PUNCT
ma-381	107	12	and	and	CCONJ
ma-381	107	13	u(x	u(x	PROPN
ma-381	107	14	,	,	PUNCT
ma-381	107	15	t	t	NOUN
ma-381	107	16	)	)	PUNCT
ma-381	108	1	=	=	SYM
ma-381	108	2	pt	pt	PROPN
ma-381	108	3	∗	∗	X
ma-381	108	4	f	f	PROPN
ma-381	108	5	(	(	PUNCT
ma-381	108	6	x	x	NOUN
ma-381	108	7	)	)	PUNCT
ma-381	108	8	.	.	PUNCT
ma-381	109	1	as	as	SCONJ
ma-381	109	2	we	we	PRON
ma-381	109	3	can	can	AUX
ma-381	109	4	see	see	VERB
ma-381	109	5	in	in	ADP
ma-381	109	6	[	[	X
ma-381	109	7	4	4	NUM
ma-381	109	8	]	]	PUNCT
ma-381	109	9	,	,	PUNCT
ma-381	109	10	u∗	u∗	PROPN
ma-381	109	11	∈	∈	PROPN
ma-381	109	12	(	(	PUNCT
ma-381	109	13	lp	lp	NOUN
ma-381	109	14	,	,	PUNCT
ma-381	109	15	`	`	PUNCT
ma-381	109	16	q)α	q)α	X
ma-381	109	17	(	(	PUNCT
ma-381	109	18	rd	rd	NOUN
ma-381	109	19	)	)	PUNCT
ma-381	109	20	whenever	whenever	SCONJ
ma-381	109	21	f	f	PROPN
ma-381	109	22	∈	∈	PROPN
ma-381	109	23	h(p	h(p	PROPN
ma-381	109	24	,	,	PUNCT
ma-381	109	25	q	q	NOUN
ma-381	109	26	,	,	PUNCT
ma-381	109	27	α)(rd	α)(rd	NOUN
ma-381	109	28	)	)	PUNCT
ma-381	109	29	.	.	PUNCT
ma-381	110	1	we	we	PRON
ma-381	110	2	give	give	VERB
ma-381	110	3	in	in	ADP
ma-381	110	4	the	the	DET
ma-381	110	5	next	next	ADJ
ma-381	110	6	result	result	NOUN
ma-381	110	7	a	a	DET
ma-381	110	8	necessary	necessary	ADJ
ma-381	110	9	and	and	CCONJ
ma-381	110	10	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	110	11	eur	eur	PROPN
ma-381	110	12	.	.	PUNCT
ma-381	111	1	j.	j.	PROPN
ma-381	111	2	math	math	PROPN
ma-381	111	3	.	.	PUNCT
ma-381	112	1	anal	anal	PROPN
ma-381	112	2	.	.	PUNCT
ma-381	113	1	10.28924	10.28924	NUM
ma-381	113	2	/	/	SYM
ma-381	113	3	ada	ada	PROPN
ma-381	113	4	/	/	SYM
ma-381	113	5	ma.5.21	ma.5.21	NOUN
ma-381	113	6	6sufficient	6sufficient	NUM
ma-381	113	7	conditions	condition	NOUN
ma-381	113	8	for	for	SCONJ
ma-381	113	9	a	a	DET
ma-381	113	10	harmonic	harmonic	ADJ
ma-381	113	11	function	function	NOUN
ma-381	113	12	u	u	NOUN
ma-381	113	13	in	in	ADP
ma-381	113	14	rd+1	rd+1	NUM
ma-381	113	15	to	to	PART
ma-381	113	16	have	have	VERB
ma-381	113	17	its	its	PRON
ma-381	113	18	non	non	ADJ
ma-381	113	19	tangential	tangential	ADJ
ma-381	113	20	maximal	maximal	ADJ
ma-381	113	21	func	func	NOUN
ma-381	113	22	-	-	PUNCT
ma-381	113	23	tion	tion	NOUN
ma-381	113	24	in	in	ADP
ma-381	113	25	(	(	PUNCT
ma-381	113	26	lp	lp	ADJ
ma-381	113	27	,	,	PUNCT
ma-381	113	28	`	`	PUNCT
ma-381	113	29	q)α(rd	q)α(rd	X
ma-381	113	30	)	)	PUNCT
ma-381	113	31	.	.	PUNCT
ma-381	114	1	the	the	DET
ma-381	114	2	proof	proof	NOUN
ma-381	114	3	is	be	AUX
ma-381	114	4	based	base	VERB
ma-381	114	5	on	on	ADP
ma-381	114	6	the	the	DET
ma-381	114	7	dilation	dilation	NOUN
ma-381	114	8	characterization	characterization	NOUN
ma-381	114	9	of	of	ADP
ma-381	114	10	hardy	hardy	ADJ
ma-381	114	11	-	-	PUNCT
ma-381	114	12	fofana	fofana	NOUN
ma-381	114	13	spacesand	spacesand	NOUN
ma-381	114	14	[	[	X
ma-381	114	15	3	3	NUM
ma-381	114	16	,	,	PUNCT
ma-381	114	17	proposition	proposition	NOUN
ma-381	114	18	2.1	2.1	NUM
ma-381	114	19	]	]	PUNCT
ma-381	114	20	.	.	PUNCT
ma-381	115	1	proposition	proposition	NOUN
ma-381	115	2	3.1	3.1	NUM
ma-381	115	3	.	.	PUNCT
ma-381	116	1	let	let	VERB
ma-381	116	2	0	0	PUNCT
ma-381	116	3	<	<	X
ma-381	116	4	p	p	X
ma-381	116	5	≤	≤	NUM
ma-381	116	6	α	α	NOUN
ma-381	116	7	≤	≤	NUM
ma-381	116	8	q	q	X
ma-381	116	9	<	<	X
ma-381	116	10	+	+	NOUN
ma-381	116	11	∞	∞	NUM
ma-381	116	12	and	and	CCONJ
ma-381	116	13	u	u	PRON
ma-381	116	14	an	an	DET
ma-381	116	15	harmonic	harmonic	ADJ
ma-381	116	16	function	function	NOUN
ma-381	116	17	on	on	ADP
ma-381	116	18	rd+1	rd+1	NOUN
ma-381	116	19	+	+	CCONJ
ma-381	116	20	.	.	PUNCT
ma-381	117	1	the	the	DET
ma-381	117	2	maximal	maximal	ADJ
ma-381	117	3	function	function	NOUN
ma-381	117	4	u∗	u∗	ADV
ma-381	117	5	belongs	belong	VERB
ma-381	117	6	to	to	ADP
ma-381	117	7	(	(	PUNCT
ma-381	117	8	lp	lp	ADJ
ma-381	117	9	,	,	PUNCT
ma-381	117	10	`	`	PUNCT
ma-381	117	11	q)α(rd	q)α(rd	X
ma-381	117	12	)	)	PUNCT
ma-381	118	1	if	if	SCONJ
ma-381	118	2	and	and	CCONJ
ma-381	118	3	only	only	ADV
ma-381	118	4	if	if	SCONJ
ma-381	118	5	there	there	PRON
ma-381	118	6	exists	exist	VERB
ma-381	118	7	f	f	PROPN
ma-381	118	8	∈	∈	PROPN
ma-381	118	9	h(p	h(p	PROPN
ma-381	118	10	,	,	PUNCT
ma-381	118	11	q	q	NOUN
ma-381	118	12	,	,	PUNCT
ma-381	118	13	α)(rd	α)(rd	NOUN
ma-381	118	14	)	)	PUNCT
ma-381	118	15	such	such	ADJ
ma-381	118	16	that	that	SCONJ
ma-381	118	17	u(x	u(x	NOUN
ma-381	118	18	,	,	PUNCT
ma-381	118	19	t	t	PROPN
ma-381	118	20	)	)	PUNCT
ma-381	118	21	:	:	PUNCT
ma-381	119	1	=	=	SYM
ma-381	119	2	f	f	X
ma-381	119	3	∗	∗	NOUN
ma-381	119	4	pt(x	pt(x	NOUN
ma-381	119	5	)	)	PUNCT
ma-381	119	6	,	,	PUNCT
ma-381	119	7	(	(	PUNCT
ma-381	119	8	x	x	X
ma-381	119	9	,	,	PUNCT
ma-381	119	10	t	t	PROPN
ma-381	119	11	)	)	PUNCT
ma-381	119	12	∈	∈	PROPN
ma-381	119	13	rd+1	rd+1	NOUN
ma-381	119	14	+	+	CCONJ
ma-381	119	15	.	.	PUNCT
ma-381	120	1	moreover	moreover	ADV
ma-381	120	2	,	,	PUNCT
ma-381	120	3	‖f	‖f	ADP
ma-381	120	4	‖h(p	‖h(p	SYM
ma-381	120	5	,	,	PUNCT
ma-381	120	6	q	q	NOUN
ma-381	120	7	,	,	PUNCT
ma-381	120	8	α	α	NOUN
ma-381	120	9	)	)	PUNCT
ma-381	120	10	≈	≈	PROPN
ma-381	120	11	‖u∗‖p	‖u∗‖p	PROPN
ma-381	120	12	,	,	PUNCT
ma-381	120	13	q	q	NOUN
ma-381	120	14	,	,	PUNCT
ma-381	120	15	α	α	NOUN
ma-381	120	16	.	.	PUNCT
ma-381	121	1	proof	proof	NOUN
ma-381	121	2	.	.	PUNCT
ma-381	122	1	let	let	VERB
ma-381	122	2	u	u	PRON
ma-381	122	3	be	be	AUX
ma-381	122	4	an	an	DET
ma-381	122	5	harmonic	harmonic	ADJ
ma-381	122	6	function	function	NOUN
ma-381	122	7	on	on	ADP
ma-381	122	8	rd+1	rd+1	ADV
ma-381	122	9	+	+	CCONJ
ma-381	122	10	,	,	PUNCT
ma-381	122	11	and	and	CCONJ
ma-381	122	12	u∗	u∗	VERB
ma-381	122	13	the	the	DET
ma-381	122	14	associate	associate	ADJ
ma-381	122	15	non	non	ADJ
ma-381	122	16	tangential	tangential	ADJ
ma-381	122	17	maximal	maximal	ADJ
ma-381	122	18	functionas	functiona	NOUN
ma-381	122	19	defined	define	VERB
ma-381	122	20	in	in	ADP
ma-381	122	21	relation	relation	NOUN
ma-381	122	22	(	(	PUNCT
ma-381	122	23	3.1).we	3.1).we	PRON
ma-381	122	24	suppose	suppose	VERB
ma-381	122	25	that	that	SCONJ
ma-381	122	26	there	there	PRON
ma-381	122	27	exists	exist	VERB
ma-381	122	28	f	f	PROPN
ma-381	122	29	∈	∈	PROPN
ma-381	122	30	h(p	h(p	PROPN
ma-381	122	31	,	,	PUNCT
ma-381	122	32	q	q	NOUN
ma-381	122	33	,	,	PUNCT
ma-381	122	34	α)(rd	α)(rd	NOUN
ma-381	122	35	)	)	PUNCT
ma-381	122	36	such	such	ADJ
ma-381	122	37	that	that	SCONJ
ma-381	122	38	u(x	u(x	NOUN
ma-381	122	39	,	,	PUNCT
ma-381	122	40	t	t	PROPN
ma-381	122	41	)	)	PUNCT
ma-381	122	42	:	:	PUNCT
ma-381	123	1	=	=	SYM
ma-381	123	2	f	f	X
ma-381	123	3	∗pt(x	∗pt(x	PROPN
ma-381	123	4	)	)	PUNCT
ma-381	123	5	for	for	ADP
ma-381	123	6	all	all	DET
ma-381	123	7	(	(	PUNCT
ma-381	123	8	x	x	NOUN
ma-381	123	9	,	,	PUNCT
ma-381	123	10	t	t	PROPN
ma-381	123	11	)	)	PUNCT
ma-381	123	12	∈	∈	PROPN
ma-381	123	13	rd+1	rd+1	NOUN
ma-381	124	1	+	+	CCONJ
ma-381	124	2	.from	.from	ADP
ma-381	124	3	the	the	DET
ma-381	124	4	poisson	poisson	NOUN
ma-381	124	5	characterization	characterization	NOUN
ma-381	124	6	of	of	ADP
ma-381	124	7	hardy	hardy	ADJ
ma-381	124	8	-	-	PUNCT
ma-381	124	9	fofana	fofana	NOUN
ma-381	124	10	spaces	space	NOUN
ma-381	124	11	(	(	PUNCT
ma-381	124	12	see	see	VERB
ma-381	124	13	[	[	X
ma-381	124	14	4	4	NUM
ma-381	124	15	,	,	PUNCT
ma-381	124	16	theorem	theorem	VERB
ma-381	124	17	2.3.8	2.3.8	NUM
ma-381	124	18	]	]	PUNCT
ma-381	124	19	)	)	PUNCT
ma-381	124	20	,	,	PUNCT
ma-381	124	21	we	we	PRON
ma-381	124	22	deducethat	deducethat	VERB
ma-381	124	23	‖u∗‖p	‖u∗‖p	VERB
ma-381	124	24	,	,	PUNCT
ma-381	124	25	q	q	NOUN
ma-381	124	26	,	,	PUNCT
ma-381	124	27	α	α	NOUN
ma-381	124	28	≤	≤	NOUN
ma-381	125	1	c	c	NOUN
ma-381	125	2	‖f	‖f	ADJ
ma-381	125	3	‖h(p	‖h(p	SYM
ma-381	125	4	,	,	PUNCT
ma-381	125	5	q	q	NOUN
ma-381	125	6	,	,	PUNCT
ma-381	125	7	α	α	NOUN
ma-381	125	8	)	)	PUNCT
ma-381	125	9	.for	.for	PUNCT
ma-381	126	1	the	the	DET
ma-381	126	2	converse	converse	NOUN
ma-381	126	3	,	,	PUNCT
ma-381	126	4	let	let	VERB
ma-381	126	5	us	we	PRON
ma-381	126	6	suppose	suppose	VERB
ma-381	126	7	that	that	SCONJ
ma-381	126	8	u∗	u∗	PROPN
ma-381	126	9	∈	∈	PROPN
ma-381	126	10	(	(	PUNCT
ma-381	126	11	lp	lp	ADJ
ma-381	126	12	,	,	PUNCT
ma-381	126	13	`	`	PUNCT
ma-381	126	14	q)α(rd	q)α(rd	X
ma-381	126	15	)	)	PUNCT
ma-381	127	1	⊂	⊂	PROPN
ma-381	127	2	(	(	PUNCT
ma-381	127	3	lp	lp	PROPN
ma-381	127	4	,	,	PUNCT
ma-381	127	5	`	`	PUNCT
ma-381	127	6	q)(rd	q)(rd	NUM
ma-381	127	7	)	)	PUNCT
ma-381	127	8	.	.	PUNCT
ma-381	128	1	it	it	PRON
ma-381	128	2	comes	come	VERB
ma-381	128	3	from	from	ADP
ma-381	128	4	[	[	X
ma-381	128	5	3,proposition	3,proposition	NUM
ma-381	128	6	2.1	2.1	NUM
ma-381	128	7	]	]	PUNCT
ma-381	128	8	that	that	SCONJ
ma-381	128	9	there	there	PRON
ma-381	128	10	exists	exist	VERB
ma-381	128	11	f	f	PROPN
ma-381	128	12	∈	∈	PROPN
ma-381	128	13	h(p	h(p	PROPN
ma-381	128	14	,	,	PUNCT
ma-381	128	15	q)(rd	q)(rd	NUM
ma-381	128	16	)	)	PUNCT
ma-381	128	17	and	and	CCONJ
ma-381	128	18	a	a	DET
ma-381	128	19	constant	constant	ADJ
ma-381	128	20	c	c	NOUN
ma-381	128	21	>	>	X
ma-381	128	22	0	0	NUM
ma-381	128	23	such	such	ADJ
ma-381	128	24	that	that	SCONJ
ma-381	128	25	u(x	u(x	NOUN
ma-381	128	26	,	,	PUNCT
ma-381	128	27	t	t	NOUN
ma-381	128	28	)	)	PUNCT
ma-381	128	29	=	=	PUNCT
ma-381	129	1	(	(	PUNCT
ma-381	129	2	f	f	PROPN
ma-381	129	3	∗	∗	NOUN
ma-381	129	4	pt)(x	pt)(x	PROPN
ma-381	129	5	)	)	PUNCT
ma-381	130	1	,	,	PUNCT
ma-381	130	2	(	(	PUNCT
ma-381	130	3	x	x	X
ma-381	130	4	,	,	PUNCT
ma-381	130	5	t	t	PROPN
ma-381	130	6	)	)	PUNCT
ma-381	130	7	∈	∈	PROPN
ma-381	130	8	rd+1	rd+1	NOUN
ma-381	130	9	+	+	CCONJ
ma-381	130	10	(	(	PUNCT
ma-381	130	11	3.2	3.2	NUM
ma-381	130	12	)	)	PUNCT
ma-381	130	13	and	and	CCONJ
ma-381	131	1	1	1	NUM
ma-381	131	2	c	c	NOUN
ma-381	131	3	‖f	‖f	ADJ
ma-381	131	4	‖h(p	‖h(p	SYM
ma-381	131	5	,	,	PUNCT
ma-381	131	6	q	q	NOUN
ma-381	131	7	)	)	PUNCT
ma-381	131	8	≤	≤	NUM
ma-381	131	9	‖u∗‖p	‖u∗‖p	VERB
ma-381	131	10	,	,	PUNCT
ma-381	131	11	q	q	PROPN
ma-381	131	12	≤	≤	NUM
ma-381	131	13	c‖f	c‖f	NOUN
ma-381	132	1	‖h(p	‖h(p	X
ma-381	132	2	,	,	PUNCT
ma-381	132	3	q	q	NOUN
ma-381	132	4	)	)	PUNCT
ma-381	132	5	.since	.since	NOUN
ma-381	132	6	stαρ	stαρ	NOUN
ma-381	132	7	f	f	PROPN
ma-381	132	8	∈	∈	PROPN
ma-381	132	9	h(p	h(p	PROPN
ma-381	132	10	,	,	PUNCT
ma-381	132	11	q)(rd	q)(rd	NUM
ma-381	132	12	)	)	PUNCT
ma-381	132	13	for	for	ADP
ma-381	132	14	all	all	DET
ma-381	132	15	ρ	ρ	PROPN
ma-381	132	16	>	>	X
ma-381	132	17	0	0	PROPN
ma-381	132	18	,	,	PUNCT
ma-381	132	19	stαρ	stαρ	NOUN
ma-381	132	20	u	u	PROPN
ma-381	132	21	harmonic	harmonic	VERB
ma-381	132	22	on	on	ADP
ma-381	132	23	rd+1	rd+1	NOUN
ma-381	132	24	+	+	ADJ
ma-381	132	25	and	and	CCONJ
ma-381	132	26	(	(	PUNCT
ma-381	132	27	stαρ	stαρ	NOUN
ma-381	132	28	u	u	PROPN
ma-381	132	29	)	)	PUNCT
ma-381	132	30	(	(	PUNCT
ma-381	132	31	x	x	X
ma-381	132	32	,	,	PUNCT
ma-381	132	33	t	t	PROPN
ma-381	132	34	)	)	PUNCT
ma-381	132	35	=	=	PUNCT
ma-381	133	1	(	(	PUNCT
ma-381	133	2	stαρ	stαρ	NOUN
ma-381	133	3	f	f	PROPN
ma-381	133	4	)	)	PUNCT
ma-381	133	5	∗pt(x),it	∗pt(x),it	NOUN
ma-381	133	6	comes	come	VERB
ma-381	133	7	that	that	SCONJ
ma-381	133	8	(	(	PUNCT
ma-381	133	9	stαρ	stαρ	NOUN
ma-381	133	10	u)∗	u)∗	PROPN
ma-381	133	11	∈	∈	PROPN
ma-381	133	12	(	(	PUNCT
ma-381	133	13	lp	lp	NOUN
ma-381	133	14	,	,	PUNCT
ma-381	133	15	lq)(rd	lq)(rd	PROPN
ma-381	133	16	)	)	PUNCT
ma-381	133	17	and	and	CCONJ
ma-381	133	18	1	1	NUM
ma-381	133	19	c	c	NOUN
ma-381	133	20	‖stαρ	‖stαρ	VERB
ma-381	133	21	f	f	NOUN
ma-381	133	22	‖h(p	‖h(p	PROPN
ma-381	133	23	,	,	PUNCT
ma-381	133	24	q	q	NOUN
ma-381	133	25	)	)	PUNCT
ma-381	133	26	≤	≤	NOUN
ma-381	133	27	‖(stαρ	‖(stαρ	NOUN
ma-381	134	1	u)∗‖p	u)∗‖p	PROPN
ma-381	134	2	,	,	PUNCT
ma-381	134	3	q	q	PROPN
ma-381	134	4	≤	≤	NUM
ma-381	134	5	c‖stαρ	c‖stαρ	ADP
ma-381	134	6	f	f	PROPN
ma-381	134	7	‖h(p	‖h(p	PROPN
ma-381	134	8	,	,	PUNCT
ma-381	134	9	q	q	NOUN
ma-381	134	10	)	)	PUNCT
ma-381	134	11	.	.	PUNCT
ma-381	135	1	this	this	DET
ma-381	135	2	relation	relation	NOUN
ma-381	135	3	being	be	AUX
ma-381	135	4	thrue	thrue	ADJ
ma-381	135	5	for	for	ADP
ma-381	135	6	all	all	DET
ma-381	135	7	ρ	ρ	PROPN
ma-381	135	8	>	>	X
ma-381	135	9	0	0	NUM
ma-381	135	10	,	,	PUNCT
ma-381	135	11	we	we	PRON
ma-381	135	12	have	have	VERB
ma-381	135	13	1	1	NUM
ma-381	135	14	c	c	NOUN
ma-381	135	15	‖f	‖f	ADJ
ma-381	135	16	‖h(p	‖h(p	SYM
ma-381	135	17	,	,	PUNCT
ma-381	135	18	q	q	NOUN
ma-381	135	19	,	,	PUNCT
ma-381	135	20	α	α	NOUN
ma-381	135	21	)	)	PUNCT
ma-381	135	22	≤	≤	NUM
ma-381	135	23	‖u∗‖p	‖u∗‖p	VERB
ma-381	135	24	,	,	PUNCT
ma-381	135	25	q	q	INTJ
ma-381	135	26	,	,	PUNCT
ma-381	135	27	α	α	PROPN
ma-381	135	28	≤	≤	PROPN
ma-381	135	29	c‖f	c‖f	NOUN
ma-381	136	1	‖h(p	‖h(p	X
ma-381	136	2	,	,	PUNCT
ma-381	136	3	q	q	NOUN
ma-381	136	4	,	,	PUNCT
ma-381	136	5	α	α	NOUN
ma-381	136	6	)	)	PUNCT
ma-381	136	7	,	,	PUNCT
ma-381	136	8	where	where	SCONJ
ma-381	136	9	we	we	PRON
ma-381	136	10	use	use	VERB
ma-381	136	11	the	the	DET
ma-381	136	12	trivial	trivial	ADJ
ma-381	136	13	identity	identity	NOUN
ma-381	136	14	(	(	PUNCT
ma-381	136	15	stαρ	stαρ	NOUN
ma-381	136	16	u)∗	u)∗	PUNCT
ma-381	136	17	=	=	PUNCT
ma-381	136	18	stαρ	stαρ	NOUN
ma-381	136	19	u	u	PROPN
ma-381	136	20	∗	∗	NOUN
ma-381	136	21	,	,	PUNCT
ma-381	136	22	ρ	ρ	PROPN
ma-381	136	23	>	>	X
ma-381	136	24	0	0	PUNCT
ma-381	137	1	and	and	CCONJ
ma-381	137	2	0	0	NUM
ma-381	137	3	<	<	X
ma-381	137	4	α	α	X
ma-381	137	5	<	<	X
ma-381	137	6	∞.	∞.	PROPN
ma-381	137	7	�	�	PROPN
ma-381	137	8	we	we	PRON
ma-381	137	9	say	say	VERB
ma-381	137	10	that	that	SCONJ
ma-381	137	11	a	a	DET
ma-381	137	12	vector	vector	NOUN
ma-381	137	13	values	value	NOUN
ma-381	137	14	function	function	VERB
ma-381	137	15	f	f	NOUN
ma-381	138	1	:	:	PUNCT
ma-381	138	2	=	=	SYM
ma-381	138	3	(	(	PUNCT
ma-381	138	4	u1	u1	PROPN
ma-381	138	5	,	,	PUNCT
ma-381	138	6	u2	u2	NOUN
ma-381	138	7	,	,	PUNCT
ma-381	138	8	...	...	PUNCT
ma-381	138	9	,	,	PUNCT
ma-381	138	10	ud+1	ud+1	PROPN
ma-381	138	11	)	)	PUNCT
ma-381	138	12	,	,	PUNCT
ma-381	138	13	with	with	ADP
ma-381	138	14	uj	uj	PROPN
ma-381	138	15	:	:	PUNCT
ma-381	138	16	rd+1	rd+1	PROPN
ma-381	139	1	+	+	CCONJ
ma-381	139	2	→	→	SYM
ma-381	139	3	r	r	NOUN
ma-381	139	4	,	,	PUNCT
ma-381	139	5	j	j	PROPN
ma-381	139	6	∈	∈	PROPN
ma-381	139	7	{	{	PUNCT
ma-381	139	8	1	1	NUM
ma-381	139	9	,	,	PUNCT
ma-381	139	10	2	2	NUM
ma-381	139	11	,	,	PUNCT
ma-381	139	12	...	...	PUNCT
ma-381	139	13	,	,	PUNCT
ma-381	139	14	d+	d+	X
ma-381	139	15	1	1	X
ma-381	139	16	}	}	PUNCT
ma-381	139	17	,	,	PUNCT
ma-381	139	18	satisfies	satisfy	VERB
ma-381	139	19	the	the	DET
ma-381	139	20	generalized	generalized	ADJ
ma-381	139	21	cauchy	cauchy	PROPN
ma-381	139	22	-	-	PUNCT
ma-381	139	23	riemann	riemann	PROPN
ma-381	139	24	equations	equation	NOUN
ma-381	139	25	(	(	PUNCT
ma-381	139	26	in	in	ADP
ma-381	139	27	short	short	ADJ
ma-381	139	28	f	f	X
ma-381	139	29	∈	∈	PROPN
ma-381	139	30	cr(rd+1	cr(rd+1	NOUN
ma-381	139	31	+	+	PUNCT
ma-381	139	32	)	)	PUNCT
ma-381	139	33	)	)	PUNCT
ma-381	140	1	if	if	SCONJ
ma-381	140	2	∂uj	∂uj	PROPN
ma-381	140	3	∂xk	∂xk	PROPN
ma-381	140	4	=	=	SYM
ma-381	140	5	∂uk	∂uk	PROPN
ma-381	140	6	∂xj	∂xj	NOUN
ma-381	140	7	,	,	PUNCT
ma-381	140	8	1	1	NUM
ma-381	140	9	≤	≤	PROPN
ma-381	140	10	j	j	PROPN
ma-381	140	11	,	,	PUNCT
ma-381	140	12	k	k	PROPN
ma-381	140	13	≤	≤	PROPN
ma-381	140	14	d	d	PROPN
ma-381	140	15	+	+	CCONJ
ma-381	140	16	1	1	NUM
ma-381	140	17	and	and	CCONJ
ma-381	140	18	d+1∑	d+1∑	PROPN
ma-381	140	19	j=1	j=1	PROPN
ma-381	140	20	∂uj	∂uj	PROPN
ma-381	140	21	∂xj	∂xj	PROPN
ma-381	140	22	=	=	SYM
ma-381	140	23	0	0	NUM
ma-381	140	24	,	,	PUNCT
ma-381	140	25	(	(	PUNCT
ma-381	140	26	3.3	3.3	NUM
ma-381	140	27	)	)	PUNCT
ma-381	140	28	where	where	SCONJ
ma-381	140	29	we	we	PRON
ma-381	140	30	set	set	VERB
ma-381	140	31	xd+1	xd+1	NOUN
ma-381	140	32	=	=	PUNCT
ma-381	140	33	t.	t.	PROPN
ma-381	140	34	also	also	ADV
ma-381	140	35	recall	recall	VERB
ma-381	140	36	that	that	PRON
ma-381	140	37	for	for	ADP
ma-381	140	38	j	j	PROPN
ma-381	140	39	∈	∈	PROPN
ma-381	140	40	{	{	PUNCT
ma-381	140	41	1	1	NUM
ma-381	140	42	,	,	PUNCT
ma-381	140	43	2	2	NUM
ma-381	140	44	,	,	PUNCT
ma-381	140	45	...	...	PUNCT
ma-381	140	46	d	d	X
ma-381	140	47	}	}	PUNCT
ma-381	140	48	,	,	PUNCT
ma-381	140	49	the	the	DET
ma-381	140	50	j	j	PROPN
ma-381	140	51	-	-	PUNCT
ma-381	140	52	th	th	VERB
ma-381	140	53	riesz	riesz	NOUN
ma-381	140	54	transform	transform	NOUN
ma-381	140	55	rj(g	rj(g	NOUN
ma-381	140	56	)	)	PUNCT
ma-381	140	57	of	of	ADP
ma-381	140	58	ameasurable	ameasurable	ADJ
ma-381	140	59	function	function	NOUN
ma-381	140	60	g	g	PROPN
ma-381	140	61	is	be	AUX
ma-381	140	62	formally	formally	ADV
ma-381	140	63	defined	define	VERB
ma-381	140	64	by	by	ADP
ma-381	140	65	rj(g)(x	rj(g)(x	NOUN
ma-381	140	66	)	)	PUNCT
ma-381	140	67	:	:	PUNCT
ma-381	141	1	=	=	PUNCT
ma-381	141	2	lim	lim	PROPN
ma-381	141	3	ε→0	ε→0	NOUN
ma-381	141	4	+	+	CCONJ
ma-381	141	5	∫	∫	NOUN
ma-381	141	6	|x−y	|x−y	VERB
ma-381	141	7	|>ε	|>ε	NUM
ma-381	141	8	kj(x	kj(x	NOUN
ma-381	141	9	−	−	NOUN
ma-381	141	10	y)g(y)dy	y)g(y)dy	VERB
ma-381	141	11	a.e	a.e	PROPN
ma-381	141	12	x	x	SYM
ma-381	141	13	∈	∈	PROPN
ma-381	141	14	rd	rd	PROPN
ma-381	141	15	.	.	PUNCT
ma-381	142	1	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	142	2	eur	eur	PROPN
ma-381	142	3	.	.	PUNCT
ma-381	143	1	j.	j.	PROPN
ma-381	143	2	math	math	PROPN
ma-381	143	3	.	.	PUNCT
ma-381	144	1	anal	anal	PROPN
ma-381	144	2	.	.	PUNCT
ma-381	145	1	10.28924	10.28924	NUM
ma-381	145	2	/	/	SYM
ma-381	145	3	ada	ada	PROPN
ma-381	145	4	/	/	SYM
ma-381	145	5	ma.5.21	ma.5.21	PROPN
ma-381	145	6	7	7	NUM
ma-381	145	7	where	where	SCONJ
ma-381	145	8	kj(x	kj(x	PUNCT
ma-381	145	9	)	)	PUNCT
ma-381	145	10	:	:	PUNCT
ma-381	146	1	=	=	SYM
ma-381	146	2	γ	γ	X
ma-381	146	3	(	(	PUNCT
ma-381	146	4	d+1	d+1	PROPN
ma-381	146	5	2	2	NUM
ma-381	146	6	)	)	PUNCT
ma-381	146	7	π	π	PROPN
ma-381	146	8	d+1	d+1	PROPN
ma-381	146	9	2	2	NUM
ma-381	146	10	xj	xj	PROPN
ma-381	146	11	|x	|x	NOUN
ma-381	146	12	|d+1	|d+1	NOUN
ma-381	146	13	,	,	PUNCT
ma-381	146	14	x	x	PUNCT
ma-381	146	15	∈	∈	NOUN
ma-381	146	16	rd\{0}.in	rd\{0}.in	PUNCT
ma-381	147	1	[	[	X
ma-381	147	2	2	2	NUM
ma-381	147	3	,	,	PUNCT
ma-381	147	4	corollary	corollary	NOUN
ma-381	147	5	4.19	4.19	NUM
ma-381	147	6	]	]	PUNCT
ma-381	147	7	,	,	PUNCT
ma-381	147	8	ablé	ablé	ADV
ma-381	147	9	and	and	CCONJ
ma-381	147	10	feuto	feuto	NOUN
ma-381	147	11	demonstrated	demonstrate	VERB
ma-381	147	12	that	that	SCONJ
ma-381	147	13	riesz	riesz	NOUN
ma-381	147	14	transformations	transformation	NOUN
ma-381	147	15	are	be	AUX
ma-381	147	16	extendableinto	extendableinto	ADP
ma-381	147	17	bounded	bound	VERB
ma-381	147	18	linear	linear	PROPN
ma-381	147	19	operators	operator	NOUN
ma-381	147	20	on	on	ADP
ma-381	147	21	hardy	hardy	ADJ
ma-381	147	22	-	-	PUNCT
ma-381	147	23	amalgam	amalgam	NOUN
ma-381	147	24	spaces	space	NOUN
ma-381	147	25	h(p	h(p	NOUN
ma-381	147	26	,	,	PUNCT
ma-381	147	27	q)(rd	q)(rd	NUM
ma-381	147	28	)	)	PUNCT
ma-381	147	29	for	for	ADP
ma-381	147	30	0	0	NUM
ma-381	147	31	<	<	X
ma-381	147	32	p	p	X
ma-381	147	33	≤	≤	NUM
ma-381	147	34	1	1	NUM
ma-381	147	35	.	.	PUNCT
ma-381	148	1	we	we	PRON
ma-381	148	2	will	will	AUX
ma-381	148	3	keepthe	keepthe	DET
ma-381	148	4	notations	notation	NOUN
ma-381	148	5	rj	rj	PROPN
ma-381	148	6	,	,	PUNCT
ma-381	148	7	j	j	PROPN
ma-381	148	8	=	=	SYM
ma-381	148	9	1	1	NUM
ma-381	148	10	,	,	PUNCT
ma-381	148	11	·	·	PUNCT
ma-381	148	12	·	·	PUNCT
ma-381	148	13	·	·	PUNCT
ma-381	148	14	,	,	PUNCT
ma-381	148	15	d	d	X
ma-381	148	16	for	for	ADP
ma-381	148	17	these	these	DET
ma-381	148	18	extentions	extention	NOUN
ma-381	148	19	.	.	PUNCT
ma-381	149	1	assaubay	assaubay	PROPN
ma-381	149	2	et	et	PROPN
ma-381	149	3	al	al	PROPN
ma-381	149	4	proved	prove	VERB
ma-381	149	5	the	the	DET
ma-381	149	6	following	follow	VERB
ma-381	149	7	result	result	NOUN
ma-381	149	8	.	.	PUNCT
ma-381	150	1	proposition	proposition	NOUN
ma-381	150	2	3.2	3.2	NUM
ma-381	150	3	(	(	PUNCT
ma-381	150	4	[	[	X
ma-381	150	5	3	3	NUM
ma-381	150	6	]	]	PUNCT
ma-381	150	7	,	,	PUNCT
ma-381	150	8	proposition	proposition	NOUN
ma-381	150	9	2.3	2.3	NUM
ma-381	150	10	)	)	PUNCT
ma-381	150	11	.	.	PUNCT
ma-381	151	1	let	let	AUX
ma-381	151	2	d−1	d−1	PROPN
ma-381	151	3	d	d	PROPN
ma-381	151	4	<	<	X
ma-381	151	5	min	min	PROPN
ma-381	151	6	{	{	PUNCT
ma-381	151	7	p	p	X
ma-381	151	8	,	,	PUNCT
ma-381	151	9	q	q	NOUN
ma-381	151	10	}	}	PUNCT
ma-381	151	11	<	<	X
ma-381	151	12	+	+	NOUN
ma-381	151	13	∞.	∞.	PROPN
ma-381	151	14	suppose	suppose	VERB
ma-381	151	15	that	that	SCONJ
ma-381	151	16	u	u	NOUN
ma-381	151	17	is	be	AUX
ma-381	151	18	harmonic	harmonic	ADJ
ma-381	151	19	function	function	NOUN
ma-381	151	20	in	in	ADP
ma-381	151	21	rd+1	rd+1	NOUN
ma-381	152	1	+	+	CCONJ
ma-381	152	2	.	.	PUNCT
ma-381	153	1	then	then	ADV
ma-381	153	2	u∗	u∗	PROPN
ma-381	153	3	∈	∈	PROPN
ma-381	153	4	(	(	PUNCT
ma-381	153	5	lp	lp	PROPN
ma-381	153	6	,	,	PUNCT
ma-381	153	7	`	`	PUNCT
ma-381	153	8	q)(rd	q)(rd	X
ma-381	153	9	)	)	PUNCT
ma-381	153	10	if	if	SCONJ
ma-381	153	11	and	and	CCONJ
ma-381	153	12	only	only	ADV
ma-381	153	13	if	if	SCONJ
ma-381	153	14	there	there	PRON
ma-381	153	15	exists	exist	VERB
ma-381	153	16	an	an	DET
ma-381	153	17	harmonic	harmonic	ADJ
ma-381	153	18	vector	vector	NOUN
ma-381	153	19	f	f	NOUN
ma-381	153	20	:	:	PUNCT
ma-381	153	21	=	=	SYM
ma-381	153	22	(	(	PUNCT
ma-381	153	23	u1	u1	PROPN
ma-381	153	24	,	,	PUNCT
ma-381	153	25	...	...	PUNCT
ma-381	153	26	,	,	PUNCT
ma-381	153	27	ud+1	ud+1	X
ma-381	153	28	)	)	PUNCT
ma-381	153	29	∈	∈	PROPN
ma-381	153	30	cr(rd+1	cr(rd+1	NOUN
ma-381	153	31	+	+	CCONJ
ma-381	153	32	)	)	PUNCT
ma-381	153	33	such	such	ADJ
ma-381	153	34	that	that	SCONJ
ma-381	153	35	ud+1	ud+1	PRON
ma-381	153	36	:	:	PUNCT
ma-381	153	37	=	=	SYM
ma-381	153	38	u	u	NOUN
ma-381	153	39	and	and	CCONJ
ma-381	153	40	supt>0	supt>0	PROPN
ma-381	153	41	‖|f	‖|f	PROPN
ma-381	153	42	(	(	PUNCT
ma-381	153	43	.	.	NUM
ma-381	153	44	,	,	PUNCT
ma-381	153	45	t)|‖p	t)|‖p	PROPN
ma-381	153	46	,	,	PUNCT
ma-381	153	47	q	q	X
ma-381	153	48	<	<	X
ma-381	153	49	+	+	NOUN
ma-381	153	50	∞.	∞.	PROPN
ma-381	153	51	furthermore	furthermore	ADV
ma-381	153	52	,	,	PUNCT
ma-381	153	53	supt>0	supt>0	PROPN
ma-381	153	54	‖|f	‖|f	PROPN
ma-381	153	55	(	(	PUNCT
ma-381	153	56	.	.	NUM
ma-381	153	57	,	,	PUNCT
ma-381	153	58	t)|‖p	t)|‖p	PROPN
ma-381	153	59	,	,	PUNCT
ma-381	153	60	q	q	PROPN
ma-381	154	1	≈	≈	PROPN
ma-381	154	2	‖u∗‖p	‖u∗‖p	PROPN
ma-381	154	3	,	,	PUNCT
ma-381	154	4	q	q	X
ma-381	154	5	.	.	PUNCT
ma-381	155	1	since	since	SCONJ
ma-381	155	2	u∗	u∗	PROPN
ma-381	155	3	∈	∈	PROPN
ma-381	155	4	(	(	PUNCT
ma-381	155	5	lp	lp	PROPN
ma-381	155	6	,	,	PUNCT
ma-381	155	7	`	`	PUNCT
ma-381	155	8	q)(rd	q)(rd	X
ma-381	155	9	)	)	PUNCT
ma-381	156	1	if	if	SCONJ
ma-381	156	2	and	and	CCONJ
ma-381	156	3	only	only	ADV
ma-381	156	4	if	if	SCONJ
ma-381	156	5	u	u	PRON
ma-381	156	6	=	=	PROPN
ma-381	156	7	f	f	PROPN
ma-381	156	8	∗	∗	NOUN
ma-381	156	9	pt	pt	NOUN
ma-381	156	10	for	for	ADP
ma-381	156	11	some	some	DET
ma-381	156	12	f	f	PROPN
ma-381	156	13	∈	∈	PROPN
ma-381	156	14	h(p	h(p	PROPN
ma-381	156	15	,	,	PUNCT
ma-381	156	16	q)(rd	q)(rd	NUM
ma-381	156	17	)	)	PUNCT
ma-381	156	18	,	,	PUNCT
ma-381	156	19	they	they	PRON
ma-381	156	20	proved	prove	VERB
ma-381	156	21	that	that	SCONJ
ma-381	156	22	onecan	onecan	PROPN
ma-381	156	23	take	take	VERB
ma-381	156	24	uj(x	uj(x	NUM
ma-381	156	25	,	,	PUNCT
ma-381	156	26	t	t	PROPN
ma-381	156	27	)	)	PUNCT
ma-381	156	28	=	=	PUNCT
ma-381	157	1	rj(f	rj(f	X
ma-381	157	2	)	)	PUNCT
ma-381	157	3	∗	∗	NOUN
ma-381	157	4	pt(x	pt(x	PRON
ma-381	157	5	)	)	PUNCT
ma-381	157	6	,	,	PUNCT
ma-381	157	7	j	j	PROPN
ma-381	157	8	=	=	SYM
ma-381	157	9	1	1	NUM
ma-381	157	10	,	,	PUNCT
ma-381	157	11	·	·	PUNCT
ma-381	157	12	·	·	PUNCT
ma-381	157	13	·	·	PUNCT
ma-381	157	14	,	,	PUNCT
ma-381	158	1	d	d	X
ma-381	158	2	.in	.in	PUNCT
ma-381	158	3	the	the	DET
ma-381	158	4	case	case	NOUN
ma-381	158	5	of	of	ADP
ma-381	158	6	hardy	hardy	ADJ
ma-381	158	7	-	-	PUNCT
ma-381	158	8	fofana	fofana	NOUN
ma-381	158	9	’s	’s	PART
ma-381	158	10	spaces	space	NOUN
ma-381	158	11	,	,	PUNCT
ma-381	158	12	we	we	PRON
ma-381	158	13	have	have	VERB
ma-381	158	14	the	the	DET
ma-381	158	15	following	following	NOUN
ma-381	158	16	.	.	PUNCT
ma-381	159	1	proposition	proposition	NOUN
ma-381	159	2	3.3	3.3	NUM
ma-381	159	3	.	.	PUNCT
ma-381	160	1	assume	assume	VERB
ma-381	160	2	that	that	SCONJ
ma-381	160	3	d−1	d−1	PROPN
ma-381	160	4	d	d	PROPN
ma-381	160	5	<	<	X
ma-381	160	6	p	p	X
ma-381	160	7	≤	≤	NUM
ma-381	160	8	α	α	NOUN
ma-381	160	9	≤	≤	NUM
ma-381	160	10	q	q	X
ma-381	160	11	<	<	X
ma-381	160	12	+	+	NOUN
ma-381	160	13	∞	∞	NUM
ma-381	160	14	and	and	CCONJ
ma-381	160	15	u	u	NOUN
ma-381	160	16	is	be	AUX
ma-381	160	17	an	an	DET
ma-381	160	18	harmonic	harmonic	ADJ
ma-381	160	19	function	function	NOUN
ma-381	160	20	in	in	ADP
ma-381	160	21	rd+1	rd+1	NOUN
ma-381	161	1	+	+	CCONJ
ma-381	161	2	.	.	PUNCT
ma-381	162	1	then	then	ADV
ma-381	162	2	u∗	u∗	PROPN
ma-381	162	3	∈	∈	PROPN
ma-381	162	4	(	(	PUNCT
ma-381	162	5	lp	lp	ADJ
ma-381	162	6	,	,	PUNCT
ma-381	162	7	`	`	PUNCT
ma-381	162	8	q)α(rd	q)α(rd	X
ma-381	162	9	)	)	PUNCT
ma-381	162	10	if	if	SCONJ
ma-381	163	1	and	and	CCONJ
ma-381	163	2	only	only	ADV
ma-381	163	3	if	if	SCONJ
ma-381	163	4	there	there	PRON
ma-381	163	5	exists	exist	VERB
ma-381	163	6	an	an	DET
ma-381	163	7	harmonic	harmonic	ADJ
ma-381	163	8	vector	vector	NOUN
ma-381	163	9	f	f	NOUN
ma-381	163	10	:	:	PUNCT
ma-381	163	11	=	=	SYM
ma-381	163	12	(	(	PUNCT
ma-381	163	13	u1	u1	PROPN
ma-381	163	14	,	,	PUNCT
ma-381	163	15	...	...	PUNCT
ma-381	163	16	,	,	PUNCT
ma-381	163	17	ud+1	ud+1	X
ma-381	163	18	)	)	PUNCT
ma-381	163	19	∈	∈	PROPN
ma-381	164	1	cr(rd+1	cr(rd+1	NOUN
ma-381	164	2	+	+	CCONJ
ma-381	164	3	)	)	PUNCT
ma-381	165	1	such	such	ADJ
ma-381	165	2	that	that	SCONJ
ma-381	165	3	ud+1	ud+1	PRON
ma-381	165	4	:	:	PUNCT
ma-381	165	5	=	=	SYM
ma-381	165	6	u	u	NOUN
ma-381	165	7	and	and	CCONJ
ma-381	165	8	supt>0	supt>0	PROPN
ma-381	165	9	‖|f	‖|f	PROPN
ma-381	165	10	(	(	PUNCT
ma-381	165	11	.	.	NUM
ma-381	165	12	,	,	PUNCT
ma-381	165	13	t)|‖p	t)|‖p	PROPN
ma-381	165	14	,	,	PUNCT
ma-381	165	15	q	q	NOUN
ma-381	165	16	,	,	PUNCT
ma-381	165	17	α	α	X
ma-381	165	18	<	<	X
ma-381	166	1	+	+	NOUN
ma-381	166	2	∞.	∞.	PROPN
ma-381	166	3	furthermore	furthermore	ADV
ma-381	166	4	sup	sup	NOUN
ma-381	166	5	t>0	t>0	NOUN
ma-381	166	6	‖|f	‖|f	NOUN
ma-381	166	7	(	(	PUNCT
ma-381	166	8	.	.	NUM
ma-381	166	9	,	,	PUNCT
ma-381	166	10	t)|‖p	t)|‖p	PROPN
ma-381	166	11	,	,	PUNCT
ma-381	166	12	q	q	NOUN
ma-381	166	13	,	,	PUNCT
ma-381	166	14	α	α	PROPN
ma-381	166	15	≈	≈	PROPN
ma-381	166	16	‖u∗‖p	‖u∗‖p	PROPN
ma-381	166	17	,	,	PUNCT
ma-381	166	18	q	q	NOUN
ma-381	166	19	,	,	PUNCT
ma-381	166	20	α	α	X
ma-381	166	21	(	(	PUNCT
ma-381	166	22	3.4	3.4	NUM
ma-381	166	23	)	)	PUNCT
ma-381	166	24	proof	proof	NOUN
ma-381	166	25	.	.	PUNCT
ma-381	167	1	let	let	VERB
ma-381	167	2	d−1	d−1	PROPN
ma-381	167	3	d	d	PROPN
ma-381	167	4	<	<	X
ma-381	167	5	p	p	X
ma-381	167	6	≤	≤	NUM
ma-381	167	7	α	α	NOUN
ma-381	167	8	≤	≤	NUM
ma-381	167	9	q	q	X
ma-381	167	10	<	<	X
ma-381	168	1	+	+	NOUN
ma-381	168	2	∞	∞	NUM
ma-381	168	3	and	and	CCONJ
ma-381	168	4	u	u	PRON
ma-381	168	5	an	an	DET
ma-381	168	6	harmonic	harmonic	ADJ
ma-381	168	7	function	function	NOUN
ma-381	168	8	on	on	ADP
ma-381	168	9	rd+1	rd+1	NOUN
ma-381	168	10	+	+	PUNCT
ma-381	168	11	.we	.we	PUNCT
ma-381	168	12	suppose	suppose	VERB
ma-381	168	13	that	that	SCONJ
ma-381	168	14	u∗	u∗	PROPN
ma-381	168	15	∈	∈	PROPN
ma-381	168	16	(	(	PUNCT
ma-381	168	17	lp	lp	ADJ
ma-381	168	18	,	,	PUNCT
ma-381	168	19	`	`	PUNCT
ma-381	168	20	q)α(rd	q)α(rd	X
ma-381	168	21	)	)	PUNCT
ma-381	168	22	.	.	PUNCT
ma-381	169	1	since	since	SCONJ
ma-381	169	2	(	(	PUNCT
ma-381	169	3	lp	lp	ADJ
ma-381	169	4	,	,	PUNCT
ma-381	169	5	`	`	PUNCT
ma-381	169	6	q)α(rd	q)α(rd	X
ma-381	169	7	)	)	PUNCT
ma-381	169	8	⊂	⊂	PROPN
ma-381	169	9	(	(	PUNCT
ma-381	169	10	lp	lp	PROPN
ma-381	169	11	,	,	PUNCT
ma-381	169	12	`	`	PUNCT
ma-381	169	13	q)(rd	q)(rd	NUM
ma-381	169	14	)	)	PUNCT
ma-381	169	15	,	,	PUNCT
ma-381	169	16	proposition	proposition	NOUN
ma-381	169	17	3.2assert	3.2assert	NUM
ma-381	169	18	that	that	SCONJ
ma-381	169	19	there	there	PRON
ma-381	169	20	exists	exist	VERB
ma-381	169	21	f	f	PROPN
ma-381	169	22	∈	∈	PROPN
ma-381	169	23	(	(	PUNCT
ma-381	169	24	lp	lp	PROPN
ma-381	169	25	,	,	PUNCT
ma-381	169	26	`	`	PUNCT
ma-381	169	27	q)(rd	q)(rd	NUM
ma-381	169	28	)	)	PUNCT
ma-381	169	29	so	so	SCONJ
ma-381	169	30	that	that	SCONJ
ma-381	169	31	:	:	PUNCT
ma-381	169	32	•	•	NUM
ma-381	169	33	u(x	u(x	PROPN
ma-381	169	34	,	,	PUNCT
ma-381	169	35	t	t	NOUN
ma-381	169	36	)	)	PUNCT
ma-381	169	37	=	=	SYM
ma-381	169	38	f	f	PROPN
ma-381	169	39	∗	∗	NOUN
ma-381	169	40	pt(x	pt(x	NOUN
ma-381	169	41	)	)	PUNCT
ma-381	169	42	,	,	PUNCT
ma-381	169	43	•	•	CCONJ
ma-381	169	44	the	the	DET
ma-381	169	45	harmonic	harmonic	ADJ
ma-381	169	46	vector	vector	NOUN
ma-381	169	47	f	f	NOUN
ma-381	169	48	=	=	PUNCT
ma-381	169	49	(	(	PUNCT
ma-381	169	50	u1	u1	PROPN
ma-381	169	51	,	,	PUNCT
ma-381	169	52	·	·	PUNCT
ma-381	169	53	·	·	PUNCT
ma-381	169	54	·	·	PUNCT
ma-381	169	55	,	,	PUNCT
ma-381	169	56	ud+1	ud+1	X
ma-381	169	57	)	)	PUNCT
ma-381	169	58	with	with	ADP
ma-381	169	59	uj(x	uj(x	PROPN
ma-381	169	60	,	,	PUNCT
ma-381	169	61	t	t	PROPN
ma-381	169	62	)	)	PUNCT
ma-381	169	63	=	=	PUNCT
ma-381	169	64	rj(f	rj(f	X
ma-381	169	65	)	)	PUNCT
ma-381	169	66	∗	∗	NOUN
ma-381	169	67	pt(x	pt(x	PRON
ma-381	169	68	)	)	PUNCT
ma-381	169	69	for	for	ADP
ma-381	169	70	j	j	PROPN
ma-381	169	71	∈	∈	PROPN
ma-381	169	72	{	{	PUNCT
ma-381	169	73	1	1	NUM
ma-381	169	74	,	,	PUNCT
ma-381	169	75	·	·	PUNCT
ma-381	169	76	·	·	PUNCT
ma-381	169	77	·	·	PUNCT
ma-381	169	78	,	,	PUNCT
ma-381	169	79	d}and	d}and	ADV
ma-381	169	80	ud+1	ud+1	NOUN
ma-381	169	81	=	=	SYM
ma-381	169	82	u	u	NOUN
ma-381	169	83	belongs	belong	VERB
ma-381	169	84	to	to	PART
ma-381	169	85	cr(rd+1	cr(rd+1	VERB
ma-381	169	86	+	+	CCONJ
ma-381	169	87	)	)	PUNCT
ma-381	169	88	,	,	PUNCT
ma-381	169	89	•	•	NUM
ma-381	169	90	supt>0	supt>0	NOUN
ma-381	169	91	‖|f	‖|f	PROPN
ma-381	169	92	(	(	PUNCT
ma-381	169	93	·	·	PUNCT
ma-381	169	94	,	,	PUNCT
ma-381	169	95	t)|‖p	t)|‖p	PROPN
ma-381	169	96	,	,	PUNCT
ma-381	169	97	q	q	PROPN
ma-381	169	98	≈	≈	PROPN
ma-381	169	99	‖u∗‖p	‖u∗‖p	PROPN
ma-381	169	100	,	,	PUNCT
ma-381	169	101	q.since	q.since	NOUN
ma-381	169	102	u∗	u∗	NOUN
ma-381	169	103	∈	∈	PROPN
ma-381	169	104	(	(	PUNCT
ma-381	169	105	lp	lp	NOUN
ma-381	169	106	,	,	PUNCT
ma-381	169	107	`	`	PUNCT
ma-381	169	108	q)α	q)α	X
ma-381	169	109	(	(	PUNCT
ma-381	169	110	rd	rd	NOUN
ma-381	169	111	)	)	PUNCT
ma-381	169	112	we	we	PRON
ma-381	169	113	have	have	VERB
ma-381	169	114	that	that	SCONJ
ma-381	169	115	the	the	DET
ma-381	169	116	tempered	temper	VERB
ma-381	169	117	distribution	distribution	NOUN
ma-381	169	118	f	f	PROPN
ma-381	169	119	belongs	belong	VERB
ma-381	169	120	to	to	ADP
ma-381	169	121	h(p	h(p	PROPN
ma-381	169	122	,	,	PUNCT
ma-381	169	123	q	q	NOUN
ma-381	169	124	,	,	PUNCT
ma-381	169	125	α)(rd	α)(rd	NOUN
ma-381	169	126	)	)	PUNCT
ma-381	169	127	,	,	PUNCT
ma-381	169	128	thanksto	thanksto	NOUN
ma-381	169	129	proposition	proposition	NOUN
ma-381	169	130	3.1	3.1	NUM
ma-381	169	131	.	.	PUNCT
ma-381	170	1	all	all	PRON
ma-381	170	2	we	we	PRON
ma-381	170	3	have	have	VERB
ma-381	170	4	to	to	PART
ma-381	170	5	prove	prove	VERB
ma-381	170	6	now	now	ADV
ma-381	170	7	is	be	AUX
ma-381	170	8	that	that	SCONJ
ma-381	170	9	x	x	PROPN
ma-381	170	10	7→	7→	NUM
ma-381	170	11	f	f	X
ma-381	170	12	(	(	PUNCT
ma-381	170	13	x	x	PROPN
ma-381	170	14	,	,	PUNCT
ma-381	170	15	t	t	PROPN
ma-381	170	16	)	)	PUNCT
ma-381	170	17	belongs	belong	VERB
ma-381	170	18	to	to	ADP
ma-381	170	19	(	(	PUNCT
ma-381	170	20	lp	lp	ADJ
ma-381	170	21	,	,	PUNCT
ma-381	170	22	`	`	PUNCT
ma-381	170	23	q)α(rd	q)α(rd	X
ma-381	170	24	)	)	PUNCT
ma-381	170	25	for	for	ADP
ma-381	170	26	t	t	PROPN
ma-381	170	27	>	>	X
ma-381	170	28	0	0	PUNCT
ma-381	170	29	and	and	CCONJ
ma-381	170	30	that	that	DET
ma-381	170	31	relation	relation	NOUN
ma-381	170	32	(	(	PUNCT
ma-381	170	33	3.4	3.4	NUM
ma-381	170	34	)	)	PUNCT
ma-381	170	35	is	be	AUX
ma-381	170	36	satisfies.fix	satisfies.fix	PROPN
ma-381	170	37	t	t	PROPN
ma-381	170	38	>	>	X
ma-381	170	39	0	0	PROPN
ma-381	170	40	and	and	CCONJ
ma-381	170	41	ρ	ρ	PROPN
ma-381	170	42	>	>	X
ma-381	170	43	0	0	PROPN
ma-381	170	44	.	.	PUNCT
ma-381	171	1	since	since	SCONJ
ma-381	171	2	u∗	u∗	PROPN
ma-381	171	3	∈	∈	PROPN
ma-381	171	4	(	(	PUNCT
ma-381	171	5	lp	lp	NOUN
ma-381	171	6	,	,	PUNCT
ma-381	171	7	`	`	PUNCT
ma-381	171	8	q)α	q)α	X
ma-381	171	9	(	(	PUNCT
ma-381	171	10	rd	rd	NOUN
ma-381	171	11	)	)	PUNCT
ma-381	171	12	and	and	CCONJ
ma-381	171	13	(	(	PUNCT
ma-381	171	14	stαρ	stαρ	NOUN
ma-381	171	15	u	u	PROPN
ma-381	171	16	)	)	PUNCT
ma-381	171	17	∗	∗	NOUN
ma-381	171	18	=	=	SYM
ma-381	171	19	stαρ	stαρ	NOUN
ma-381	171	20	(	(	PUNCT
ma-381	171	21	u∗	u∗	PROPN
ma-381	171	22	)	)	PUNCT
ma-381	171	23	,	,	PUNCT
ma-381	171	24	we	we	PRON
ma-381	171	25	have	have	VERB
ma-381	171	26	that	that	PRON
ma-381	171	27	for	for	ADP
ma-381	171	28	ρ	ρ	PROPN
ma-381	171	29	>	>	SYM
ma-381	171	30	0	0	NUM
ma-381	171	31	,	,	PUNCT
ma-381	171	32	‖	‖	PROPN
ma-381	171	33	(	(	PUNCT
ma-381	171	34	stαρ	stαρ	NOUN
ma-381	171	35	u	u	PROPN
ma-381	171	36	)	)	PUNCT
ma-381	171	37	∗	∗	NOUN
ma-381	171	38	‖p	‖p	PROPN
ma-381	171	39	,	,	PUNCT
ma-381	171	40	q	q	PROPN
ma-381	171	41	≤	≤	NUM
ma-381	171	42	‖u∗‖p	‖u∗‖p	VERB
ma-381	171	43	,	,	PUNCT
ma-381	171	44	q	q	NOUN
ma-381	171	45	,	,	PUNCT
ma-381	171	46	α	α	X
ma-381	171	47	.	.	PUNCT
ma-381	172	1	hence	hence	ADV
ma-381	172	2	(	(	PUNCT
ma-381	172	3	stαρ	stαρ	NOUN
ma-381	172	4	u	u	PROPN
ma-381	172	5	)	)	PUNCT
ma-381	172	6	∗	∗	NOUN
ma-381	172	7	∈	∈	PROPN
ma-381	172	8	(	(	PUNCT
ma-381	172	9	lp	lp	NOUN
ma-381	172	10	,	,	PUNCT
ma-381	172	11	`	`	PUNCT
ma-381	172	12	q	q	X
ma-381	172	13	)	)	PUNCT
ma-381	172	14	(	(	PUNCT
ma-381	172	15	rd	rd	NOUN
ma-381	172	16	)	)	PUNCT
ma-381	172	17	so	so	SCONJ
ma-381	172	18	that	that	SCONJ
ma-381	172	19	there	there	PRON
ma-381	172	20	exists	exist	VERB
ma-381	172	21	fρ	fρ	ADP
ma-381	172	22	∈	∈	PROPN
ma-381	172	23	h(p	h(p	NOUN
ma-381	172	24	,	,	PUNCT
ma-381	172	25	q)(rd)satisfying	q)(rd)satisfye	VERB
ma-381	172	26	(	(	PUNCT
ma-381	172	27	stαρ	stαρ	PROPN
ma-381	172	28	u)(x	u)(x	PROPN
ma-381	172	29	,	,	PUNCT
ma-381	172	30	t	t	PROPN
ma-381	172	31	)	)	PUNCT
ma-381	173	1	=	=	PUNCT
ma-381	174	1	(	(	PUNCT
ma-381	174	2	fρ	fρ	NOUN
ma-381	174	3	∗	∗	NOUN
ma-381	174	4	pt)(x),with	pt)(x),with	PROPN
ma-381	174	5	fρ(x	fρ(x	NOUN
ma-381	174	6	,	,	PUNCT
ma-381	174	7	t	t	PROPN
ma-381	174	8	)	)	PUNCT
ma-381	174	9	:	:	PUNCT
ma-381	175	1	=	=	SYM
ma-381	175	2	(	(	PUNCT
ma-381	175	3	r1(fρ	r1(fρ	PROPN
ma-381	175	4	)	)	PUNCT
ma-381	175	5	∗	∗	NOUN
ma-381	175	6	pt(x	pt(x	PRON
ma-381	175	7	)	)	PUNCT
ma-381	175	8	,	,	PUNCT
ma-381	175	9	·	·	PUNCT
ma-381	175	10	·	·	PUNCT
ma-381	175	11	·	·	PUNCT
ma-381	175	12	,	,	PUNCT
ma-381	175	13	rd(f	rd(f	X
ma-381	175	14	ρ	ρ	NOUN
ma-381	175	15	)	)	PUNCT
ma-381	175	16	∗	∗	NOUN
ma-381	175	17	pt(x	pt(x	PRON
ma-381	175	18	)	)	PUNCT
ma-381	175	19	,	,	PUNCT
ma-381	175	20	(	(	PUNCT
ma-381	175	21	fρ	fρ	NOUN
ma-381	175	22	)	)	PUNCT
ma-381	175	23	∗	∗	NOUN
ma-381	175	24	pt)(x	pt)(x	PROPN
ma-381	175	25	)	)	PUNCT
ma-381	175	26	)	)	PUNCT
ma-381	176	1	(	(	PUNCT
ma-381	176	2	3.5)belonging	3.5)belonging	NUM
ma-381	176	3	to	to	ADP
ma-381	176	4	cr+(rd+1	cr+(rd+1	PROPN
ma-381	176	5	+	+	CCONJ
ma-381	176	6	)	)	PUNCT
ma-381	176	7	and	and	CCONJ
ma-381	176	8	sup	sup	NOUN
ma-381	176	9	t>0	t>0	NOUN
ma-381	176	10	‖|fρ	‖|fρ	ADJ
ma-381	176	11	(	(	PUNCT
ma-381	176	12	·	·	PUNCT
ma-381	176	13	,	,	PUNCT
ma-381	176	14	t)|‖p	t)|‖p	PROPN
ma-381	176	15	,	,	PUNCT
ma-381	176	16	q	q	PROPN
ma-381	177	1	≈	≈	NOUN
ma-381	177	2	‖stαρ	‖stαρ	NOUN
ma-381	177	3	(	(	PUNCT
ma-381	177	4	u∗)‖p	u∗)‖p	PROPN
ma-381	177	5	,	,	PUNCT
ma-381	177	6	q.	q.	PROPN
ma-381	177	7	(	(	PUNCT
ma-381	177	8	3.6	3.6	NUM
ma-381	177	9	)	)	PUNCT
ma-381	177	10	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	177	11	eur	eur	PROPN
ma-381	177	12	.	.	PUNCT
ma-381	178	1	j.	j.	PROPN
ma-381	178	2	math	math	PROPN
ma-381	178	3	.	.	PUNCT
ma-381	179	1	anal	anal	PROPN
ma-381	179	2	.	.	PUNCT
ma-381	180	1	10.28924	10.28924	NUM
ma-381	180	2	/	/	SYM
ma-381	180	3	ada	ada	PROPN
ma-381	180	4	/	/	SYM
ma-381	180	5	ma.5.21	ma.5.21	PROPN
ma-381	180	6	8moreover	8moreover	PROPN
ma-381	180	7	(	(	PUNCT
ma-381	180	8	stαρ	stαρ	PROPN
ma-381	180	9	u)(x	u)(x	PROPN
ma-381	180	10	,	,	PUNCT
ma-381	180	11	t	t	PROPN
ma-381	180	12	)	)	PUNCT
ma-381	180	13	=	=	PUNCT
ma-381	181	1	(	(	PUNCT
ma-381	181	2	stαρ	stαρ	NOUN
ma-381	181	3	f	f	PROPN
ma-381	181	4	)	)	PUNCT
ma-381	181	5	∗pt(x	∗pt(x	PROPN
ma-381	181	6	)	)	PUNCT
ma-381	181	7	,	,	PUNCT
ma-381	181	8	thanks	thank	NOUN
ma-381	181	9	to	to	ADP
ma-381	181	10	relation	relation	NOUN
ma-381	181	11	(	(	PUNCT
ma-381	181	12	2.5	2.5	NUM
ma-381	181	13	)	)	PUNCT
ma-381	181	14	.	.	PUNCT
ma-381	182	1	it	it	PRON
ma-381	182	2	follows	follow	VERB
ma-381	182	3	that	that	SCONJ
ma-381	182	4	fρ∗pt	fρ∗pt	PROPN
ma-381	182	5	=	=	PUNCT
ma-381	182	6	(	(	PUNCT
ma-381	182	7	stαρ	stαρ	NOUN
ma-381	182	8	f	f	PROPN
ma-381	182	9	)	)	PUNCT
ma-381	182	10	∗ptfor	∗ptfor	ADP
ma-381	182	11	all	all	DET
ma-381	182	12	t	t	X
ma-381	182	13	>	>	X
ma-381	182	14	0	0	PUNCT
ma-381	183	1	so	so	SCONJ
ma-381	183	2	that	that	SCONJ
ma-381	183	3	fρ	fρ	ADJ
ma-381	183	4	=	=	PUNCT
ma-381	183	5	stαρ	stαρ	NOUN
ma-381	183	6	f	f	PROPN
ma-381	183	7	.	.	PUNCT
ma-381	184	1	we	we	PRON
ma-381	184	2	recall	recall	VERB
ma-381	184	3	that	that	SCONJ
ma-381	184	4	the	the	DET
ma-381	184	5	last	last	ADJ
ma-381	184	6	equality	equality	NOUN
ma-381	184	7	comes	come	VERB
ma-381	184	8	from	from	ADP
ma-381	184	9	the	the	DET
ma-381	184	10	fact	fact	NOUN
ma-381	184	11	that	that	SCONJ
ma-381	184	12	for	for	ADP
ma-381	184	13	f	f	PROPN
ma-381	184	14	∈	∈	PROPN
ma-381	184	15	h(p	h(p	PROPN
ma-381	184	16	,	,	PUNCT
ma-381	184	17	q)(rd	q)(rd	NUM
ma-381	184	18	)	)	PUNCT
ma-381	184	19	,	,	PUNCT
ma-381	184	20	f	f	PROPN
ma-381	184	21	∗	∗	NOUN
ma-381	184	22	pt	pt	PROPN
ma-381	184	23	tends	tend	VERB
ma-381	184	24	to	to	ADP
ma-381	184	25	f	f	PROPN
ma-381	184	26	in	in	ADP
ma-381	184	27	s	s	PROPN
ma-381	184	28	′(rd	′(rd	NOUN
ma-381	184	29	)	)	PUNCT
ma-381	184	30	as	as	SCONJ
ma-381	184	31	t	t	PROPN
ma-381	184	32	goes	go	VERB
ma-381	184	33	to	to	ADP
ma-381	184	34	0	0	NUM
ma-381	184	35	.	.	PUNCT
ma-381	184	36	replacing	replace	VERB
ma-381	184	37	fρ	fρ	NOUN
ma-381	184	38	by	by	ADP
ma-381	184	39	stαρ	stαρ	PROPN
ma-381	184	40	f	f	PROPN
ma-381	184	41	in	in	ADP
ma-381	184	42	relation	relation	NOUN
ma-381	184	43	(	(	PUNCT
ma-381	184	44	3.5)yields	3.5)yields	NUM
ma-381	184	45	fρ(x	fρ(x	NUM
ma-381	184	46	,	,	PUNCT
ma-381	184	47	t	t	PROPN
ma-381	184	48	)	)	PUNCT
ma-381	184	49	=	=	PUNCT
ma-381	184	50	(	(	PUNCT
ma-381	184	51	r1(stαρ	r1(stαρ	NOUN
ma-381	184	52	f	f	PROPN
ma-381	184	53	)	)	PUNCT
ma-381	184	54	∗	∗	NOUN
ma-381	184	55	pt(x	pt(x	PRON
ma-381	184	56	)	)	PUNCT
ma-381	184	57	,	,	PUNCT
ma-381	184	58	·	·	PUNCT
ma-381	184	59	·	·	PUNCT
ma-381	184	60	·	·	PUNCT
ma-381	184	61	,	,	PUNCT
ma-381	184	62	rd(stαρ	rd(stαρ	PROPN
ma-381	184	63	f	f	PROPN
ma-381	184	64	)	)	PUNCT
ma-381	184	65	∗	∗	NOUN
ma-381	184	66	pt(x	pt(x	PRON
ma-381	184	67	)	)	PUNCT
ma-381	184	68	,	,	PUNCT
ma-381	184	69	(	(	PUNCT
ma-381	184	70	stαρ	stαρ	NOUN
ma-381	184	71	f	f	PROPN
ma-381	184	72	)	)	PUNCT
ma-381	184	73	∗	∗	NOUN
ma-381	184	74	pt)(x	pt)(x	PROPN
ma-381	184	75	)	)	PUNCT
ma-381	184	76	)	)	PUNCT
ma-381	184	77	.	.	PUNCT
ma-381	185	1	since	since	SCONJ
ma-381	185	2	the	the	DET
ma-381	185	3	operator	operator	NOUN
ma-381	185	4	stαρ	stαρ	NOUN
ma-381	185	5	commute	commute	VERB
ma-381	185	6	with	with	ADP
ma-381	185	7	rj	rj	PROPN
ma-381	185	8	we	we	PRON
ma-381	185	9	have	have	VERB
ma-381	185	10	that	that	DET
ma-381	185	11	fρ	fρ	PROPN
ma-381	185	12	(	(	PUNCT
ma-381	185	13	·	·	PUNCT
ma-381	185	14	,	,	PUNCT
ma-381	185	15	t	t	PROPN
ma-381	185	16	)	)	PUNCT
ma-381	185	17	=	=	SYM
ma-381	186	1	(	(	PUNCT
ma-381	186	2	stαρ	stαρ	NOUN
ma-381	186	3	(	(	PUNCT
ma-381	186	4	r1f	r1f	PROPN
ma-381	186	5	)	)	PUNCT
ma-381	186	6	∗	∗	PROPN
ma-381	186	7	pt	pt	PROPN
ma-381	186	8	(	(	PUNCT
ma-381	186	9	·	·	PUNCT
ma-381	186	10	)	)	PUNCT
ma-381	186	11	,	,	PUNCT
ma-381	186	12	·	·	PUNCT
ma-381	186	13	·	·	PUNCT
ma-381	186	14	·	·	PUNCT
ma-381	186	15	,	,	PUNCT
ma-381	186	16	stαρ	stαρ	NOUN
ma-381	186	17	(	(	PUNCT
ma-381	186	18	rd	rd	NOUN
ma-381	186	19	f	f	PROPN
ma-381	186	20	)	)	PUNCT
ma-381	186	21	∗	∗	PROPN
ma-381	186	22	pt	pt	PROPN
ma-381	186	23	(	(	PUNCT
ma-381	186	24	·	·	PUNCT
ma-381	186	25	)	)	PUNCT
ma-381	186	26	,	,	PUNCT
ma-381	186	27	(	(	PUNCT
ma-381	186	28	stαρ	stαρ	NOUN
ma-381	186	29	f	f	PROPN
ma-381	186	30	)	)	PUNCT
ma-381	186	31	∗	∗	PROPN
ma-381	186	32	pt	pt	PROPN
ma-381	186	33	)	)	PUNCT
ma-381	186	34	(	(	PUNCT
ma-381	186	35	·	·	PUNCT
ma-381	186	36	)	)	PUNCT
ma-381	186	37	)	)	PUNCT
ma-381	187	1	=	=	PUNCT
ma-381	187	2	(	(	PUNCT
ma-381	187	3	stαρ	stαρ	NOUN
ma-381	187	4	(	(	PUNCT
ma-381	187	5	u1	u1	PROPN
ma-381	187	6	(	(	PUNCT
ma-381	187	7	·	·	PUNCT
ma-381	187	8	,	,	PUNCT
ma-381	187	9	ρ−1	ρ−1	PROPN
ma-381	187	10	t	t	PROPN
ma-381	187	11	)	)	PUNCT
ma-381	187	12	)	)	PUNCT
ma-381	187	13	,	,	PUNCT
ma-381	187	14	·	·	PUNCT
ma-381	187	15	·	·	PUNCT
ma-381	187	16	·	·	PUNCT
ma-381	187	17	,	,	PUNCT
ma-381	187	18	stαρ	stαρ	NOUN
ma-381	187	19	(	(	PUNCT
ma-381	187	20	ud	ud	INTJ
ma-381	187	21	(	(	PUNCT
ma-381	187	22	·	·	PUNCT
ma-381	187	23	,	,	PUNCT
ma-381	187	24	ρ−1	ρ−1	PROPN
ma-381	187	25	t	t	PROPN
ma-381	187	26	)	)	PUNCT
ma-381	187	27	)	)	PUNCT
ma-381	187	28	,	,	PUNCT
ma-381	187	29	stαρ	stαρ	NOUN
ma-381	187	30	(	(	PUNCT
ma-381	187	31	ud+1	ud+1	PROPN
ma-381	187	32	(	(	PUNCT
ma-381	187	33	·	·	PUNCT
ma-381	187	34	,	,	PUNCT
ma-381	187	35	ρ−1	ρ−1	PROPN
ma-381	187	36	t	t	PROPN
ma-381	187	37	)	)	PUNCT
ma-381	187	38	)	)	PUNCT
ma-381	187	39	)	)	PUNCT
ma-381	188	1	=	=	PUNCT
ma-381	188	2	stαρ	stαρ	NOUN
ma-381	188	3	(	(	PUNCT
ma-381	188	4	f	f	PROPN
ma-381	188	5	(	(	PUNCT
ma-381	188	6	·	·	PUNCT
ma-381	188	7	,	,	PUNCT
ma-381	188	8	ρ−1	ρ−1	PROPN
ma-381	188	9	t	t	PROPN
ma-381	188	10	)	)	PUNCT
ma-381	188	11	)	)	PUNCT
ma-381	188	12	.	.	PUNCT
ma-381	189	1	if	if	SCONJ
ma-381	189	2	we	we	PRON
ma-381	189	3	take	take	VERB
ma-381	189	4	this	this	DET
ma-381	189	5	expression	expression	NOUN
ma-381	189	6	of	of	ADP
ma-381	189	7	fρ	fρ	PROPN
ma-381	189	8	in	in	ADP
ma-381	189	9	relation	relation	NOUN
ma-381	189	10	(	(	PUNCT
ma-381	189	11	3.6	3.6	NUM
ma-381	189	12	)	)	PUNCT
ma-381	189	13	we	we	PRON
ma-381	189	14	obtain	obtain	VERB
ma-381	189	15	that	that	DET
ma-381	189	16	sup	sup	NOUN
ma-381	189	17	t>0	t>0	NOUN
ma-381	189	18	‖|stαρ	‖|stαρ	PRON
ma-381	190	1	(	(	PUNCT
ma-381	190	2	f	f	X
ma-381	190	3	(	(	PUNCT
ma-381	190	4	·	·	PUNCT
ma-381	190	5	,	,	PUNCT
ma-381	190	6	ρ−1t)|‖p	ρ−1t)|‖p	NOUN
ma-381	190	7	,	,	PUNCT
ma-381	190	8	q	q	PROPN
ma-381	191	1	≈	≈	PROPN
ma-381	191	2	‖stαρ	‖stαρ	PUNCT
ma-381	191	3	(	(	PUNCT
ma-381	191	4	u∗)‖p	u∗)‖p	PROPN
ma-381	191	5	,	,	PUNCT
ma-381	191	6	q.	q.	PROPN
ma-381	191	7	but	but	CCONJ
ma-381	191	8	supt>0	supt>0	PROPN
ma-381	191	9	‖|stαρ	‖|stαρ	PROPN
ma-381	192	1	(	(	PUNCT
ma-381	192	2	f	f	PROPN
ma-381	192	3	(	(	PUNCT
ma-381	192	4	·	·	PUNCT
ma-381	192	5	,	,	PUNCT
ma-381	192	6	ρ−1t))|‖p	ρ−1t))|‖p	NOUN
ma-381	192	7	,	,	PUNCT
ma-381	192	8	q	q	NOUN
ma-381	192	9	=	=	SYM
ma-381	192	10	supt>0	supt>0	NOUN
ma-381	192	11	‖|stαρ	‖|stαρ	PROPN
ma-381	192	12	(	(	PUNCT
ma-381	192	13	f	f	PROPN
ma-381	192	14	(	(	PUNCT
ma-381	192	15	·	·	PUNCT
ma-381	192	16	,	,	PUNCT
ma-381	192	17	t))|‖p	t))|‖p	PROPN
ma-381	192	18	,	,	PUNCT
ma-381	192	19	q	q	NOUN
ma-381	192	20	and	and	CCONJ
ma-381	192	21	the	the	DET
ma-381	192	22	result	result	NOUN
ma-381	192	23	follow	follow	VERB
ma-381	192	24	from	from	ADP
ma-381	192	25	the	the	DET
ma-381	192	26	defi	defi	NOUN
ma-381	192	27	-	-	PUNCT
ma-381	192	28	nition	nition	NOUN
ma-381	192	29	of	of	ADP
ma-381	192	30	hardy	hardy	ADJ
ma-381	192	31	-	-	PUNCT
ma-381	192	32	fofana	fofana	NOUN
ma-381	192	33	space	space	NOUN
ma-381	192	34	.	.	PUNCT
ma-381	193	1	�	�	PROPN
ma-381	193	2	the	the	DET
ma-381	193	3	next	next	ADJ
ma-381	193	4	result	result	NOUN
ma-381	193	5	gives	give	VERB
ma-381	193	6	a	a	DET
ma-381	193	7	characterization	characterization	NOUN
ma-381	193	8	of	of	ADP
ma-381	193	9	h(p	h(p	NOUN
ma-381	193	10	,	,	PUNCT
ma-381	193	11	q	q	NOUN
ma-381	193	12	,	,	PUNCT
ma-381	193	13	α)(rd	α)(rd	ADJ
ma-381	193	14	)	)	PUNCT
ma-381	193	15	via	via	ADP
ma-381	193	16	riesz	riesz	PROPN
ma-381	193	17	transforms	transform	VERB
ma-381	193	18	rj(f	rj(f	PROPN
ma-381	193	19	∗	∗	PROPN
ma-381	193	20	φ	φ	NUM
ma-381	193	21	)	)	PUNCT
ma-381	193	22	.	.	PUNCT
ma-381	194	1	sincewe	sincewe	NOUN
ma-381	194	2	need	need	VERB
ma-381	194	3	to	to	PART
ma-381	194	4	use	use	VERB
ma-381	194	5	the	the	DET
ma-381	194	6	characterization	characterization	NOUN
ma-381	194	7	of	of	ADP
ma-381	194	8	h(p	h(p	PROPN
ma-381	194	9	,	,	PUNCT
ma-381	194	10	q)(rd	q)(rd	NUM
ma-381	194	11	)	)	PUNCT
ma-381	194	12	given	give	VERB
ma-381	194	13	in	in	ADP
ma-381	194	14	[	[	X
ma-381	194	15	3	3	NUM
ma-381	194	16	]	]	PUNCT
ma-381	194	17	,	,	PUNCT
ma-381	194	18	we	we	PRON
ma-381	194	19	give	give	VERB
ma-381	194	20	the	the	DET
ma-381	194	21	following	follow	VERB
ma-381	194	22	definition	definition	NOUN
ma-381	194	23	.	.	PUNCT
ma-381	195	1	definition	definition	NOUN
ma-381	195	2	3.4	3.4	NUM
ma-381	195	3	.	.	PUNCT
ma-381	196	1	let	let	VERB
ma-381	196	2	0	0	PUNCT
ma-381	196	3	<	<	X
ma-381	196	4	p	p	X
ma-381	196	5	≤	≤	NUM
ma-381	196	6	α	α	NOUN
ma-381	196	7	≤	≤	NUM
ma-381	196	8	q	q	NOUN
ma-381	197	1	<	<	X
ma-381	197	2	∞.	∞.	PROPN
ma-381	197	3	a	a	DET
ma-381	197	4	tempered	temper	VERB
ma-381	197	5	distribution	distribution	NOUN
ma-381	197	6	f	f	NOUN
ma-381	197	7	is	be	AUX
ma-381	197	8	said	say	VERB
ma-381	197	9	to	to	PART
ma-381	197	10	be	be	AUX
ma-381	197	11	:	:	PUNCT
ma-381	197	12	•	•	X
ma-381	197	13	(	(	PUNCT
ma-381	197	14	p	p	NOUN
ma-381	197	15	,	,	PUNCT
ma-381	197	16	q)-restricted	q)-restricte	VERB
ma-381	197	17	at	at	ADP
ma-381	197	18	infinity	infinity	NOUN
ma-381	197	19	if	if	SCONJ
ma-381	197	20	there	there	PRON
ma-381	197	21	exists	exist	VERB
ma-381	197	22	µ0	µ0	NOUN
ma-381	197	23	≥	≥	NUM
ma-381	197	24	1	1	NUM
ma-381	197	25	such	such	ADJ
ma-381	197	26	that	that	PRON
ma-381	197	27	for	for	ADP
ma-381	197	28	µ	µ	PRON
ma-381	197	29	≥	≥	NUM
ma-381	197	30	µ0	µ0	NOUN
ma-381	197	31	,	,	PUNCT
ma-381	197	32	we	we	PRON
ma-381	197	33	have	have	VERB
ma-381	197	34	f	f	PROPN
ma-381	197	35	∗	∗	PROPN
ma-381	197	36	φ	φ	PROPN
ma-381	197	37	∈	∈	PROPN
ma-381	197	38	(	(	PUNCT
ma-381	197	39	lpµ	lpµ	X
ma-381	197	40	,	,	PUNCT
ma-381	197	41	`	`	PUNCT
ma-381	197	42	qµ)(rd	qµ)(rd	NOUN
ma-381	197	43	)	)	PUNCT
ma-381	197	44	,	,	PUNCT
ma-381	197	45	φ	φ	PROPN
ma-381	197	46	∈	∈	PROPN
ma-381	197	47	s(rd	s(rd	NUM
ma-381	197	48	)	)	PUNCT
ma-381	197	49	.	.	PUNCT
ma-381	198	1	•	•	NOUN
ma-381	198	2	(	(	PUNCT
ma-381	198	3	p	p	X
ma-381	198	4	,	,	PUNCT
ma-381	198	5	q	q	X
ma-381	198	6	,	,	PUNCT
ma-381	198	7	α)-restricted	α)-restricte	VERB
ma-381	198	8	at	at	ADP
ma-381	198	9	infinity	infinity	NOUN
ma-381	198	10	if	if	SCONJ
ma-381	198	11	there	there	PRON
ma-381	198	12	exists	exist	VERB
ma-381	198	13	µ0	µ0	NOUN
ma-381	198	14	≥	≥	NUM
ma-381	198	15	1	1	NUM
ma-381	198	16	such	such	ADJ
ma-381	198	17	that	that	PRON
ma-381	198	18	for	for	ADP
ma-381	198	19	µ	µ	PRON
ma-381	198	20	≥	≥	NUM
ma-381	198	21	µ0	µ0	NOUN
ma-381	198	22	,	,	PUNCT
ma-381	198	23	we	we	PRON
ma-381	198	24	have	have	VERB
ma-381	198	25	f	f	PROPN
ma-381	198	26	∗	∗	PROPN
ma-381	198	27	φ	φ	PROPN
ma-381	198	28	∈	∈	PROPN
ma-381	198	29	(	(	PUNCT
ma-381	198	30	lpµ	lpµ	X
ma-381	198	31	,	,	PUNCT
ma-381	198	32	`	`	PUNCT
ma-381	198	33	qµ)αµ(rd	qµ)αµ(rd	X
ma-381	198	34	)	)	PUNCT
ma-381	198	35	,	,	PUNCT
ma-381	198	36	φ	φ	PROPN
ma-381	198	37	∈	∈	PROPN
ma-381	198	38	s(rd	s(rd	NUM
ma-381	198	39	)	)	PUNCT
ma-381	198	40	.	.	PUNCT
ma-381	199	1	it	it	PRON
ma-381	199	2	is	be	AUX
ma-381	199	3	easy	easy	ADJ
ma-381	199	4	to	to	PART
ma-381	199	5	see	see	VERB
ma-381	199	6	that	that	DET
ma-381	199	7	tempered	temper	VERB
ma-381	199	8	distributions	distribution	NOUN
ma-381	199	9	which	which	PRON
ma-381	199	10	are	be	AUX
ma-381	199	11	(	(	PUNCT
ma-381	199	12	p	p	X
ma-381	199	13	,	,	PUNCT
ma-381	199	14	q	q	X
ma-381	199	15	,	,	PUNCT
ma-381	199	16	α)-restricted	α)-restricte	VERB
ma-381	199	17	for	for	ADP
ma-381	199	18	p	p	NOUN
ma-381	199	19	≤	≤	NUM
ma-381	199	20	α	α	NOUN
ma-381	199	21	≤	≤	NOUN
ma-381	200	1	q	q	X
ma-381	200	2	,	,	PUNCT
ma-381	200	3	are	be	AUX
ma-381	200	4	also	also	ADV
ma-381	200	5	(	(	PUNCT
ma-381	200	6	p	p	X
ma-381	200	7	,	,	PUNCT
ma-381	200	8	q)-restricted	q)-restricted	ADJ
ma-381	200	9	.	.	PUNCT
ma-381	201	1	theorem	theorem	VERB
ma-381	201	2	1.1	1.1	NUM
ma-381	201	3	in	in	ADP
ma-381	201	4	[	[	X
ma-381	201	5	3	3	NUM
ma-381	201	6	]	]	PUNCT
ma-381	201	7	assert	assert	VERB
ma-381	201	8	that	that	SCONJ
ma-381	201	9	a	a	DET
ma-381	201	10	tempered	temper	VERB
ma-381	201	11	distribution	distribution	NOUN
ma-381	201	12	f	f	PROPN
ma-381	201	13	belongs	belong	VERB
ma-381	201	14	to	to	ADP
ma-381	201	15	h(p	h(p	NOUN
ma-381	201	16	,	,	PUNCT
ma-381	201	17	q)(rd)for	q)(rd)for	ADP
ma-381	201	18	d−1	d−1	PROPN
ma-381	201	19	d	d	PROPN
ma-381	201	20	<	<	X
ma-381	201	21	min	min	PROPN
ma-381	201	22	(	(	PUNCT
ma-381	201	23	p	p	X
ma-381	201	24	,	,	PUNCT
ma-381	201	25	q	q	NOUN
ma-381	201	26	)	)	PUNCT
ma-381	201	27	<	<	X
ma-381	201	28	∞	∞	PROPN
ma-381	201	29	,	,	PUNCT
ma-381	201	30	if	if	SCONJ
ma-381	201	31	and	and	CCONJ
ma-381	201	32	only	only	ADV
ma-381	201	33	if	if	SCONJ
ma-381	201	34	it	it	PRON
ma-381	201	35	is	be	AUX
ma-381	201	36	(	(	PUNCT
ma-381	201	37	p	p	X
ma-381	201	38	,	,	PUNCT
ma-381	201	39	q)-restricted	q)-restricte	VERB
ma-381	201	40	at	at	ADP
ma-381	201	41	infty	infty	NOUN
ma-381	201	42	and	and	CCONJ
ma-381	201	43	,	,	PUNCT
ma-381	201	44	for	for	ADP
ma-381	201	45	φ	φ	PROPN
ma-381	201	46	∈	∈	PROPN
ma-381	201	47	s(rd	s(rd	PROPN
ma-381	201	48	)	)	PUNCT
ma-381	201	49	with	with	ADP
ma-381	201	50	nonvanish	nonvanish	PROPN
ma-381	201	51	integral	integral	ADJ
ma-381	201	52	,	,	PUNCT
ma-381	201	53	sup	sup	NUM
ma-381	201	54	t>0	t>0	NOUN
ma-381	202	1	‖f	‖f	VERB
ma-381	202	2	∗	∗	NOUN
ma-381	202	3	φt‖p	φt‖p	NOUN
ma-381	202	4	,	,	PUNCT
ma-381	202	5	q	q	NOUN
ma-381	203	1	+	+	NUM
ma-381	203	2	d∑	d∑	PROPN
ma-381	203	3	j=1	j=1	ADJ
ma-381	203	4	‖(rj	‖(rj	NOUN
ma-381	203	5	f	f	PROPN
ma-381	203	6	)	)	PUNCT
ma-381	203	7	∗	∗	NOUN
ma-381	203	8	φt‖p	φt‖p	NOUN
ma-381	203	9	,	,	PUNCT
ma-381	203	10	q	q	PUNCT
ma-381	203	11			PROPN
ma-381	203	12	<	<	X
ma-381	203	13	∞.	∞.	PROPN
ma-381	203	14	when	when	SCONJ
ma-381	203	15	this	this	PRON
ma-381	203	16	is	be	AUX
ma-381	203	17	the	the	DET
ma-381	203	18	case	case	NOUN
ma-381	203	19	,	,	PUNCT
ma-381	203	20	‖f	‖f	ADP
ma-381	203	21	‖h(p	‖h(p	NOUN
ma-381	203	22	,	,	PUNCT
ma-381	203	23	q	q	X
ma-381	203	24	)	)	PUNCT
ma-381	203	25	≈	≈	NOUN
ma-381	203	26	sup	sup	PROPN
ma-381	203	27	t>0	t>0	NOUN
ma-381	204	1	‖f	‖f	VERB
ma-381	204	2	∗	∗	NOUN
ma-381	204	3	φt‖p	φt‖p	NOUN
ma-381	204	4	,	,	PUNCT
ma-381	204	5	q	q	NOUN
ma-381	205	1	+	+	NUM
ma-381	205	2	d∑	d∑	PROPN
ma-381	205	3	j=1	j=1	ADJ
ma-381	205	4	‖(rj	‖(rj	NOUN
ma-381	205	5	f	f	PROPN
ma-381	205	6	)	)	PUNCT
ma-381	205	7	∗	∗	NOUN
ma-381	205	8	φt‖p	φt‖p	NOUN
ma-381	205	9	,	,	PUNCT
ma-381	205	10	q	q	NOUN
ma-381	205	11			PROPN
ma-381	205	12	.	.	PUNCT
ma-381	206	1	in	in	ADP
ma-381	206	2	the	the	DET
ma-381	206	3	case	case	NOUN
ma-381	206	4	of	of	ADP
ma-381	206	5	hardy	hardy	ADJ
ma-381	206	6	-	-	PUNCT
ma-381	206	7	fofana	fofana	NOUN
ma-381	206	8	space	space	NOUN
ma-381	206	9	,	,	PUNCT
ma-381	206	10	we	we	PRON
ma-381	206	11	have	have	VERB
ma-381	206	12	the	the	DET
ma-381	206	13	following	follow	VERB
ma-381	206	14	result	result	NOUN
ma-381	206	15	.	.	PUNCT
ma-381	207	1	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	207	2	eur	eur	PROPN
ma-381	207	3	.	.	PUNCT
ma-381	208	1	j.	j.	PROPN
ma-381	208	2	math	math	PROPN
ma-381	208	3	.	.	PUNCT
ma-381	209	1	anal	anal	PROPN
ma-381	209	2	.	.	PUNCT
ma-381	210	1	10.28924	10.28924	NUM
ma-381	210	2	/	/	SYM
ma-381	210	3	ada	ada	PROPN
ma-381	210	4	/	/	SYM
ma-381	210	5	ma.5.21	ma.5.21	NOUN
ma-381	210	6	9	9	NUM
ma-381	210	7	theorem	theorem	NOUN
ma-381	210	8	3.5	3.5	NUM
ma-381	210	9	.	.	PUNCT
ma-381	211	1	let	let	VERB
ma-381	211	2	d−1	d−1	PROPN
ma-381	211	3	d	d	PROPN
ma-381	211	4	<	<	X
ma-381	211	5	p	p	X
ma-381	211	6	≤	≤	NUM
ma-381	211	7	α	α	NOUN
ma-381	211	8	≤	≤	NUM
ma-381	211	9	q	q	X
ma-381	211	10	<	<	X
ma-381	211	11	∞	∞	PROPN
ma-381	211	12	,	,	PUNCT
ma-381	211	13	f	f	PROPN
ma-381	211	14	∈	∈	PROPN
ma-381	211	15	s	s	VERB
ma-381	211	16	′(rd).then	′(rd).then	NOUN
ma-381	211	17	f	f	PROPN
ma-381	211	18	∈	∈	PROPN
ma-381	211	19	h(p	h(p	PROPN
ma-381	211	20	,	,	PUNCT
ma-381	211	21	q	q	NOUN
ma-381	211	22	,	,	PUNCT
ma-381	211	23	α)(rd	α)(rd	ADJ
ma-381	211	24	)	)	PUNCT
ma-381	211	25	if	if	SCONJ
ma-381	212	1	and	and	CCONJ
ma-381	212	2	only	only	ADV
ma-381	212	3	if	if	SCONJ
ma-381	212	4	f	f	PROPN
ma-381	212	5	is	be	AUX
ma-381	212	6	(	(	PUNCT
ma-381	212	7	p	p	X
ma-381	212	8	,	,	PUNCT
ma-381	212	9	q	q	X
ma-381	212	10	,	,	PUNCT
ma-381	212	11	α)-restricted	α)-restricte	VERB
ma-381	212	12	at	at	ADP
ma-381	212	13	infinity	infinity	NOUN
ma-381	212	14	and	and	CCONJ
ma-381	212	15	,	,	PUNCT
ma-381	212	16	for	for	ADP
ma-381	212	17	φ	φ	PROPN
ma-381	212	18	∈	∈	PROPN
ma-381	212	19	s(rd	s(rd	PROPN
ma-381	212	20	)	)	PUNCT
ma-381	212	21	with	with	ADP
ma-381	212	22	non	non	ADJ
ma-381	212	23	vanish	vanish	VERB
ma-381	212	24	integral	integral	ADJ
ma-381	212	25	,	,	PUNCT
ma-381	212	26	sup	sup	NUM
ma-381	212	27	t>0	t>0	NOUN
ma-381	212	28	‖f	‖f	VERB
ma-381	212	29	∗	∗	NOUN
ma-381	212	30	φt‖p	φt‖p	NOUN
ma-381	212	31	,	,	PUNCT
ma-381	212	32	q	q	NOUN
ma-381	212	33	,	,	PUNCT
ma-381	212	34	α	α	PROPN
ma-381	213	1	+	+	X
ma-381	213	2	d∑	d∑	PROPN
ma-381	213	3	j=1	j=1	NOUN
ma-381	213	4	∥∥(rj	∥∥(rj	PROPN
ma-381	213	5	f	f	PROPN
ma-381	213	6	)	)	PUNCT
ma-381	214	1	∗	∗	NOUN
ma-381	214	2	φt	φt	NOUN
ma-381	214	3	∥∥	∥∥	X
ma-381	214	4	p	p	NOUN
ma-381	214	5	,	,	PUNCT
ma-381	214	6	q	q	NOUN
ma-381	214	7	,	,	PUNCT
ma-381	214	8	α	α	PRON
ma-381	215	1			PROPN
ma-381	215	2	<	<	X
ma-381	216	1	+	+	PROPN
ma-381	216	2	∞.	∞.	PROPN
ma-381	216	3	(	(	PUNCT
ma-381	216	4	3.7	3.7	NUM
ma-381	216	5	)	)	PUNCT
ma-381	216	6	moreover	moreover	ADV
ma-381	216	7	,	,	PUNCT
ma-381	216	8	‖f	‖f	ADP
ma-381	216	9	‖h(p	‖h(p	SYM
ma-381	216	10	,	,	PUNCT
ma-381	216	11	q	q	NOUN
ma-381	216	12	,	,	PUNCT
ma-381	216	13	α	α	NOUN
ma-381	216	14	)	)	PUNCT
ma-381	216	15	≈	≈	NOUN
ma-381	216	16	sup	sup	PROPN
ma-381	216	17	t>0	t>0	NOUN
ma-381	216	18	‖f	‖f	VERB
ma-381	216	19	∗	∗	NOUN
ma-381	216	20	φt‖p	φt‖p	NOUN
ma-381	216	21	,	,	PUNCT
ma-381	216	22	q	q	NOUN
ma-381	216	23	,	,	PUNCT
ma-381	216	24	α	α	PROPN
ma-381	217	1	+	+	X
ma-381	217	2	d∑	d∑	PROPN
ma-381	217	3	j=1	j=1	NOUN
ma-381	217	4	∥∥(rj	∥∥(rj	PROPN
ma-381	217	5	f	f	PROPN
ma-381	217	6	)	)	PUNCT
ma-381	218	1	∗	∗	NOUN
ma-381	218	2	φt	φt	NOUN
ma-381	218	3	∥∥	∥∥	X
ma-381	218	4	p	p	NOUN
ma-381	218	5	,	,	PUNCT
ma-381	218	6	q	q	NOUN
ma-381	218	7	,	,	PUNCT
ma-381	218	8	α	α	NOUN
ma-381	218	9			PROPN
ma-381	218	10	.	.	PUNCT
ma-381	219	1	(	(	PUNCT
ma-381	219	2	3.8	3.8	NUM
ma-381	219	3	)	)	PUNCT
ma-381	219	4	proof	proof	NOUN
ma-381	219	5	.	.	PUNCT
ma-381	220	1	let	let	VERB
ma-381	220	2	d−1	d−1	PROPN
ma-381	220	3	d	d	PROPN
ma-381	220	4	<	<	X
ma-381	220	5	p	p	X
ma-381	220	6	≤	≤	NUM
ma-381	220	7	α	α	NOUN
ma-381	220	8	≤	≤	NUM
ma-381	220	9	q	q	X
ma-381	220	10	<	<	X
ma-381	220	11	∞	∞	NUM
ma-381	220	12	and	and	CCONJ
ma-381	220	13	f	f	PROPN
ma-381	220	14	∈	∈	PROPN
ma-381	220	15	s	s	PART
ma-381	220	16	′(rd).we	′(rd).we	NOUN
ma-381	220	17	suppose	suppose	VERB
ma-381	220	18	that	that	SCONJ
ma-381	220	19	f	f	PROPN
ma-381	220	20	is	be	AUX
ma-381	220	21	(	(	PUNCT
ma-381	220	22	p	p	X
ma-381	220	23	,	,	PUNCT
ma-381	220	24	q	q	X
ma-381	220	25	,	,	PUNCT
ma-381	220	26	α)-restricted	α)-restricte	VERB
ma-381	220	27	at	at	ADP
ma-381	220	28	infinity	infinity	NOUN
ma-381	220	29	and	and	CCONJ
ma-381	220	30	satisfies	satisfie	NOUN
ma-381	220	31	(	(	PUNCT
ma-381	220	32	3.7	3.7	NUM
ma-381	220	33	)	)	PUNCT
ma-381	220	34	for	for	ADP
ma-381	220	35	non	non	ADJ
ma-381	220	36	vanishing	vanish	VERB
ma-381	220	37	schwartzfunction	schwartzfunction	NOUN
ma-381	220	38	φ	φ	NOUN
ma-381	220	39	.	.	PUNCT
ma-381	221	1	there	there	PRON
ma-381	221	2	exists	exist	VERB
ma-381	221	3	µ0	µ0	NOUN
ma-381	221	4	>	>	X
ma-381	221	5	1	1	NUM
ma-381	221	6	(	(	PUNCT
ma-381	221	7	large	large	ADJ
ma-381	221	8	enought	enought	ADJ
ma-381	221	9	)	)	PUNCT
ma-381	221	10	such	such	ADJ
ma-381	221	11	that	that	PRON
ma-381	221	12	for	for	ADP
ma-381	221	13	µ	µ	X
ma-381	221	14	>	>	X
ma-381	221	15	µ0	µ0	NOUN
ma-381	221	16	,	,	PUNCT
ma-381	221	17	we	we	PRON
ma-381	221	18	have	have	VERB
ma-381	221	19	f	f	PROPN
ma-381	221	20	∗	∗	PROPN
ma-381	221	21	φ	φ	PROPN
ma-381	221	22	∈	∈	PROPN
ma-381	221	23	(	(	PUNCT
ma-381	221	24	lpµ	lpµ	ADJ
ma-381	221	25	;	;	PUNCT
ma-381	221	26	`	`	PUNCT
ma-381	221	27	qµ)αµ	qµ)αµ	X
ma-381	221	28	(	(	PUNCT
ma-381	221	29	rd	rd	NOUN
ma-381	221	30	)	)	PUNCT
ma-381	221	31	,	,	PUNCT
ma-381	221	32	φ	φ	PROPN
ma-381	221	33	∈	∈	PROPN
ma-381	221	34	s(rd	s(rd	NUM
ma-381	221	35	)	)	PUNCT
ma-381	221	36	.	.	PUNCT
ma-381	222	1	(	(	PUNCT
ma-381	222	2	3.9	3.9	NUM
ma-381	222	3	)	)	PUNCT
ma-381	222	4	it	it	PRON
ma-381	222	5	comes	come	VERB
ma-381	222	6	from	from	ADP
ma-381	222	7	the	the	DET
ma-381	222	8	definition	definition	NOUN
ma-381	222	9	of	of	ADP
ma-381	222	10	fofana	fofana	PROPN
ma-381	222	11	spaces	space	NOUN
ma-381	222	12	that	that	SCONJ
ma-381	222	13	stαµρ	stαµρ	NOUN
ma-381	222	14	(	(	PUNCT
ma-381	222	15	f	f	PROPN
ma-381	222	16	∗	∗	PROPN
ma-381	222	17	φ	φ	PROPN
ma-381	222	18	)	)	PUNCT
ma-381	222	19	∈	∈	PROPN
ma-381	222	20	(	(	PUNCT
ma-381	222	21	lpµ	lpµ	X
ma-381	222	22	,	,	PUNCT
ma-381	222	23	`	`	PUNCT
ma-381	222	24	µq	µq	PROPN
ma-381	222	25	)	)	PUNCT
ma-381	222	26	(	(	PUNCT
ma-381	222	27	rd	rd	NOUN
ma-381	222	28	)	)	PUNCT
ma-381	222	29	φ	φ	PROPN
ma-381	222	30	∈	∈	PROPN
ma-381	222	31	s(rd	s(rd	PROPN
ma-381	222	32	)	)	PUNCT
ma-381	222	33	,	,	PUNCT
ma-381	222	34	ρ	ρ	PROPN
ma-381	222	35	>	>	X
ma-381	222	36	0	0	X
ma-381	222	37	.	.	PUNCT
ma-381	223	1	taking	take	VERB
ma-381	223	2	ρ	ρ	PROPN
ma-381	223	3	=	=	SYM
ma-381	223	4	1	1	NUM
ma-381	223	5	,	,	PUNCT
ma-381	223	6	we	we	PRON
ma-381	223	7	obtain	obtain	VERB
ma-381	223	8	that	that	SCONJ
ma-381	223	9	f	f	PROPN
ma-381	223	10	is	be	AUX
ma-381	223	11	(	(	PUNCT
ma-381	223	12	p	p	X
ma-381	223	13	,	,	PUNCT
ma-381	223	14	q)-restricted	q)-restricte	VERB
ma-381	223	15	at	at	ADP
ma-381	223	16	infinity	infinity	NOUN
ma-381	223	17	.	.	PUNCT
ma-381	224	1	since	since	SCONJ
ma-381	224	2	for	for	ADP
ma-381	224	3	all	all	DET
ma-381	224	4	φ	φ	NUM
ma-381	224	5	∈	∈	PROPN
ma-381	224	6	s(rd	s(rd	NUM
ma-381	224	7	)	)	PUNCT
ma-381	224	8	with	with	ADP
ma-381	224	9	nonvanishing	nonvanishe	VERB
ma-381	224	10	integral	integral	ADJ
ma-381	224	11	we	we	PRON
ma-381	224	12	also	also	ADV
ma-381	224	13	have	have	VERB
ma-381	224	14	that	that	DET
ma-381	224	15	a	a	DET
ma-381	224	16	=	=	NOUN
ma-381	224	17	sup	sup	NOUN
ma-381	224	18	t>0	t>0	NOUN
ma-381	224	19	sup	sup	PROPN
ma-381	224	20	ρ	ρ	X
ma-381	224	21	∥∥stαρ	∥∥stαρ	PROPN
ma-381	224	22	(	(	PUNCT
ma-381	224	23	f	f	PROPN
ma-381	224	24	∗	∗	X
ma-381	224	25	φt	φt	PROPN
ma-381	224	26	)	)	PUNCT
ma-381	224	27	∥∥	∥∥	PROPN
ma-381	224	28	p	p	NOUN
ma-381	224	29	,	,	PUNCT
ma-381	224	30	q	q	X
ma-381	225	1	+	+	CCONJ
ma-381	225	2	d∑	d∑	ADJ
ma-381	225	3	j=1	j=1	NOUN
ma-381	225	4	sup	sup	NOUN
ma-381	225	5	ρ>0	ρ>0	VERB
ma-381	225	6	∥∥stαρ	∥∥stαρ	PROPN
ma-381	225	7	(	(	PUNCT
ma-381	225	8	(	(	PUNCT
ma-381	225	9	rj	rj	PROPN
ma-381	225	10	f	f	PROPN
ma-381	225	11	)	)	PUNCT
ma-381	225	12	∗	∗	NOUN
ma-381	225	13	φt	φt	NOUN
ma-381	225	14	)	)	PUNCT
ma-381	225	15	∥∥	∥∥	PROPN
ma-381	226	1	p	p	NOUN
ma-381	226	2	,	,	PUNCT
ma-381	226	3	q	q	PROPN
ma-381	226	4			PROPN
ma-381	226	5	<	<	X
ma-381	226	6	∞	∞	PROPN
ma-381	226	7	,	,	PUNCT
ma-381	226	8	it	it	PRON
ma-381	226	9	follows	follow	VERB
ma-381	226	10	that	that	SCONJ
ma-381	226	11	sup	sup	NOUN
ma-381	226	12	t>0	t>0	NOUN
ma-381	226	13	‖f	‖f	VERB
ma-381	226	14	∗	∗	NOUN
ma-381	226	15	φt‖p	φt‖p	NOUN
ma-381	226	16	,	,	PUNCT
ma-381	226	17	q	q	NOUN
ma-381	227	1	+	+	NUM
ma-381	227	2	d∑	d∑	PROPN
ma-381	227	3	j=1	j=1	NOUN
ma-381	227	4	∥∥(rj	∥∥(rj	PROPN
ma-381	227	5	f	f	PROPN
ma-381	227	6	)	)	PUNCT
ma-381	228	1	∗	∗	NOUN
ma-381	228	2	φt	φt	NOUN
ma-381	229	1	∥∥	∥∥	X
ma-381	229	2	p	p	NOUN
ma-381	229	3	,	,	PUNCT
ma-381	229	4	q	q	PROPN
ma-381	229	5			PROPN
ma-381	229	6	≤	≤	NOUN
ma-381	229	7	a.	a.	NOUN
ma-381	229	8	thus	thus	ADV
ma-381	229	9	f	f	PROPN
ma-381	229	10	∈	∈	PROPN
ma-381	229	11	h(p	h(p	PROPN
ma-381	229	12	,	,	PUNCT
ma-381	229	13	q)(rd	q)(rd	NUM
ma-381	229	14	)	)	PUNCT
ma-381	229	15	thanks	thank	NOUN
ma-381	229	16	to	to	ADP
ma-381	229	17	[	[	X
ma-381	229	18	3	3	NUM
ma-381	229	19	,	,	PUNCT
ma-381	229	20	theorem	theorem	VERB
ma-381	229	21	1.1	1.1	NUM
ma-381	229	22	]	]	PUNCT
ma-381	229	23	.	.	PUNCT
ma-381	230	1	it	it	PRON
ma-381	230	2	remains	remain	VERB
ma-381	230	3	to	to	PART
ma-381	230	4	prove	prove	VERB
ma-381	230	5	that	that	SCONJ
ma-381	230	6	the	the	DET
ma-381	230	7	familly	familly	ADV
ma-381	230	8	{	{	PUNCT
ma-381	230	9	stαρ	stαρ	PROPN
ma-381	230	10	f	f	AUX
ma-381	230	11	}	}	PUNCT
ma-381	230	12	ρ>0	ρ>0	VERB
ma-381	230	13	isuniformly	isuniformly	ADV
ma-381	230	14	bounded	bound	VERB
ma-381	230	15	in	in	ADP
ma-381	230	16	h(p	h(p	PROPN
ma-381	230	17	,	,	PUNCT
ma-381	230	18	q)(rd).fix	q)(rd).fix	PROPN
ma-381	230	19	ρ	ρ	NOUN
ma-381	230	20	>	>	X
ma-381	230	21	0	0	PROPN
ma-381	230	22	.	.	PUNCT
ma-381	231	1	we	we	PRON
ma-381	231	2	have	have	VERB
ma-381	231	3	stαρ	stαρ	PROPN
ma-381	231	4	f	f	PROPN
ma-381	231	5	∈	∈	PROPN
ma-381	231	6	h(p	h(p	PROPN
ma-381	231	7	,	,	PUNCT
ma-381	231	8	q)(rd	q)(rd	NUM
ma-381	231	9	)	)	PUNCT
ma-381	232	1	so	so	SCONJ
ma-381	232	2	that	that	SCONJ
ma-381	232	3	‖stαρ	‖stαρ	PRON
ma-381	232	4	f	f	NOUN
ma-381	232	5	‖h(p	‖h(p	PROPN
ma-381	232	6	,	,	PUNCT
ma-381	232	7	q	q	NOUN
ma-381	232	8	)	)	PUNCT
ma-381	232	9	≈	≈	NOUN
ma-381	232	10	sup	sup	NOUN
ma-381	232	11	t>0	t>0	NOUN
ma-381	232	12	∥∥stαρ	∥∥stαρ	PROPN
ma-381	232	13	(	(	PUNCT
ma-381	232	14	f	f	PROPN
ma-381	232	15	)	)	PUNCT
ma-381	232	16	∗	∗	NOUN
ma-381	232	17	φt	φt	NOUN
ma-381	233	1	∥∥	∥∥	X
ma-381	233	2	p	p	NOUN
ma-381	233	3	,	,	PUNCT
ma-381	233	4	q	q	X
ma-381	234	1	+	+	CCONJ
ma-381	234	2	d∑	d∑	PROPN
ma-381	234	3	j=1	j=1	NOUN
ma-381	234	4	∥∥rj(stαρ	∥∥rj(stαρ	PROPN
ma-381	234	5	f	f	PROPN
ma-381	234	6	)	)	PUNCT
ma-381	234	7	∗	∗	NOUN
ma-381	234	8	φt	φt	NOUN
ma-381	234	9	∥∥	∥∥	X
ma-381	234	10	p	p	NOUN
ma-381	234	11	,	,	PUNCT
ma-381	234	12	q	q	PROPN
ma-381	234	13			PROPN
ma-381	234	14	thanks	thank	NOUN
ma-381	234	15	once	once	ADV
ma-381	234	16	more	more	ADJ
ma-381	234	17	to	to	ADP
ma-381	234	18	[	[	X
ma-381	234	19	3	3	NUM
ma-381	234	20	,	,	PUNCT
ma-381	234	21	theorem	theorem	VERB
ma-381	234	22	1.1	1.1	NUM
ma-381	234	23	]	]	PUNCT
ma-381	234	24	.	.	PUNCT
ma-381	235	1	but	but	CCONJ
ma-381	235	2	we	we	PRON
ma-381	235	3	have	have	VERB
ma-381	235	4	in	in	ADP
ma-381	235	5	one	one	NUM
ma-381	235	6	hand	hand	NOUN
ma-381	235	7	that	that	PRON
ma-381	235	8	rj(f	rj(f	VERB
ma-381	235	9	)	)	PUNCT
ma-381	235	10	∗	∗	NOUN
ma-381	235	11	φt	φt	NOUN
ma-381	236	1	=	=	SYM
ma-381	236	2	rj	rj	PROPN
ma-381	236	3	(	(	PUNCT
ma-381	236	4	f	f	PROPN
ma-381	236	5	∗	∗	X
ma-381	236	6	φt	φt	PROPN
ma-381	236	7	)	)	PUNCT
ma-381	236	8	,	,	PUNCT
ma-381	236	9	so	so	SCONJ
ma-381	236	10	that	that	SCONJ
ma-381	236	11	stαρ	stαρ	NOUN
ma-381	237	1	[	[	X
ma-381	237	2	(	(	PUNCT
ma-381	237	3	rj	rj	PROPN
ma-381	237	4	f	f	PROPN
ma-381	237	5	)	)	PUNCT
ma-381	237	6	∗	∗	NOUN
ma-381	237	7	φt	φt	NOUN
ma-381	237	8	]	]	PUNCT
ma-381	237	9	=	=	PUNCT
ma-381	237	10	stαρ	stαρ	NOUN
ma-381	237	11	[	[	PUNCT
ma-381	237	12	rj	rj	PROPN
ma-381	237	13	(	(	PUNCT
ma-381	237	14	f	f	PROPN
ma-381	237	15	∗	∗	X
ma-381	237	16	φt	φt	PROPN
ma-381	237	17	)	)	PUNCT
ma-381	237	18	]	]	PUNCT
ma-381	238	1	=	=	SYM
ma-381	238	2	rj	rj	PROPN
ma-381	238	3	[	[	PUNCT
ma-381	238	4	stαρ	stαρ	NOUN
ma-381	238	5	(	(	PUNCT
ma-381	238	6	f	f	PROPN
ma-381	238	7	∗	∗	X
ma-381	238	8	φt	φt	PROPN
ma-381	238	9	)	)	PUNCT
ma-381	238	10	]	]	PUNCT
ma-381	238	11	,	,	PUNCT
ma-381	238	12	(	(	PUNCT
ma-381	238	13	3.10	3.10	NUM
ma-381	238	14	)	)	PUNCT
ma-381	238	15	where	where	SCONJ
ma-381	238	16	the	the	DET
ma-381	238	17	last	last	ADJ
ma-381	238	18	equality	equality	NOUN
ma-381	238	19	comes	come	VERB
ma-381	238	20	from	from	ADP
ma-381	238	21	the	the	DET
ma-381	238	22	fact	fact	NOUN
ma-381	238	23	that	that	SCONJ
ma-381	238	24	dilation	dilation	NOUN
ma-381	238	25	comute	comute	VERB
ma-381	238	26	with	with	ADP
ma-381	238	27	riesz	riesz	NOUN
ma-381	238	28	transforms	transform	VERB
ma-381	238	29	.	.	PUNCT
ma-381	239	1	in	in	ADP
ma-381	239	2	the	the	DET
ma-381	239	3	otherhand	otherhand	NOUN
ma-381	239	4	we	we	PRON
ma-381	239	5	have	have	VERB
ma-381	239	6	that	that	DET
ma-381	239	7	sup	sup	NOUN
ma-381	239	8	t>0	t>0	NOUN
ma-381	239	9	‖stαρ	‖stαρ	NUM
ma-381	239	10	(	(	PUNCT
ma-381	239	11	f	f	PROPN
ma-381	239	12	∗	∗	X
ma-381	239	13	φt	φt	NOUN
ma-381	239	14	)	)	PUNCT
ma-381	239	15	‖p	‖p	PROPN
ma-381	239	16	,	,	PUNCT
ma-381	239	17	q	q	NOUN
ma-381	239	18	=	=	NOUN
ma-381	239	19	sup	sup	NOUN
ma-381	239	20	t>0	t>0	NOUN
ma-381	239	21	‖stαρ	‖stαρ	NUM
ma-381	239	22	(	(	PUNCT
ma-381	239	23	f	f	NOUN
ma-381	239	24	)	)	PUNCT
ma-381	239	25	∗	∗	NOUN
ma-381	239	26	φt‖p	φt‖p	NOUN
ma-381	239	27	,	,	PUNCT
ma-381	239	28	q	q	NOUN
ma-381	239	29	,	,	PUNCT
ma-381	239	30	(	(	PUNCT
ma-381	239	31	3.11	3.11	NUM
ma-381	239	32	)	)	PUNCT
ma-381	239	33	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	239	34	eur	eur	PROPN
ma-381	239	35	.	.	PUNCT
ma-381	240	1	j.	j.	PROPN
ma-381	240	2	math	math	PROPN
ma-381	240	3	.	.	PUNCT
ma-381	241	1	anal	anal	PROPN
ma-381	241	2	.	.	PUNCT
ma-381	242	1	10.28924	10.28924	NUM
ma-381	242	2	/	/	SYM
ma-381	242	3	ada	ada	PROPN
ma-381	242	4	/	/	SYM
ma-381	242	5	ma.5.21	ma.5.21	NOUN
ma-381	242	6	10thanks	10thanks	NUM
ma-381	242	7	to	to	PART
ma-381	242	8	lemma	lemma	VERB
ma-381	242	9	2.3	2.3	NUM
ma-381	242	10	.	.	PUNCT
ma-381	243	1	therefore	therefore	ADV
ma-381	243	2	,	,	PUNCT
ma-381	243	3	we	we	PRON
ma-381	243	4	have	have	VERB
ma-381	243	5	sup	sup	NOUN
ma-381	243	6	ρ>0	ρ>0	NOUN
ma-381	243	7	sup	sup	NOUN
ma-381	243	8	t>0	t>0	NOUN
ma-381	244	1	‖stαρ	‖stαρ	NUM
ma-381	244	2	(	(	PUNCT
ma-381	244	3	f	f	NOUN
ma-381	244	4	)	)	PUNCT
ma-381	244	5	∗	∗	NOUN
ma-381	244	6	φt‖p	φt‖p	NOUN
ma-381	244	7	,	,	PUNCT
ma-381	244	8	q	q	NOUN
ma-381	244	9	=	=	NOUN
ma-381	244	10	sup	sup	NOUN
ma-381	244	11	ρ>0	ρ>0	NOUN
ma-381	244	12	sup	sup	NOUN
ma-381	244	13	t>0	t>0	NOUN
ma-381	244	14	‖stαρ	‖stαρ	NUM
ma-381	244	15	(	(	PUNCT
ma-381	244	16	f	f	PROPN
ma-381	244	17	∗	∗	X
ma-381	244	18	φt	φt	NOUN
ma-381	244	19	)	)	PUNCT
ma-381	244	20	‖p	‖p	PROPN
ma-381	244	21	,	,	PUNCT
ma-381	244	22	q	q	PROPN
ma-381	244	23	≤	≤	NOUN
ma-381	244	24	a	a	PRON
ma-381	244	25	and	and	CCONJ
ma-381	244	26	sup	sup	NOUN
ma-381	244	27	ρ>0	ρ>0	NOUN
ma-381	244	28	sup	sup	NOUN
ma-381	244	29	t>0	t>0	NOUN
ma-381	244	30	d∑	d∑	NOUN
ma-381	244	31	j=1	j=1	X
ma-381	244	32	‖rj	‖rj	PUNCT
ma-381	244	33	[	[	PUNCT
ma-381	244	34	stαρ	stαρ	NOUN
ma-381	244	35	(	(	PUNCT
ma-381	244	36	f	f	PROPN
ma-381	244	37	∗	∗	X
ma-381	244	38	φt	φt	PROPN
ma-381	244	39	)	)	PUNCT
ma-381	244	40	]	]	PUNCT
ma-381	245	1	‖p	‖p	PROPN
ma-381	245	2	,	,	PUNCT
ma-381	245	3	q	q	NOUN
ma-381	245	4	=	=	NOUN
ma-381	245	5	sup	sup	NOUN
ma-381	245	6	ρ>0	ρ>0	NOUN
ma-381	245	7	sup	sup	NOUN
ma-381	245	8	t>0	t>0	NOUN
ma-381	245	9	d∑	d∑	INTJ
ma-381	245	10	j=1	j=1	NOUN
ma-381	245	11	‖stαρ	‖stαρ	PUNCT
ma-381	246	1	[	[	X
ma-381	246	2	(	(	PUNCT
ma-381	246	3	rj	rj	PROPN
ma-381	246	4	f	f	PROPN
ma-381	246	5	)	)	PUNCT
ma-381	246	6	∗	∗	NOUN
ma-381	246	7	φt	φt	NOUN
ma-381	246	8	]	]	PUNCT
ma-381	246	9	‖p	‖p	PROPN
ma-381	246	10	,	,	PUNCT
ma-381	246	11	q	q	PROPN
ma-381	246	12	≤	≤	NUM
ma-381	246	13	a.	a.	NOUN
ma-381	246	14	we	we	PRON
ma-381	246	15	deduce	deduce	VERB
ma-381	246	16	that	that	SCONJ
ma-381	246	17	supρ>0	supρ>0	VERB
ma-381	247	1	‖stαρ	‖stαρ	X
ma-381	247	2	f	f	X
ma-381	247	3	‖h(p	‖h(p	PROPN
ma-381	247	4	,	,	PUNCT
ma-381	247	5	q	q	X
ma-381	247	6	)	)	PUNCT
ma-381	247	7	<	<	X
ma-381	247	8	∞	∞	PROPN
ma-381	247	9	,	,	PUNCT
ma-381	247	10	which	which	PRON
ma-381	247	11	prove	prove	VERB
ma-381	247	12	that	that	SCONJ
ma-381	247	13	f	f	PROPN
ma-381	247	14	∈	∈	PROPN
ma-381	247	15	h(p	h(p	PROPN
ma-381	247	16	,	,	PUNCT
ma-381	247	17	q	q	X
ma-381	247	18	,	,	PUNCT
ma-381	247	19	α)(rd).for	α)(rd).for	ADP
ma-381	247	20	the	the	DET
ma-381	247	21	converse	converse	NOUN
ma-381	247	22	,	,	PUNCT
ma-381	247	23	we	we	PRON
ma-381	247	24	suppose	suppose	VERB
ma-381	247	25	that	that	SCONJ
ma-381	247	26	f	f	PROPN
ma-381	247	27	∈	∈	PROPN
ma-381	247	28	h(p	h(p	PROPN
ma-381	247	29	,	,	PUNCT
ma-381	247	30	q	q	NOUN
ma-381	247	31	,	,	PUNCT
ma-381	247	32	α)(rd	α)(rd	NOUN
ma-381	247	33	)	)	PUNCT
ma-381	247	34	.	.	PUNCT
ma-381	248	1	it	it	PRON
ma-381	248	2	follows	follow	VERB
ma-381	248	3	that	that	SCONJ
ma-381	248	4	stαρ	stαρ	PROPN
ma-381	248	5	f	f	PROPN
ma-381	248	6	∈	∈	PROPN
ma-381	248	7	h(p	h(p	PROPN
ma-381	248	8	,	,	PUNCT
ma-381	248	9	q)(rd	q)(rd	NUM
ma-381	248	10	)	)	PUNCT
ma-381	248	11	with	with	ADP
ma-381	248	12	‖stαρ	‖stαρ	PROPN
ma-381	248	13	f	f	PROPN
ma-381	248	14	‖h(p	‖h(p	PROPN
ma-381	248	15	,	,	PUNCT
ma-381	248	16	q	q	NOUN
ma-381	248	17	)	)	PUNCT
ma-381	248	18	≤	≤	NOUN
ma-381	248	19	‖f	‖f	ADP
ma-381	248	20	‖h(p	‖h(p	NOUN
ma-381	248	21	,	,	PUNCT
ma-381	248	22	q	q	NOUN
ma-381	248	23	,	,	PUNCT
ma-381	248	24	α	α	NOUN
ma-381	248	25	)	)	PUNCT
ma-381	248	26	<	<	X
ma-381	248	27	∞	∞	PROPN
ma-381	248	28	for	for	ADP
ma-381	248	29	all	all	DET
ma-381	248	30	ρ	ρ	NOUN
ma-381	248	31	>	>	X
ma-381	248	32	0	0	NUM
ma-381	248	33	.	.	PUNCT
ma-381	249	1	it	it	PRON
ma-381	249	2	comes	come	VERB
ma-381	249	3	from	from	ADP
ma-381	249	4	[	[	X
ma-381	249	5	3	3	NUM
ma-381	249	6	,	,	PUNCT
ma-381	249	7	theorem	theorem	VERB
ma-381	249	8	1.1	1.1	NUM
ma-381	249	9	]	]	PUNCT
ma-381	249	10	that	that	SCONJ
ma-381	249	11	stαρ	stαρ	PROPN
ma-381	249	12	f	f	PROPN
ma-381	249	13	is	be	AUX
ma-381	249	14	(	(	PUNCT
ma-381	249	15	p	p	X
ma-381	249	16	,	,	PUNCT
ma-381	249	17	q)-resticted	q)-resticte	VERB
ma-381	249	18	at	at	ADP
ma-381	249	19	infinity	infinity	NOUN
ma-381	249	20	and	and	CCONJ
ma-381	249	21	‖stαρ	‖stαρ	PRON
ma-381	249	22	f	f	PROPN
ma-381	249	23	‖h(p	‖h(p	PROPN
ma-381	249	24	,	,	PUNCT
ma-381	249	25	q	q	NOUN
ma-381	249	26	)	)	PUNCT
ma-381	249	27	≈	≈	NOUN
ma-381	249	28	sup	sup	NOUN
ma-381	249	29	t>0	t>0	NOUN
ma-381	249	30	∥∥stαρ	∥∥stαρ	PROPN
ma-381	249	31	(	(	PUNCT
ma-381	249	32	f	f	PROPN
ma-381	249	33	)	)	PUNCT
ma-381	249	34	∗	∗	NOUN
ma-381	249	35	φt	φt	NOUN
ma-381	250	1	∥∥	∥∥	X
ma-381	250	2	p	p	NOUN
ma-381	250	3	,	,	PUNCT
ma-381	250	4	q	q	X
ma-381	251	1	+	+	CCONJ
ma-381	251	2	d∑	d∑	PROPN
ma-381	251	3	j=1	j=1	PROPN
ma-381	251	4	∥∥stαρ	∥∥stαρ	PROPN
ma-381	251	5	(	(	PUNCT
ma-381	251	6	rj	rj	PROPN
ma-381	251	7	f	f	PROPN
ma-381	251	8	)	)	PUNCT
ma-381	251	9	∗	∗	NOUN
ma-381	251	10	φt	φt	NOUN
ma-381	251	11	∥∥	∥∥	X
ma-381	251	12	p	p	NOUN
ma-381	251	13	,	,	PUNCT
ma-381	251	14	q	q	X
ma-381	251	15			PROPN
ma-381	251	16	for	for	ADP
ma-381	251	17	all	all	DET
ma-381	251	18	φ	φ	NUM
ma-381	251	19	∈	∈	PROPN
ma-381	251	20	s(rd	s(rd	NUM
ma-381	251	21	)	)	PUNCT
ma-381	251	22	with	with	ADP
ma-381	251	23	non	non	ADJ
ma-381	251	24	vanish	vanish	VERB
ma-381	251	25	integral.from	integral.from	ADP
ma-381	251	26	relations	relation	NOUN
ma-381	251	27	(	(	PUNCT
ma-381	251	28	3.11	3.11	NUM
ma-381	251	29	)	)	PUNCT
ma-381	251	30	and	and	CCONJ
ma-381	251	31	(	(	PUNCT
ma-381	251	32	3.10	3.10	NUM
ma-381	251	33	)	)	PUNCT
ma-381	251	34	,	,	PUNCT
ma-381	251	35	and	and	CCONJ
ma-381	251	36	the	the	DET
ma-381	251	37	definitions	definition	NOUN
ma-381	251	38	of	of	ADP
ma-381	251	39	‖	‖	PROPN
ma-381	251	40	·	·	PUNCT
ma-381	251	41	‖p	‖p	PROPN
ma-381	251	42	,	,	PUNCT
ma-381	251	43	q	q	NOUN
ma-381	251	44	,	,	PUNCT
ma-381	251	45	α	α	NOUN
ma-381	251	46	and	and	CCONJ
ma-381	251	47	of	of	ADP
ma-381	251	48	‖	‖	PROPN
ma-381	251	49	·	·	PUNCT
ma-381	251	50	‖h(p	‖h(p	NOUN
ma-381	251	51	,	,	PUNCT
ma-381	251	52	q	q	NOUN
ma-381	251	53	,	,	PUNCT
ma-381	251	54	α	α	NOUN
ma-381	251	55	)	)	PUNCT
ma-381	251	56	,	,	PUNCT
ma-381	251	57	we	we	PRON
ma-381	251	58	have	have	VERB
ma-381	251	59	that	that	PRON
ma-381	251	60	‖f	‖f	ADP
ma-381	251	61	‖h(p	‖h(p	NOUN
ma-381	251	62	,	,	PUNCT
ma-381	251	63	q	q	NOUN
ma-381	251	64	,	,	PUNCT
ma-381	251	65	α	α	NOUN
ma-381	251	66	)	)	PUNCT
ma-381	251	67	≈	≈	PROPN
ma-381	251	68	sup	sup	NOUN
ma-381	251	69	t>0	t>0	NOUN
ma-381	251	70	(	(	PUNCT
ma-381	251	71	‖f	‖f	ADP
ma-381	251	72	∗	∗	NOUN
ma-381	251	73	φt‖p	φt‖p	NOUN
ma-381	251	74	,	,	PUNCT
ma-381	251	75	q	q	NOUN
ma-381	251	76	,	,	PUNCT
ma-381	251	77	α	α	PROPN
ma-381	251	78	+	+	X
ma-381	252	1	d∑	d∑	PROPN
ma-381	252	2	j=1	j=1	NOUN
ma-381	252	3	∥∥(rj	∥∥(rj	PROPN
ma-381	252	4	f	f	PROPN
ma-381	252	5	)	)	PUNCT
ma-381	253	1	∗	∗	NOUN
ma-381	253	2	φt	φt	NOUN
ma-381	253	3	∥∥	∥∥	X
ma-381	253	4	p	p	NOUN
ma-381	253	5	,	,	PUNCT
ma-381	253	6	q	q	NOUN
ma-381	253	7	,	,	PUNCT
ma-381	253	8	α	α	NOUN
ma-381	253	9	)	)	PUNCT
ma-381	254	1	<	<	X
ma-381	254	2	∞.	∞.	PROPN
ma-381	254	3	let	let	VERB
ma-381	254	4	φ	φ	NUM
ma-381	254	5	∈	∈	PROPN
ma-381	254	6	s(rd	s(rd	PROPN
ma-381	254	7	)	)	PUNCT
ma-381	254	8	.	.	PUNCT
ma-381	255	1	we	we	PRON
ma-381	255	2	have	have	VERB
ma-381	255	3	‖f	‖f	ADP
ma-381	255	4	∗	∗	NOUN
ma-381	255	5	φ‖p	φ‖p	NOUN
ma-381	255	6	,	,	PUNCT
ma-381	255	7	q	q	NOUN
ma-381	255	8	,	,	PUNCT
ma-381	255	9	α	α	NOUN
ma-381	255	10	≤	≤	NOUN
ma-381	255	11	c	c	NOUN
ma-381	255	12	‖f	‖f	ADJ
ma-381	255	13	‖h(p	‖h(p	SYM
ma-381	255	14	,	,	PUNCT
ma-381	255	15	q	q	NOUN
ma-381	255	16	,	,	PUNCT
ma-381	255	17	α	α	NOUN
ma-381	255	18	)	)	PUNCT
ma-381	255	19	.	.	PUNCT
ma-381	256	1	for	for	ADP
ma-381	256	2	µ	µ	PRON
ma-381	256	3	≥	≥	NUM
ma-381	256	4	1	1	NUM
ma-381	256	5	we	we	PRON
ma-381	256	6	have	have	VERB
ma-381	256	7	f	f	PROPN
ma-381	256	8	∗	∗	PROPN
ma-381	256	9	φ	φ	PROPN
ma-381	256	10	∈	∈	PROPN
ma-381	256	11	(	(	PUNCT
ma-381	256	12	lpµ	lpµ	ADJ
ma-381	256	13	,	,	PUNCT
ma-381	256	14	`	`	PUNCT
ma-381	256	15	qµ)αµ.in	qµ)αµ.in	AUX
ma-381	256	16	fact	fact	NOUN
ma-381	256	17	assuming	assume	VERB
ma-381	256	18	that	that	SCONJ
ma-381	256	19	‖f	‖f	ADP
ma-381	256	20	∗	∗	NOUN
ma-381	256	21	ϕ‖∞	ϕ‖∞	NUM
ma-381	256	22	6=	6=	SYM
ma-381	256	23	0	0	PUNCT
ma-381	256	24	we	we	PRON
ma-381	256	25	have	have	VERB
ma-381	256	26	f	f	PROPN
ma-381	256	27	∗	∗	PROPN
ma-381	256	28	φ	φ	PROPN
ma-381	256	29	∈	∈	PROPN
ma-381	256	30	(	(	PUNCT
ma-381	256	31	lp	lp	NOUN
ma-381	256	32	,	,	PUNCT
ma-381	256	33	`	`	PUNCT
ma-381	256	34	q)α	q)α	PUNCT
ma-381	256	35	and	and	CCONJ
ma-381	256	36	‖f	‖f	ADP
ma-381	256	37	∗	∗	NOUN
ma-381	256	38	φ‖pµ,qµ,αµ	φ‖pµ,qµ,αµ	VERB
ma-381	256	39	≤	≤	ADJ
ma-381	256	40	c	c	NOUN
ma-381	256	41	‖f	‖f	PRON
ma-381	256	42	∗	∗	NOUN
ma-381	256	43	φ‖1−	φ‖1−	NOUN
ma-381	256	44	1	1	NUM
ma-381	256	45	µ	µ	NOUN
ma-381	256	46	∞	∞	NUM
ma-381	256	47	‖f	‖f	ADP
ma-381	256	48	∗	∗	NOUN
ma-381	256	49	φ‖	φ‖	X
ma-381	256	50	1	1	NUM
ma-381	256	51	µ	µ	PROPN
ma-381	256	52	p	p	X
ma-381	256	53	,	,	PUNCT
ma-381	256	54	q	q	NOUN
ma-381	256	55	,	,	PUNCT
ma-381	256	56	α	α	NOUN
ma-381	256	57	and	and	CCONJ
ma-381	256	58	then	then	ADV
ma-381	256	59	f	f	PROPN
ma-381	256	60	is	be	AUX
ma-381	256	61	(	(	PUNCT
ma-381	256	62	q	q	INTJ
ma-381	256	63	,	,	PUNCT
ma-381	256	64	p	p	X
ma-381	256	65	,	,	PUNCT
ma-381	256	66	α)-restricted	α)-restricte	VERB
ma-381	256	67	at	at	ADP
ma-381	256	68	infinity	infinity	NOUN
ma-381	256	69	.	.	PUNCT
ma-381	257	1	�	�	PROPN
ma-381	257	2	4	4	NUM
ma-381	257	3	.	.	PUNCT
ma-381	257	4	temperature	temperature	NOUN
ma-381	257	5	cauchy	cauchy	PROPN
ma-381	257	6	-	-	PUNCT
ma-381	257	7	riemann	riemann	PROPN
ma-381	257	8	equations	equation	NOUN
ma-381	257	9	and	and	CCONJ
ma-381	257	10	hardy	hardy	ADJ
ma-381	257	11	-	-	PUNCT
ma-381	257	12	fofana	fofana	NOUN
ma-381	257	13	spaces	space	VERB
ma-381	257	14	a	a	DET
ma-381	257	15	vector	vector	NOUN
ma-381	257	16	f	f	NOUN
ma-381	257	17	=	=	PUNCT
ma-381	257	18	(	(	PUNCT
ma-381	257	19	u1	u1	PROPN
ma-381	257	20	,	,	PUNCT
ma-381	257	21	u2	u2	PROPN
ma-381	257	22	,	,	PUNCT
ma-381	257	23	·	·	PUNCT
ma-381	257	24	·	·	PUNCT
ma-381	257	25	·	·	PUNCT
ma-381	257	26	,	,	PUNCT
ma-381	257	27	ud+1	ud+1	NOUN
ma-381	257	28	)	)	PUNCT
ma-381	257	29	of	of	ADP
ma-381	257	30	functions	function	NOUN
ma-381	257	31	in	in	ADP
ma-381	257	32	rd+1	rd+1	ADV
ma-381	257	33	+	+	CCONJ
ma-381	257	34	satisfy	satisfy	VERB
ma-381	257	35	the	the	DET
ma-381	257	36	generalized	generalized	ADJ
ma-381	257	37	temperature	temperature	NOUN
ma-381	257	38	cauchy	cauchy	PROPN
ma-381	257	39	-	-	PUNCT
ma-381	257	40	riemann	riemann	PROPN
ma-381	257	41	equations	equation	NOUN
ma-381	257	42	,	,	PUNCT
ma-381	257	43	if	if	SCONJ
ma-381	257	44	it	it	PRON
ma-381	257	45	satisfies	satisfy	VERB
ma-381	257	46	the	the	DET
ma-381	257	47	following	follow	VERB
ma-381	257	48	conditions	condition	NOUN
ma-381	257	49	:(	:(	NOUN
ma-381	257	50	1	1	X
ma-381	257	51	)	)	PUNCT
ma-381	257	52	∑d	∑d	PROPN
ma-381	258	1	j=1	j=1	PROPN
ma-381	258	2	∂uj	∂uj	PROPN
ma-381	258	3	∂xj	∂xj	PROPN
ma-381	258	4	=	=	PUNCT
ma-381	258	5	i∂	i∂	VERB
ma-381	258	6	1/2	1/2	NUM
ma-381	258	7	t	t	NOUN
ma-381	258	8	ud+1(2	ud+1(2	NOUN
ma-381	258	9	)	)	PUNCT
ma-381	258	10	∂uj	∂uj	PROPN
ma-381	258	11	∂xk	∂xk	PROPN
ma-381	258	12	=	=	SYM
ma-381	258	13	∂uk	∂uk	PROPN
ma-381	258	14	∂xj	∂xj	NOUN
ma-381	258	15	for	for	ADP
ma-381	258	16	j	j	PROPN
ma-381	258	17	,	,	PUNCT
ma-381	258	18	k	k	PROPN
ma-381	258	19	=	=	SYM
ma-381	258	20	1	1	NUM
ma-381	258	21	,	,	PUNCT
ma-381	258	22	2	2	NUM
ma-381	258	23	,	,	PUNCT
ma-381	258	24	·	·	PUNCT
ma-381	258	25	·	·	PUNCT
ma-381	258	26	·	·	PUNCT
ma-381	258	27	,	,	PUNCT
ma-381	258	28	d(3	d(3	PROPN
ma-381	258	29	)	)	PUNCT
ma-381	258	30	∂ud+1	∂ud+1	VERB
ma-381	259	1	∂xj	∂xj	NOUN
ma-381	259	2	=	=	SYM
ma-381	259	3	−i∂1/2	−i∂1/2	PROPN
ma-381	259	4	t	t	PROPN
ma-381	259	5	uj	uj	PROPN
ma-381	259	6	,	,	PUNCT
ma-381	259	7	j	j	PROPN
ma-381	259	8	=	=	SYM
ma-381	259	9	1	1	NUM
ma-381	259	10	,	,	PUNCT
ma-381	259	11	2	2	NUM
ma-381	259	12	,	,	PUNCT
ma-381	259	13	·	·	PUNCT
ma-381	259	14	·	·	PUNCT
ma-381	259	15	·	·	PUNCT
ma-381	259	16	,	,	PUNCT
ma-381	259	17	d	d	X
ma-381	259	18	,	,	PUNCT
ma-381	259	19	with	with	ADP
ma-381	259	20	(	(	PUNCT
ma-381	259	21	∂	∂	NUM
ma-381	259	22	1/2	1/2	NUM
ma-381	259	23	t	t	NOUN
ma-381	259	24	g)(t	g)(t	PUNCT
ma-381	259	25	)	)	PUNCT
ma-381	259	26	:	:	PUNCT
ma-381	260	1	=	=	PUNCT
ma-381	260	2	e	e	NOUN
ma-381	260	3	iπ/2	iπ/2	NOUN
ma-381	260	4	√	√	PROPN
ma-381	260	5	π	π	PROPN
ma-381	260	6	∫	∫	PROPN
ma-381	260	7	∞	∞	PROPN
ma-381	260	8	t	t	PROPN
ma-381	260	9	g′(s)√	g′(s)√	PROPN
ma-381	260	10	s	s	PROPN
ma-381	260	11	−	−	PROPN
ma-381	260	12	t	t	NOUN
ma-381	260	13	ds	ds	PROPN
ma-381	260	14	,	,	PUNCT
ma-381	260	15	t	t	X
ma-381	260	16	>	>	X
ma-381	260	17	0	0	PUNCT
ma-381	261	1	when	when	SCONJ
ma-381	261	2	g	g	PROPN
ma-381	261	3	is	be	AUX
ma-381	261	4	a	a	DET
ma-381	261	5	smooth	smooth	ADJ
ma-381	261	6	enough	enough	ADJ
ma-381	261	7	function	function	NOUN
ma-381	261	8	on	on	ADP
ma-381	261	9	(	(	PUNCT
ma-381	261	10	0,∞)in	0,∞)in	INTJ
ma-381	262	1	[	[	X
ma-381	262	2	3	3	NUM
ma-381	262	3	]	]	PUNCT
ma-381	262	4	,	,	PUNCT
ma-381	262	5	the	the	DET
ma-381	262	6	authors	author	NOUN
ma-381	262	7	defined	define	VERB
ma-381	262	8	the	the	DET
ma-381	262	9	space	space	NOUN
ma-381	262	10	hp	hp	NOUN
ma-381	262	11	,	,	PUNCT
ma-381	262	12	q(rd+1	q(rd+1	PROPN
ma-381	262	13	+	+	CCONJ
ma-381	262	14	)	)	PUNCT
ma-381	262	15	(	(	PUNCT
ma-381	262	16	0	0	PUNCT
ma-381	262	17	<	<	X
ma-381	262	18	p	p	X
ma-381	262	19	,	,	PUNCT
ma-381	262	20	q	q	X
ma-381	262	21	<	<	X
ma-381	262	22	∞	∞	NUM
ma-381	262	23	)	)	PUNCT
ma-381	262	24	as	as	ADP
ma-381	262	25	the	the	DET
ma-381	262	26	vector	vector	NOUN
ma-381	262	27	space	space	NOUN
ma-381	262	28	of	of	ADP
ma-381	262	29	vectorfunctions	vectorfunction	NOUN
ma-381	262	30	f	f	X
ma-381	263	1	=	=	SYM
ma-381	263	2	(	(	PUNCT
ma-381	263	3	u1	u1	PROPN
ma-381	263	4	,	,	PUNCT
ma-381	263	5	u2	u2	PROPN
ma-381	263	6	,	,	PUNCT
ma-381	263	7	·	·	PUNCT
ma-381	263	8	·	·	PUNCT
ma-381	263	9	·	·	PUNCT
ma-381	263	10	,	,	PUNCT
ma-381	263	11	ud+1	ud+1	X
ma-381	263	12	)	)	PUNCT
ma-381	263	13	satisfying	satisfy	VERB
ma-381	263	14	generalized	generalized	ADJ
ma-381	263	15	temperature	temperature	NOUN
ma-381	263	16	cauchy	cauchy	PROPN
ma-381	263	17	-	-	PUNCT
ma-381	263	18	riemann	riemann	PROPN
ma-381	263	19	equationsand	equationsand	NOUN
ma-381	263	20	such	such	ADJ
ma-381	263	21	that	that	DET
ma-381	263	22	‖f‖hp	‖f‖hp	NOUN
ma-381	263	23	,	,	PUNCT
ma-381	263	24	q(rd+1	q(rd+1	PROPN
ma-381	263	25	+	+	NUM
ma-381	263	26	)	)	PUNCT
ma-381	263	27	:	:	PUNCT
ma-381	264	1	=	=	PUNCT
ma-381	264	2	sup	sup	NOUN
ma-381	264	3	t>0	t>0	NOUN
ma-381	264	4	‖|f	‖|f	NOUN
ma-381	264	5	(	(	PUNCT
ma-381	264	6	·	·	PUNCT
ma-381	264	7	,	,	PUNCT
ma-381	264	8	t)|‖p	t)|‖p	PROPN
ma-381	264	9	,	,	PUNCT
ma-381	264	10	q	q	X
ma-381	264	11	<	<	X
ma-381	264	12	∞.	∞.	PROPN
ma-381	264	13	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	264	14	eur	eur	PROPN
ma-381	264	15	.	.	PUNCT
ma-381	265	1	j.	j.	PROPN
ma-381	265	2	math	math	PROPN
ma-381	265	3	.	.	PUNCT
ma-381	266	1	anal	anal	PROPN
ma-381	266	2	.	.	PUNCT
ma-381	267	1	10.28924	10.28924	NUM
ma-381	267	2	/	/	SYM
ma-381	267	3	ada	ada	PROPN
ma-381	267	4	/	/	SYM
ma-381	267	5	ma.5.21	ma.5.21	NOUN
ma-381	267	6	11	11	NUM
ma-381	267	7	they	they	PRON
ma-381	267	8	also	also	ADV
ma-381	267	9	proved	prove	VERB
ma-381	267	10	that	that	SCONJ
ma-381	267	11	under	under	ADP
ma-381	267	12	appropriate	appropriate	ADJ
ma-381	267	13	conditions	condition	NOUN
ma-381	267	14	on	on	ADP
ma-381	267	15	the	the	DET
ma-381	267	16	exponents	exponent	NOUN
ma-381	267	17	p	p	NOUN
ma-381	267	18	and	and	CCONJ
ma-381	267	19	q	q	NOUN
ma-381	267	20	,	,	PUNCT
ma-381	267	21	the	the	DET
ma-381	267	22	space	space	NOUN
ma-381	267	23	hp	hp	PROPN
ma-381	267	24	,	,	PUNCT
ma-381	267	25	q(rd+1	q(rd+1	PROPN
ma-381	267	26	+	+	CCONJ
ma-381	267	27	)	)	PUNCT
ma-381	267	28	is	be	AUX
ma-381	267	29	topologically	topologically	ADV
ma-381	267	30	isomorphic	isomorphic	ADJ
ma-381	267	31	to	to	ADP
ma-381	267	32	hp	hp	PROPN
ma-381	267	33	,	,	PUNCT
ma-381	267	34	q(rd	q(rd	NOUN
ma-381	267	35	)	)	PUNCT
ma-381	267	36	.	.	PUNCT
ma-381	268	1	to	to	PART
ma-381	268	2	carry	carry	VERB
ma-381	268	3	out	out	ADP
ma-381	268	4	the	the	DET
ma-381	268	5	proof	proof	NOUN
ma-381	268	6	of	of	ADP
ma-381	268	7	this	this	DET
ma-381	268	8	result	result	NOUN
ma-381	268	9	,	,	PUNCT
ma-381	268	10	they	they	PRON
ma-381	268	11	use	use	VERB
ma-381	268	12	a	a	DET
ma-381	268	13	subspaceof	subspaceof	NOUN
ma-381	268	14	what	what	PRON
ma-381	268	15	they	they	PRON
ma-381	268	16	call	call	VERB
ma-381	268	17	the	the	DET
ma-381	268	18	temperature	temperature	NOUN
ma-381	268	19	space	space	NOUN
ma-381	268	20	t	t	NOUN
ma-381	268	21	(	(	PUNCT
ma-381	268	22	rd+1	rd+1	NOUN
ma-381	269	1	+	+	PUNCT
ma-381	269	2	)	)	PUNCT
ma-381	269	3	;	;	PUNCT
ma-381	269	4	that	that	PRON
ma-381	269	5	is	be	AUX
ma-381	269	6	the	the	DET
ma-381	269	7	space	space	NOUN
ma-381	269	8	of	of	ADP
ma-381	269	9	functions	function	NOUN
ma-381	269	10	u	u	PROPN
ma-381	269	11	∈	∈	PROPN
ma-381	269	12	c2(rd+1	c2(rd+1	NOUN
ma-381	269	13	+	+	X
ma-381	269	14	)	)	PUNCT
ma-381	269	15	,	,	PUNCT
ma-381	269	16	satisfying	satisfy	VERB
ma-381	269	17	∂u	∂u	PROPN
ma-381	269	18	∂t	∂t	PROPN
ma-381	269	19	=	=	SYM
ma-381	269	20	d∑	d∑	PROPN
ma-381	269	21	j=1	j=1	PROPN
ma-381	269	22	∂2u	∂2u	PROPN
ma-381	269	23	∂x2	∂x2	PROPN
ma-381	269	24	j	j	PROPN
ma-381	269	25	in	in	ADP
ma-381	269	26	rd+1	rd+1	NOUN
ma-381	269	27	+	+	CCONJ
ma-381	269	28	.	.	PUNCT
ma-381	270	1	more	more	ADV
ma-381	270	2	precisely	precisely	ADV
ma-381	270	3	,	,	PUNCT
ma-381	270	4	for	for	ADP
ma-381	270	5	0	0	NUM
ma-381	270	6	<	<	X
ma-381	270	7	p	p	X
ma-381	270	8	,	,	PUNCT
ma-381	270	9	q	q	X
ma-381	270	10	<	<	X
ma-381	270	11	∞	∞	PROPN
ma-381	270	12	,	,	PUNCT
ma-381	270	13	they	they	PRON
ma-381	270	14	put	put	VERB
ma-381	270	15	t	t	PROPN
ma-381	270	16	p	p	X
ma-381	270	17	,	,	PUNCT
ma-381	270	18	q(rd+1	q(rd+1	PROPN
ma-381	270	19	+	+	NUM
ma-381	270	20	)	)	PUNCT
ma-381	270	21	:	:	PUNCT
ma-381	271	1	=	=	SYM
ma-381	271	2	{	{	PUNCT
ma-381	271	3	u	u	NOUN
ma-381	271	4	∈	∈	PROPN
ma-381	271	5	t	t	PROPN
ma-381	271	6	(	(	PUNCT
ma-381	271	7	rd+1	rd+1	NOUN
ma-381	271	8	+	+	PUNCT
ma-381	271	9	)	)	PUNCT
ma-381	271	10	:	:	PUNCT
ma-381	272	1	||u||t	||u||t	PROPN
ma-381	272	2	(	(	PUNCT
ma-381	272	3	p	p	X
ma-381	272	4	,	,	PUNCT
ma-381	272	5	q	q	NOUN
ma-381	272	6	)	)	PUNCT
ma-381	272	7	<	<	X
ma-381	272	8	∞	∞	NUM
ma-381	272	9	}	}	PUNCT
ma-381	272	10	where	where	SCONJ
ma-381	272	11	||u||t	||u||t	PROPN
ma-381	272	12	(	(	PUNCT
ma-381	272	13	p	p	X
ma-381	272	14	,	,	PUNCT
ma-381	272	15	q	q	NOUN
ma-381	272	16	)	)	PUNCT
ma-381	272	17	:	:	PUNCT
ma-381	272	18	=	=	PUNCT
ma-381	272	19	sup	sup	NOUN
ma-381	272	20	t>0	t>0	NOUN
ma-381	272	21	||u	||u	NOUN
ma-381	272	22	(	(	PUNCT
ma-381	272	23	.	.	PUNCT
ma-381	272	24	,	,	PUNCT
ma-381	272	25	t)||q	t)||q	ADV
ma-381	272	26	,	,	PUNCT
ma-381	272	27	p.	p.	NOUN
ma-381	273	1	they	they	PRON
ma-381	273	2	proved	prove	VERB
ma-381	273	3	[	[	X
ma-381	273	4	3	3	NUM
ma-381	273	5	,	,	PUNCT
ma-381	273	6	proposition	proposition	NOUN
ma-381	273	7	3.2	3.2	NUM
ma-381	273	8	(	(	PUNCT
ma-381	273	9	i	i	NOUN
ma-381	273	10	)	)	PUNCT
ma-381	273	11	]	]	PUNCT
ma-381	273	12	that	that	SCONJ
ma-381	273	13	for	for	ADP
ma-381	273	14	d−1	d−1	PROPN
ma-381	273	15	d	d	PROPN
ma-381	273	16	<	<	X
ma-381	273	17	p	p	X
ma-381	273	18	,	,	PUNCT
ma-381	273	19	q	q	X
ma-381	273	20	<	<	X
ma-381	273	21	∞	∞	PROPN
ma-381	273	22	,	,	PUNCT
ma-381	273	23	f	f	X
ma-381	273	24	=	=	PUNCT
ma-381	273	25	(	(	PUNCT
ma-381	273	26	u1	u1	PROPN
ma-381	273	27	,	,	PUNCT
ma-381	273	28	u2	u2	PROPN
ma-381	273	29	,	,	PUNCT
ma-381	273	30	·	·	PUNCT
ma-381	273	31	·	·	PUNCT
ma-381	273	32	·	·	PUNCT
ma-381	273	33	,	,	PUNCT
ma-381	273	34	ud+1	ud+1	X
ma-381	273	35	)	)	PUNCT
ma-381	273	36	∈	∈	PROPN
ma-381	273	37	hp	hp	PROPN
ma-381	273	38	,	,	PUNCT
ma-381	273	39	q(rd+1	q(rd+1	PROPN
ma-381	273	40	+	+	NUM
ma-381	273	41	)	)	PUNCT
ma-381	273	42	implies	imply	VERB
ma-381	273	43	that	that	SCONJ
ma-381	273	44	u	u	NOUN
ma-381	273	45	:	:	PUNCT
ma-381	273	46	=	=	SYM
ma-381	273	47	ud+1	ud+1	NUM
ma-381	273	48	∈	∈	PROPN
ma-381	273	49	t	t	NOUN
ma-381	273	50	p	p	X
ma-381	273	51	,	,	PUNCT
ma-381	273	52	q(rd+1	q(rd+1	PROPN
ma-381	273	53	+	+	CCONJ
ma-381	273	54	)	)	PUNCT
ma-381	273	55	and	and	CCONJ
ma-381	273	56	uj	uj	X
ma-381	273	57	(	(	PUNCT
ma-381	273	58	·	·	PROPN
ma-381	273	59	,	,	PUNCT
ma-381	273	60	t	t	PROPN
ma-381	273	61	)	)	PUNCT
ma-381	273	62	=	=	SYM
ma-381	273	63	rj(u	rj(u	X
ma-381	273	64	(	(	PUNCT
ma-381	273	65	·	·	PUNCT
ma-381	273	66	,	,	PUNCT
ma-381	273	67	t	t	PROPN
ma-381	273	68	)	)	PUNCT
ma-381	273	69	)	)	PUNCT
ma-381	273	70	,	,	PUNCT
ma-381	273	71	t	t	X
ma-381	273	72	>	>	X
ma-381	273	73	0	0	PROPN
ma-381	273	74	,	,	PUNCT
ma-381	273	75	j	j	PROPN
ma-381	273	76	=	=	SYM
ma-381	273	77	1	1	NUM
ma-381	273	78	,	,	PUNCT
ma-381	273	79	·	·	PUNCT
ma-381	273	80	·	·	PUNCT
ma-381	273	81	·	·	PUNCT
ma-381	273	82	,	,	PUNCT
ma-381	273	83	d	d	X
ma-381	273	84	.we	.we	PUNCT
ma-381	273	85	claim	claim	VERB
ma-381	273	86	that	that	SCONJ
ma-381	273	87	for	for	ADP
ma-381	273	88	0	0	NUM
ma-381	273	89	<	<	X
ma-381	273	90	p	p	X
ma-381	273	91	≤	≤	NUM
ma-381	273	92	α	α	NOUN
ma-381	273	93	≤	≤	NOUN
ma-381	273	94	q	q	X
ma-381	273	95	<	<	X
ma-381	273	96	∞	∞	PROPN
ma-381	273	97	and	and	CCONJ
ma-381	273	98	r	r	NOUN
ma-381	273	99	>	>	X
ma-381	273	100	0	0	NUM
ma-381	273	101	,	,	PUNCT
ma-381	273	102	the	the	DET
ma-381	273	103	space	space	NOUN
ma-381	273	104	t	t	NOUN
ma-381	273	105	p	p	X
ma-381	273	106	,	,	PUNCT
ma-381	273	107	q(rd+1	q(rd+1	PROPN
ma-381	273	108	+	+	CCONJ
ma-381	273	109	)	)	PUNCT
ma-381	273	110	is	be	AUX
ma-381	273	111	stable	stable	ADJ
ma-381	273	112	under	under	ADP
ma-381	273	113	thedilation	thedilation	NOUN
ma-381	273	114	stαr	stαr	NOUN
ma-381	273	115	.	.	PUNCT
ma-381	274	1	this	this	PRON
ma-381	274	2	is	be	AUX
ma-381	274	3	due	due	ADJ
ma-381	274	4	to	to	ADP
ma-381	274	5	the	the	DET
ma-381	274	6	fact	fact	NOUN
ma-381	274	7	that	that	SCONJ
ma-381	274	8	for	for	ADP
ma-381	274	9	f	f	PROPN
ma-381	274	10	∈	∈	PROPN
ma-381	274	11	(	(	PUNCT
ma-381	274	12	lp	lp	PROPN
ma-381	274	13	,	,	PUNCT
ma-381	274	14	`	`	PUNCT
ma-381	274	15	q	q	X
ma-381	274	16	)	)	PUNCT
ma-381	274	17	(	(	PUNCT
ma-381	274	18	rd	rd	NOUN
ma-381	274	19	)	)	PUNCT
ma-381	274	20	,	,	PUNCT
ma-381	274	21	there	there	PRON
ma-381	274	22	exists	exist	VERB
ma-381	274	23	a	a	DET
ma-381	274	24	constant	constant	ADJ
ma-381	274	25	c(α	c(α	NOUN
ma-381	274	26	,	,	PUNCT
ma-381	274	27	r	r	NOUN
ma-381	274	28	,	,	PUNCT
ma-381	274	29	p	p	X
ma-381	274	30	,	,	PUNCT
ma-381	274	31	q	q	PROPN
ma-381	274	32	)	)	PUNCT
ma-381	274	33	>	>	X
ma-381	274	34	0	0	NUM
ma-381	275	1	such	such	ADJ
ma-381	275	2	that	that	SCONJ
ma-381	275	3	c(α	c(α	NOUN
ma-381	275	4	,	,	PUNCT
ma-381	275	5	r	r	NOUN
ma-381	275	6	,	,	PUNCT
ma-381	275	7	p	p	X
ma-381	275	8	,	,	PUNCT
ma-381	275	9	q)−1‖f	q)−1‖f	PROPN
ma-381	275	10	‖p	‖p	PROPN
ma-381	275	11	,	,	PUNCT
ma-381	275	12	q	q	PROPN
ma-381	275	13	≤	≤	NUM
ma-381	275	14	‖stαr	‖stαr	PUNCT
ma-381	275	15	f	f	X
ma-381	275	16	‖p	‖p	PROPN
ma-381	275	17	,	,	PUNCT
ma-381	275	18	q	q	PROPN
ma-381	275	19	≤	≤	NUM
ma-381	275	20	c(α	c(α	NOUN
ma-381	275	21	,	,	PUNCT
ma-381	275	22	r	r	NOUN
ma-381	275	23	,	,	PUNCT
ma-381	275	24	p	p	NOUN
ma-381	275	25	,	,	PUNCT
ma-381	275	26	q)‖f	q)‖f	NOUN
ma-381	275	27	‖p	‖p	PROPN
ma-381	275	28	,	,	PUNCT
ma-381	275	29	q	q	NOUN
ma-381	275	30	,	,	PUNCT
ma-381	275	31	and	and	CCONJ
ma-381	275	32	this	this	DET
ma-381	275	33	dilation	dilation	NOUN
ma-381	275	34	commute	commute	NOUN
ma-381	275	35	with	with	ADP
ma-381	275	36	riesz	riesz	NOUN
ma-381	275	37	transforms	transform	VERB
ma-381	275	38	.	.	PUNCT
ma-381	276	1	it	it	PRON
ma-381	276	2	follows	follow	VERB
ma-381	276	3	that	that	SCONJ
ma-381	276	4	if	if	SCONJ
ma-381	276	5	f	f	PROPN
ma-381	276	6	=	=	SYM
ma-381	276	7	(	(	PUNCT
ma-381	276	8	u1	u1	PROPN
ma-381	276	9	,	,	PUNCT
ma-381	276	10	·	·	PUNCT
ma-381	276	11	·	·	PUNCT
ma-381	276	12	·	·	PUNCT
ma-381	276	13	,	,	PUNCT
ma-381	276	14	ud+1	ud+1	X
ma-381	276	15	)	)	PUNCT
ma-381	276	16	∈	∈	PROPN
ma-381	276	17	hp	hp	PROPN
ma-381	276	18	,	,	PUNCT
ma-381	276	19	q(rd+1	q(rd+1	PROPN
ma-381	276	20	+	+	NUM
ma-381	276	21	)	)	PUNCT
ma-381	276	22	then	then	ADV
ma-381	276	23	stαr	stαr	VERB
ma-381	276	24	f	f	PROPN
ma-381	276	25	∈	∈	PROPN
ma-381	276	26	hp	hp	PROPN
ma-381	276	27	,	,	PUNCT
ma-381	276	28	q(rd+1	q(rd+1	PROPN
ma-381	276	29	+	+	NUM
ma-381	276	30	)	)	PUNCT
ma-381	276	31	.we	.we	PUNCT
ma-381	277	1	put	put	VERB
ma-381	277	2	‖f‖h(p	‖f‖h(p	NOUN
ma-381	277	3	,	,	PUNCT
ma-381	277	4	q	q	NOUN
ma-381	277	5	,	,	PUNCT
ma-381	277	6	α	α	NOUN
ma-381	277	7	)	)	PUNCT
ma-381	277	8	:	:	PUNCT
ma-381	278	1	=	=	SYM
ma-381	278	2	sup	sup	NOUN
ma-381	278	3	r>0	r>0	ADV
ma-381	278	4	‖stαr	‖stαr	NOUN
ma-381	278	5	f‖hp	f‖hp	PROPN
ma-381	278	6	,	,	PUNCT
ma-381	278	7	q(rd+1	q(rd+1	PROPN
ma-381	278	8	+	+	CCONJ
ma-381	278	9	)	)	PUNCT
ma-381	278	10	and	and	CCONJ
ma-381	278	11	defined	define	VERB
ma-381	278	12	the	the	DET
ma-381	278	13	space	space	NOUN
ma-381	278	14	h(p	h(p	NOUN
ma-381	278	15	,	,	PUNCT
ma-381	278	16	q	q	NOUN
ma-381	278	17	,	,	PUNCT
ma-381	278	18	α)(rd+1	α)(rd+1	PROPN
ma-381	278	19	+	+	CCONJ
ma-381	278	20	)	)	PUNCT
ma-381	278	21	as	as	ADP
ma-381	278	22	the	the	DET
ma-381	278	23	subspace	subspace	NOUN
ma-381	278	24	of	of	ADP
ma-381	278	25	hp	hp	PROPN
ma-381	278	26	,	,	PUNCT
ma-381	278	27	q(rd+1	q(rd+1	PROPN
ma-381	278	28	+	+	NUM
ma-381	278	29	)	)	PUNCT
ma-381	278	30	consits	consit	NOUN
ma-381	278	31	of	of	ADP
ma-381	278	32	f	f	PROPN
ma-381	278	33	satisfying	satisfy	VERB
ma-381	278	34	‖f‖h(p	‖f‖h(p	NOUN
ma-381	278	35	,	,	PUNCT
ma-381	278	36	q	q	NOUN
ma-381	278	37	,	,	PUNCT
ma-381	278	38	α	α	NOUN
ma-381	278	39	)	)	PUNCT
ma-381	278	40	<	<	X
ma-381	278	41	∞.	∞.	PROPN
ma-381	278	42	we	we	PRON
ma-381	278	43	have	have	VERB
ma-381	278	44	the	the	DET
ma-381	278	45	following	following	ADJ
ma-381	278	46	result	result	NOUN
ma-381	278	47	in	in	ADP
ma-381	278	48	hardy	hardy	ADJ
ma-381	278	49	-	-	PUNCT
ma-381	278	50	fofana	fofana	ADJ
ma-381	278	51	spaces	space	NOUN
ma-381	278	52	.	.	PUNCT
ma-381	279	1	theorem	theorem	VERB
ma-381	279	2	4.1	4.1	NUM
ma-381	279	3	.	.	PUNCT
ma-381	280	1	let	let	VERB
ma-381	280	2	d−1	d−1	PROPN
ma-381	280	3	d	d	PROPN
ma-381	280	4	<	<	X
ma-381	280	5	p	p	X
ma-381	280	6	≤	≤	NUM
ma-381	280	7	α	α	NOUN
ma-381	280	8	≤	≤	NUM
ma-381	280	9	q	q	X
ma-381	280	10	<	<	X
ma-381	280	11	∞	∞	PROPN
ma-381	280	12	,	,	PUNCT
ma-381	280	13	and	and	CCONJ
ma-381	280	14	wt	wt	ADP
ma-381	280	15	the	the	DET
ma-381	280	16	heat	heat	NOUN
ma-381	280	17	kernel	kernel	NOUN
ma-381	280	18	defined	define	VERB
ma-381	280	19	by	by	ADP
ma-381	280	20	wt(x	wt(x	NOUN
ma-381	280	21	)	)	PUNCT
ma-381	281	1	=	=	PUNCT
ma-381	281	2	e−|x	e−|x	VERB
ma-381	281	3	|	|	ADP
ma-381	281	4	2/4	2/4	NUM
ma-381	281	5	t	t	NOUN
ma-381	281	6	(	(	PUNCT
ma-381	281	7	4πt)d/2	4πt)d/2	NUM
ma-381	281	8	.	.	PUNCT
ma-381	282	1	the	the	DET
ma-381	282	2	map	map	NOUN
ma-381	282	3	l	l	NOUN
ma-381	282	4	define	define	NOUN
ma-381	282	5	on	on	ADP
ma-381	282	6	h(p	h(p	NOUN
ma-381	282	7	,	,	PUNCT
ma-381	282	8	q	q	NOUN
ma-381	282	9	,	,	PUNCT
ma-381	282	10	α)(rd	α)(rd	NOUN
ma-381	282	11	)	)	PUNCT
ma-381	282	12	by	by	ADP
ma-381	282	13	l(f	l(f	PROPN
ma-381	282	14	)	)	PUNCT
ma-381	282	15	(	(	PUNCT
ma-381	282	16	x	x	X
ma-381	282	17	,	,	PUNCT
ma-381	282	18	t	t	PROPN
ma-381	282	19	)	)	PUNCT
ma-381	282	20	:	:	PUNCT
ma-381	283	1	=	=	SYM
ma-381	283	2	(	(	PUNCT
ma-381	283	3	(	(	PUNCT
ma-381	283	4	(	(	PUNCT
ma-381	283	5	r1f	r1f	PROPN
ma-381	283	6	)	)	PUNCT
ma-381	283	7	∗wt)(x	∗wt)(x	NUM
ma-381	283	8	)	)	PUNCT
ma-381	283	9	,	,	PUNCT
ma-381	283	10	·	·	PUNCT
ma-381	283	11	·	·	PUNCT
ma-381	283	12	·	·	PUNCT
ma-381	283	13	,	,	PUNCT
ma-381	283	14	(	(	PUNCT
ma-381	283	15	(	(	PUNCT
ma-381	283	16	rd	rd	NOUN
ma-381	283	17	f	f	PROPN
ma-381	283	18	)	)	PUNCT
ma-381	283	19	∗wt)(x	∗wt)(x	NUM
ma-381	283	20	)	)	PUNCT
ma-381	283	21	,	,	PUNCT
ma-381	283	22	(	(	PUNCT
ma-381	283	23	f	f	PROPN
ma-381	283	24	∗wt)(x	∗wt)(x	NUM
ma-381	283	25	)	)	PUNCT
ma-381	283	26	)	)	PUNCT
ma-381	284	1	for	for	ADP
ma-381	284	2	all	all	DET
ma-381	284	3	x	x	SYM
ma-381	284	4	∈	∈	PROPN
ma-381	284	5	rd	rd	PROPN
ma-381	284	6	and	and	CCONJ
ma-381	284	7	t	t	PROPN
ma-381	284	8	>	>	X
ma-381	284	9	0	0	PROPN
ma-381	284	10	,	,	PUNCT
ma-381	284	11	is	be	AUX
ma-381	284	12	a	a	DET
ma-381	284	13	topological	topological	ADJ
ma-381	284	14	isomorphism	isomorphism	NOUN
ma-381	284	15	from	from	ADP
ma-381	284	16	h(p	h(p	PROPN
ma-381	284	17	,	,	PUNCT
ma-381	284	18	q	q	NOUN
ma-381	284	19	,	,	PUNCT
ma-381	284	20	α)(rd	α)(rd	NOUN
ma-381	284	21	)	)	PUNCT
ma-381	284	22	onto	onto	ADP
ma-381	284	23	h(p	h(p	NOUN
ma-381	284	24	,	,	PUNCT
ma-381	284	25	q	q	NOUN
ma-381	284	26	,	,	PUNCT
ma-381	284	27	α)(rd+1	α)(rd+1	PROPN
ma-381	284	28	+	+	PUNCT
ma-381	284	29	)	)	PUNCT
ma-381	284	30	.	.	PUNCT
ma-381	285	1	proof	proof	NOUN
ma-381	285	2	.	.	PUNCT
ma-381	286	1	let	let	VERB
ma-381	286	2	f	f	PRON
ma-381	286	3	∈	∈	PROPN
ma-381	286	4	h(p	h(p	PROPN
ma-381	286	5	,	,	PUNCT
ma-381	286	6	q	q	NOUN
ma-381	286	7	,	,	PUNCT
ma-381	286	8	α)(rd	α)(rd	NOUN
ma-381	286	9	)	)	PUNCT
ma-381	286	10	.	.	PUNCT
ma-381	287	1	for	for	ADP
ma-381	287	2	r	r	NOUN
ma-381	287	3	>	>	X
ma-381	287	4	0	0	NUM
ma-381	287	5	we	we	PRON
ma-381	287	6	have	have	VERB
ma-381	287	7	stαr	stαr	NOUN
ma-381	287	8	f	f	PROPN
ma-381	287	9	∈	∈	PROPN
ma-381	287	10	hp	hp	PROPN
ma-381	287	11	,	,	PUNCT
ma-381	287	12	q(rd	q(rd	NOUN
ma-381	287	13	)	)	PUNCT
ma-381	287	14	,	,	PUNCT
ma-381	287	15	thanks	thank	NOUN
ma-381	287	16	to	to	ADP
ma-381	287	17	the	the	DET
ma-381	287	18	definition	definition	NOUN
ma-381	287	19	of	of	ADP
ma-381	287	20	h(p	h(p	PROPN
ma-381	287	21	,	,	PUNCT
ma-381	287	22	q	q	NOUN
ma-381	287	23	,	,	PUNCT
ma-381	287	24	α)(rd	α)(rd	NOUN
ma-381	287	25	)	)	PUNCT
ma-381	287	26	.	.	PUNCT
ma-381	288	1	it	it	PRON
ma-381	288	2	comes	come	VERB
ma-381	288	3	from	from	ADP
ma-381	288	4	[	[	X
ma-381	288	5	3	3	NUM
ma-381	288	6	,	,	PUNCT
ma-381	288	7	theorem	theorem	VERB
ma-381	288	8	1.3	1.3	NUM
ma-381	288	9	]	]	PUNCT
ma-381	288	10	that	that	PRON
ma-381	288	11	l(stαr	l(stαr	PROPN
ma-381	288	12	f	f	X
ma-381	288	13	)	)	PUNCT
ma-381	288	14	∈	∈	PROPN
ma-381	288	15	hp	hp	PROPN
ma-381	288	16	,	,	PUNCT
ma-381	288	17	q(rd+1	q(rd+1	PROPN
ma-381	288	18	+	+	NUM
ma-381	288	19	)	)	PUNCT
ma-381	288	20	,	,	PUNCT
ma-381	288	21	with	with	ADP
ma-381	288	22	‖l(stαr	‖l(stαr	X
ma-381	288	23	f	f	NOUN
ma-381	288	24	)	)	PUNCT
ma-381	288	25	‖hp	‖hp	PROPN
ma-381	288	26	,	,	PUNCT
ma-381	288	27	q(rd+1	q(rd+1	PROPN
ma-381	288	28	+	+	NUM
ma-381	288	29	)	)	PUNCT
ma-381	288	30	≤	≤	NUM
ma-381	288	31	c‖stαr	c‖stαr	X
ma-381	288	32	f	f	PROPN
ma-381	288	33	‖hp	‖hp	PROPN
ma-381	288	34	,	,	PUNCT
ma-381	288	35	q(rd	q(rd	NUM
ma-381	288	36	)	)	PUNCT
ma-381	288	37	(	(	PUNCT
ma-381	288	38	4.1	4.1	NUM
ma-381	288	39	)	)	PUNCT
ma-381	288	40	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	288	41	eur	eur	PROPN
ma-381	288	42	.	.	PUNCT
ma-381	289	1	j.	j.	PROPN
ma-381	289	2	math	math	PROPN
ma-381	289	3	.	.	PUNCT
ma-381	290	1	anal	anal	PROPN
ma-381	290	2	.	.	PUNCT
ma-381	291	1	10.28924	10.28924	NUM
ma-381	291	2	/	/	SYM
ma-381	291	3	ada	ada	PROPN
ma-381	291	4	/	/	SYM
ma-381	291	5	ma.5.21	ma.5.21	PROPN
ma-381	291	6	12for	12for	ADP
ma-381	291	7	all	all	DET
ma-381	291	8	r	r	NOUN
ma-381	291	9	>	>	X
ma-381	291	10	0	0	X
ma-381	291	11	.	.	PUNCT
ma-381	292	1	since	since	SCONJ
ma-381	292	2	rj(f	rj(f	NOUN
ma-381	292	3	∗wt	∗wt	NOUN
ma-381	292	4	)	)	PUNCT
ma-381	292	5	=	=	SYM
ma-381	292	6	rj(f	rj(f	X
ma-381	292	7	)	)	PUNCT
ma-381	292	8	∗wt	∗wt	PUNCT
ma-381	292	9	and	and	CCONJ
ma-381	292	10	stαr	stαr	VERB
ma-381	292	11	commute	commute	VERB
ma-381	292	12	with	with	ADP
ma-381	292	13	riesz	riesz	NOUN
ma-381	292	14	transforms	transform	VERB
ma-381	292	15	,	,	PUNCT
ma-381	292	16	we	we	PRON
ma-381	292	17	havethat	havethat	ADV
ma-381	292	18	l(stαr	l(stαr	VERB
ma-381	292	19	f	f	PROPN
ma-381	292	20	)	)	PUNCT
ma-381	293	1	=	=	PRON
ma-381	293	2	stαr	stαr	NOUN
ma-381	293	3	(	(	PUNCT
ma-381	293	4	l(f	l(f	PROPN
ma-381	293	5	)	)	PUNCT
ma-381	293	6	)	)	PUNCT
ma-381	293	7	,	,	PUNCT
ma-381	293	8	r	r	NOUN
ma-381	293	9	>	>	X
ma-381	293	10	0	0	X
ma-381	293	11	.	.	PUNCT
ma-381	293	12	thaking	thake	VERB
ma-381	293	13	this	this	DET
ma-381	293	14	remark	remark	NOUN
ma-381	293	15	in	in	ADP
ma-381	293	16	(	(	PUNCT
ma-381	293	17	4.1	4.1	NUM
ma-381	293	18	)	)	PUNCT
ma-381	293	19	we	we	PRON
ma-381	293	20	obtain	obtain	VERB
ma-381	293	21	that	that	PRON
ma-381	293	22	‖l(f	‖l(f	PROPN
ma-381	293	23	)	)	PUNCT
ma-381	293	24	‖h(p	‖h(p	PROPN
ma-381	293	25	,	,	PUNCT
ma-381	293	26	q	q	NOUN
ma-381	293	27	,	,	PUNCT
ma-381	293	28	α)(rd+1	α)(rd+1	PROPN
ma-381	293	29	+	+	CCONJ
ma-381	293	30	)	)	PUNCT
ma-381	293	31	≤	≤	NUM
ma-381	293	32	c‖f	c‖f	NOUN
ma-381	294	1	‖h(p	‖h(p	X
ma-381	294	2	,	,	PUNCT
ma-381	294	3	q	q	NOUN
ma-381	294	4	,	,	PUNCT
ma-381	294	5	α)(rd	α)(rd	NUM
ma-381	294	6	)	)	PUNCT
ma-381	294	7	.let	.let	PUNCT
ma-381	295	1	now	now	ADV
ma-381	295	2	f	f	X
ma-381	295	3	=	=	PUNCT
ma-381	295	4	(	(	PUNCT
ma-381	295	5	u1	u1	PROPN
ma-381	295	6	,	,	PUNCT
ma-381	295	7	u2	u2	PROPN
ma-381	295	8	,	,	PUNCT
ma-381	295	9	·	·	PUNCT
ma-381	295	10	·	·	PUNCT
ma-381	295	11	·	·	PUNCT
ma-381	295	12	,	,	PUNCT
ma-381	295	13	ud	ud	INTJ
ma-381	295	14	,	,	PUNCT
ma-381	295	15	ud+1	ud+1	PROPN
ma-381	295	16	)	)	PUNCT
ma-381	295	17	belonging	belong	VERB
ma-381	295	18	to	to	ADP
ma-381	295	19	h(p	h(p	NOUN
ma-381	295	20	,	,	PUNCT
ma-381	295	21	q	q	NOUN
ma-381	295	22	,	,	PUNCT
ma-381	295	23	α)(rd+1	α)(rd+1	PROPN
ma-381	295	24	+	+	PUNCT
ma-381	295	25	)	)	PUNCT
ma-381	295	26	.	.	PUNCT
ma-381	296	1	this	this	PRON
ma-381	296	2	implies	imply	VERB
ma-381	296	3	that	that	PRON
ma-381	296	4	stαr	stαr	VERB
ma-381	296	5	f	f	NOUN
ma-381	296	6	=	=	PUNCT
ma-381	296	7	(	(	PUNCT
ma-381	296	8	stαr	stαr	VERB
ma-381	296	9	u1,stαr	u1,stαr	NUM
ma-381	296	10	u2	u2	PROPN
ma-381	296	11	,	,	PUNCT
ma-381	296	12	·	·	PUNCT
ma-381	296	13	·	·	PUNCT
ma-381	296	14	·	·	PUNCT
ma-381	296	15	,	,	PUNCT
ma-381	296	16	stαr	stαr	VERB
ma-381	296	17	ud	ud	INTJ
ma-381	296	18	,	,	PUNCT
ma-381	296	19	stαr	stαr	VERB
ma-381	296	20	ud+1	ud+1	NOUN
ma-381	296	21	)	)	PUNCT
ma-381	296	22	∈	∈	PROPN
ma-381	296	23	hp	hp	PROPN
ma-381	296	24	,	,	PUNCT
ma-381	296	25	q(rd+1	q(rd+1	PROPN
ma-381	296	26	+	+	CCONJ
ma-381	296	27	)	)	PUNCT
ma-381	296	28	for	for	ADP
ma-381	296	29	all	all	DET
ma-381	296	30	r	r	NOUN
ma-381	296	31	>	>	X
ma-381	296	32	0	0	NUM
ma-381	296	33	.	.	PUNCT
ma-381	297	1	as	as	SCONJ
ma-381	297	2	we	we	PRON
ma-381	297	3	can	can	AUX
ma-381	297	4	see	see	VERB
ma-381	297	5	in	in	ADP
ma-381	297	6	the	the	DET
ma-381	297	7	proofof	proofof	NOUN
ma-381	298	1	[	[	X
ma-381	298	2	3	3	NUM
ma-381	298	3	,	,	PUNCT
ma-381	298	4	theorem	theorem	VERB
ma-381	298	5	3.1	3.1	NUM
ma-381	298	6	]	]	PUNCT
ma-381	298	7	this	this	PRON
ma-381	298	8	implies	imply	VERB
ma-381	298	9	that	that	SCONJ
ma-381	298	10	for	for	ADP
ma-381	298	11	all	all	DET
ma-381	298	12	r	r	NOUN
ma-381	298	13	>	>	X
ma-381	298	14	0	0	NUM
ma-381	298	15	,	,	PUNCT
ma-381	298	16	there	there	PRON
ma-381	298	17	exists	exist	VERB
ma-381	298	18	fr	fr	PROPN
ma-381	298	19	∈	∈	PROPN
ma-381	298	20	hp	hp	PROPN
ma-381	298	21	,	,	PUNCT
ma-381	298	22	q(rd	q(rd	NOUN
ma-381	298	23	)	)	PUNCT
ma-381	298	24	so	so	SCONJ
ma-381	298	25	that	that	PRON
ma-381	298	26	stαr	stαr	VERB
ma-381	298	27	ud+1	ud+1	NUM
ma-381	298	28	∈	∈	PROPN
ma-381	298	29	t	t	NOUN
ma-381	298	30	p	p	X
ma-381	298	31	,	,	PUNCT
ma-381	298	32	q(rd+1	q(rd+1	PROPN
ma-381	298	33	+	+	CCONJ
ma-381	298	34	)	)	PUNCT
ma-381	298	35	with	with	ADP
ma-381	298	36	stαr	stαr	PROPN
ma-381	298	37	ud+1(x	ud+1(x	PROPN
ma-381	298	38	,	,	PUNCT
ma-381	298	39	t	t	PROPN
ma-381	298	40	)	)	PUNCT
ma-381	299	1	=	=	NOUN
ma-381	299	2	fr	fr	PROPN
ma-381	299	3	∗wt(x	∗wt(x	PROPN
ma-381	299	4	)	)	PUNCT
ma-381	299	5	and	and	CCONJ
ma-381	299	6	‖fr‖hp	‖fr‖hp	NOUN
ma-381	299	7	,	,	PUNCT
ma-381	299	8	q	q	PROPN
ma-381	299	9	≤	≤	NUM
ma-381	299	10	c	c	NOUN
ma-381	299	11	sup	sup	NOUN
ma-381	299	12	t>0	t>0	NOUN
ma-381	299	13	‖stαr	‖stαr	PUNCT
ma-381	299	14	f	f	NOUN
ma-381	299	15	(	(	PUNCT
ma-381	299	16	·	·	PROPN
ma-381	299	17	,	,	PUNCT
ma-381	299	18	t)‖p	t)‖p	PROPN
ma-381	299	19	,	,	PUNCT
ma-381	299	20	q	q	PROPN
ma-381	299	21	≤	≤	PROPN
ma-381	299	22	c‖f‖h(p	c‖f‖h(p	PROPN
ma-381	299	23	,	,	PUNCT
ma-381	299	24	q	q	NOUN
ma-381	299	25	,	,	PUNCT
ma-381	299	26	α	α	NOUN
ma-381	299	27	)	)	PUNCT
ma-381	299	28	,	,	PUNCT
ma-381	299	29	(	(	PUNCT
ma-381	299	30	4.2	4.2	NUM
ma-381	299	31	)	)	PUNCT
ma-381	299	32	and	and	CCONJ
ma-381	299	33	stαr	stαr	VERB
ma-381	299	34	uj	uj	PROPN
ma-381	299	35	(	(	PUNCT
ma-381	299	36	·	·	PROPN
ma-381	299	37	,	,	PUNCT
ma-381	299	38	t	t	PROPN
ma-381	299	39	)	)	PUNCT
ma-381	299	40	=	=	SYM
ma-381	300	1	rj(stαr	rj(stαr	PROPN
ma-381	300	2	ud+1	ud+1	NUM
ma-381	300	3	(	(	PUNCT
ma-381	300	4	·	·	PUNCT
ma-381	300	5	,	,	PUNCT
ma-381	300	6	t	t	PROPN
ma-381	300	7	)	)	PUNCT
ma-381	300	8	)	)	PUNCT
ma-381	300	9	,	,	PUNCT
ma-381	300	10	t	t	X
ma-381	300	11	>	>	X
ma-381	300	12	0	0	PROPN
ma-381	300	13	,	,	PUNCT
ma-381	300	14	j	j	PROPN
ma-381	300	15	=	=	SYM
ma-381	300	16	1	1	NUM
ma-381	300	17	,	,	PUNCT
ma-381	300	18	·	·	PUNCT
ma-381	300	19	·	·	PUNCT
ma-381	300	20	·	·	PUNCT
ma-381	300	21	,	,	PUNCT
ma-381	300	22	d.we	d.we	NOUN
ma-381	300	23	put	put	VERB
ma-381	300	24	f	f	NOUN
ma-381	300	25	:	:	PUNCT
ma-381	301	1	=	=	SYM
ma-381	301	2	f	f	PROPN
ma-381	301	3	1	1	X
ma-381	301	4	.	.	PUNCT
ma-381	302	1	we	we	PRON
ma-381	302	2	have	have	VERB
ma-381	302	3	stαr	stαr	NOUN
ma-381	302	4	f	f	X
ma-381	303	1	=	=	PUNCT
ma-381	303	2	fr	fr	PROPN
ma-381	303	3	for	for	ADP
ma-381	303	4	all	all	DET
ma-381	303	5	r	r	NOUN
ma-381	303	6	>	>	X
ma-381	303	7	0	0	X
ma-381	303	8	.	.	PUNCT
ma-381	304	1	taking	take	VERB
ma-381	304	2	this	this	PRON
ma-381	304	3	in	in	ADP
ma-381	304	4	estimate	estimate	NOUN
ma-381	304	5	(	(	PUNCT
ma-381	304	6	4.2	4.2	NUM
ma-381	304	7	)	)	PUNCT
ma-381	304	8	yields	yield	NOUN
ma-381	304	9	‖stαr	‖stαr	PUNCT
ma-381	304	10	f	f	PROPN
ma-381	304	11	‖hp	‖hp	PROPN
ma-381	304	12	,	,	PUNCT
ma-381	304	13	q	q	PROPN
ma-381	304	14	≤	≤	PROPN
ma-381	304	15	c‖f‖h(p	c‖f‖h(p	PROPN
ma-381	304	16	,	,	PUNCT
ma-381	304	17	q	q	NOUN
ma-381	304	18	,	,	PUNCT
ma-381	304	19	α	α	NOUN
ma-381	304	20	)	)	PUNCT
ma-381	304	21	wich	wich	PRON
ma-381	304	22	prove	prove	VERB
ma-381	304	23	that	that	SCONJ
ma-381	304	24	f	f	PROPN
ma-381	304	25	∈	∈	PROPN
ma-381	304	26	h(p	h(p	PROPN
ma-381	304	27	,	,	PUNCT
ma-381	304	28	q	q	X
ma-381	304	29	,	,	PUNCT
ma-381	304	30	α)(rd).the	α)(rd).the	DET
ma-381	304	31	vector	vector	NOUN
ma-381	304	32	g(x	g(x	PROPN
ma-381	304	33	,	,	PUNCT
ma-381	304	34	t	t	PROPN
ma-381	304	35	)	)	PUNCT
ma-381	304	36	=	=	SYM
ma-381	304	37	(	(	PUNCT
ma-381	304	38	(	(	PUNCT
ma-381	304	39	r1(f	r1(f	NUM
ma-381	304	40	)	)	PUNCT
ma-381	304	41	∗	∗	NOUN
ma-381	304	42	pt)(x	pt)(x	PROPN
ma-381	304	43	)	)	PUNCT
ma-381	304	44	,	,	PUNCT
ma-381	304	45	·	·	PUNCT
ma-381	304	46	·	·	PUNCT
ma-381	304	47	·	·	PUNCT
ma-381	304	48	,	,	PUNCT
ma-381	304	49	rd(f	rd(f	X
ma-381	304	50	)	)	PUNCT
ma-381	304	51	∗	∗	NOUN
ma-381	304	52	pt)(x	pt)(x	PROPN
ma-381	304	53	)	)	PUNCT
ma-381	304	54	,	,	PUNCT
ma-381	304	55	(	(	PUNCT
ma-381	304	56	f	f	PROPN
ma-381	304	57	∗	∗	NOUN
ma-381	304	58	pt)(x	pt)(x	PROPN
ma-381	304	59	)	)	PUNCT
ma-381	304	60	)	)	PUNCT
ma-381	304	61	,	,	PUNCT
ma-381	304	62	x	x	PUNCT
ma-381	304	63	∈	∈	PROPN
ma-381	304	64	rd	rd	PROPN
ma-381	304	65	,	,	PUNCT
ma-381	304	66	t	t	PROPN
ma-381	304	67	>	>	X
ma-381	304	68	0	0	PUNCT
ma-381	304	69	is	be	AUX
ma-381	304	70	harmonic	harmonic	ADJ
ma-381	304	71	,	,	PUNCT
ma-381	304	72	satisfies	satisfy	VERB
ma-381	304	73	the	the	DET
ma-381	304	74	generalized	generalized	ADJ
ma-381	304	75	cauchy	cauchy	PROPN
ma-381	304	76	-	-	PUNCT
ma-381	304	77	riemann	riemann	PROPN
ma-381	304	78	equation	equation	NOUN
ma-381	304	79	,	,	PUNCT
ma-381	304	80	and	and	CCONJ
ma-381	304	81	sup	sup	NOUN
ma-381	304	82	t>0	t>0	NOUN
ma-381	304	83	‖|g	‖|g	NOUN
ma-381	304	84	(	(	PUNCT
ma-381	304	85	·	·	PUNCT
ma-381	304	86	,	,	PUNCT
ma-381	304	87	t)|‖p	t)|‖p	PRON
ma-381	304	88	,	,	PUNCT
ma-381	304	89	q	q	NOUN
ma-381	304	90	,	,	PUNCT
ma-381	304	91	α	α	PROPN
ma-381	304	92	≤	≤	PUNCT
ma-381	304	93	c‖f‖h(p	c‖f‖h(p	PROPN
ma-381	304	94	,	,	PUNCT
ma-381	304	95	q	q	NOUN
ma-381	304	96	,	,	PUNCT
ma-381	304	97	α	α	NOUN
ma-381	304	98	)	)	PUNCT
ma-381	304	99	.	.	PUNCT
ma-381	305	1	�	�	PROPN
ma-381	305	2	references	reference	NOUN
ma-381	305	3	[	[	X
ma-381	305	4	1	1	NUM
ma-381	305	5	]	]	X
ma-381	305	6	z.v.d.p	z.v.d.p	PROPN
ma-381	305	7	.	.	PROPN
ma-381	305	8	ablé	ablé	PROPN
ma-381	305	9	and	and	CCONJ
ma-381	305	10	j.	j.	PROPN
ma-381	305	11	feuto	feuto	PROPN
ma-381	305	12	,	,	PUNCT
ma-381	305	13	atomic	atomic	ADJ
ma-381	305	14	decomposition	decomposition	NOUN
ma-381	305	15	of	of	ADP
ma-381	305	16	hardy	hardy	ADJ
ma-381	305	17	-	-	PUNCT
ma-381	305	18	amalgam	amalgam	NOUN
ma-381	305	19	spaces	space	NOUN
ma-381	305	20	,	,	PUNCT
ma-381	305	21	j.	j.	PROPN
ma-381	305	22	math	math	PROPN
ma-381	305	23	.	.	PUNCT
ma-381	306	1	anal	anal	PROPN
ma-381	306	2	.	.	PUNCT
ma-381	307	1	appl	appl	PROPN
ma-381	307	2	.	.	PUNCT
ma-381	308	1	455	455	NUM
ma-381	308	2	(	(	PUNCT
ma-381	308	3	2017	2017	NUM
ma-381	308	4	)	)	PUNCT
ma-381	308	5	,	,	PUNCT
ma-381	308	6	1899–1936.[2	1899–1936.[2	NUM
ma-381	308	7	]	]	X
ma-381	308	8	z.v.d.p	z.v.d.p	PROPN
ma-381	308	9	.	.	PROPN
ma-381	308	10	ablé	ablé	PROPN
ma-381	308	11	and	and	CCONJ
ma-381	308	12	j.	j.	PROPN
ma-381	308	13	feuto	feuto	PROPN
ma-381	308	14	,	,	PUNCT
ma-381	308	15	duals	dual	NOUN
ma-381	308	16	of	of	ADP
ma-381	308	17	hardy	hardy	ADJ
ma-381	308	18	amalgam	amalgam	NOUN
ma-381	308	19	spaces	space	NOUN
ma-381	308	20	and	and	CCONJ
ma-381	308	21	norm	norm	NOUN
ma-381	308	22	inequalities	inequality	NOUN
ma-381	308	23	,	,	PUNCT
ma-381	308	24	anal	anal	PROPN
ma-381	308	25	.	.	PUNCT
ma-381	309	1	math	math	NOUN
ma-381	309	2	.	.	PUNCT
ma-381	310	1	45	45	NUM
ma-381	310	2	(	(	PUNCT
ma-381	310	3	2019	2019	NUM
ma-381	310	4	)	)	PUNCT
ma-381	310	5	,	,	PUNCT
ma-381	310	6	647–686.[3	647–686.[3	NUM
ma-381	310	7	]	]	PUNCT
ma-381	310	8	a.-t	a.-t	NOUN
ma-381	310	9	.	.	PUNCT
ma-381	311	1	assaubay	assaubay	PROPN
ma-381	311	2	,	,	PUNCT
ma-381	311	3	j.j	j.j	PROPN
ma-381	311	4	.	.	PROPN
ma-381	311	5	betancor	betancor	PROPN
ma-381	311	6	,	,	PUNCT
ma-381	311	7	a.j	a.j	PROPN
ma-381	311	8	.	.	PROPN
ma-381	311	9	castro	castro	PROPN
ma-381	311	10	and	and	CCONJ
ma-381	311	11	j.c	j.c	PROPN
ma-381	311	12	.	.	PROPN
ma-381	311	13	farina	farina	PROPN
ma-381	311	14	,	,	PUNCT
ma-381	311	15	riesz	riesz	PROPN
ma-381	311	16	transforms	transform	VERB
ma-381	311	17	,	,	PUNCT
ma-381	311	18	cauchy	cauchy	NOUN
ma-381	311	19	-	-	PUNCT
ma-381	311	20	riemann	riemann	PROPN
ma-381	311	21	systems	system	NOUN
ma-381	311	22	,	,	PUNCT
ma-381	311	23	and	and	CCONJ
ma-381	311	24	hardy	hardy	ADJ
ma-381	311	25	-	-	PUNCT
ma-381	311	26	amalgam	amalgam	NOUN
ma-381	311	27	spaces	space	NOUN
ma-381	311	28	,	,	PUNCT
ma-381	311	29	banach	banach	NOUN
ma-381	311	30	j.	j.	PROPN
ma-381	311	31	math	math	PROPN
ma-381	311	32	.	.	PUNCT
ma-381	312	1	anal	anal	ADJ
ma-381	312	2	.	.	PUNCT
ma-381	313	1	3	3	NUM
ma-381	313	2	(	(	PUNCT
ma-381	313	3	2019	2019	NUM
ma-381	313	4	)	)	PUNCT
ma-381	313	5	,	,	PUNCT
ma-381	313	6	697–725.[4	697–725.[4	NUM
ma-381	313	7	]	]	X
ma-381	313	8	m.a	m.a	PROPN
ma-381	313	9	.	.	PROPN
ma-381	313	10	dakoury	dakoury	PROPN
ma-381	313	11	and	and	CCONJ
ma-381	313	12	j.	j.	PROPN
ma-381	313	13	feuto	feuto	PROPN
ma-381	313	14	,	,	PUNCT
ma-381	313	15	norm	norm	NOUN
ma-381	313	16	inequality	inequality	NOUN
ma-381	313	17	for	for	ADP
ma-381	313	18	intrinsic	intrinsic	ADJ
ma-381	313	19	square	square	ADJ
ma-381	313	20	functions	function	NOUN
ma-381	313	21	in	in	ADP
ma-381	313	22	generalized	generalized	ADJ
ma-381	313	23	hardy	hardy	ADJ
ma-381	313	24	-	-	PUNCT
ma-381	313	25	morrey	morrey	NOUN
ma-381	313	26	spaces	space	NOUN
ma-381	313	27	,	,	PUNCT
ma-381	313	28	open	open	ADJ
ma-381	313	29	access	access	NOUN
ma-381	313	30	libr	libr	NOUN
ma-381	313	31	.	.	PUNCT
ma-381	314	1	j.	j.	PROPN
ma-381	314	2	9	9	NUM
ma-381	314	3	(	(	PUNCT
ma-381	314	4	2022	2022	NUM
ma-381	314	5	)	)	PUNCT
ma-381	314	6	,	,	PUNCT
ma-381	314	7	e8463.[5	e8463.[5	PRON
ma-381	314	8	]	]	X
ma-381	314	9	m.a	m.a	PROPN
ma-381	314	10	.	.	PROPN
ma-381	314	11	dakoury	dakoury	PROPN
ma-381	314	12	and	and	CCONJ
ma-381	314	13	j.	j.	PROPN
ma-381	314	14	feuto	feuto	PROPN
ma-381	314	15	,	,	PUNCT
ma-381	314	16	norm	norm	NOUN
ma-381	314	17	inequalities	inequality	NOUN
ma-381	314	18	for	for	ADP
ma-381	314	19	calderón	calderón	NOUN
ma-381	314	20	–	–	PUNCT
ma-381	314	21	zygmund	zygmund	ADJ
ma-381	314	22	operators	operator	NOUN
ma-381	314	23	in	in	ADP
ma-381	314	24	some	some	DET
ma-381	314	25	generalized	generalized	ADJ
ma-381	314	26	hardy	hardy	ADJ
ma-381	314	27	–	–	PUNCT
ma-381	314	28	morreyspaces	morreyspace	NOUN
ma-381	314	29	,	,	PUNCT
ma-381	314	30	vietnam	vietnam	PROPN
ma-381	314	31	j.	j.	PROPN
ma-381	314	32	math	math	PROPN
ma-381	314	33	.	.	PUNCT
ma-381	315	1	(	(	PUNCT
ma-381	315	2	2024	2024	NUM
ma-381	315	3	)	)	PUNCT
ma-381	315	4	.	.	PUNCT
ma-381	316	1	https://doi.org/10.1007/s10013-024-00703-0.[6	https://doi.org/10.1007/s10013-024-00703-0.[6	PROPN
ma-381	316	2	]	]	X
ma-381	316	3	j.	j.	PROPN
ma-381	316	4	feuto	feuto	PROPN
ma-381	316	5	,	,	PUNCT
ma-381	316	6	norm	norm	NOUN
ma-381	316	7	inequalities	inequality	NOUN
ma-381	316	8	in	in	ADP
ma-381	316	9	some	some	DET
ma-381	316	10	subspaces	subspace	NOUN
ma-381	316	11	of	of	ADP
ma-381	316	12	morrey	morrey	PROPN
ma-381	316	13	space	space	NOUN
ma-381	316	14	,	,	PUNCT
ma-381	316	15	ann	ann	PROPN
ma-381	316	16	.	.	PROPN
ma-381	316	17	math	math	PROPN
ma-381	316	18	.	.	PUNCT
ma-381	317	1	blaise	blaise	PROPN
ma-381	317	2	pascal	pascal	PROPN
ma-381	317	3	21	21	NUM
ma-381	317	4	(	(	PUNCT
ma-381	317	5	2014	2014	NUM
ma-381	317	6	)	)	PUNCT
ma-381	317	7	,	,	PUNCT
ma-381	317	8	21–37.[7	21–37.[7	NUM
ma-381	317	9	]	]	X
ma-381	317	10	i.	i.	PROPN
ma-381	317	11	fofana	fofana	PROPN
ma-381	317	12	,	,	PUNCT
ma-381	317	13	étude	étude	PROPN
ma-381	317	14	d’une	d’une	PROPN
ma-381	317	15	classe	classe	NOUN
ma-381	317	16	d’espaces	d’espace	NOUN
ma-381	317	17	de	de	X
ma-381	317	18	fonctions	fonction	NOUN
ma-381	317	19	contenant	contenant	X
ma-381	317	20	les	les	X
ma-381	317	21	espaces	espaces	X
ma-381	317	22	de	de	X
ma-381	317	23	lorentz	lorentz	PROPN
ma-381	317	24	,	,	PUNCT
ma-381	317	25	afrika	afrika	X
ma-381	317	26	mat	mat	NOUN
ma-381	317	27	.	.	PROPN
ma-381	317	28	2	2	NUM
ma-381	317	29	(	(	PUNCT
ma-381	317	30	1988	1988	NUM
ma-381	317	31	)	)	PUNCT
ma-381	317	32	,	,	PUNCT
ma-381	317	33	29–50.[8	29–50.[8	NUM
ma-381	317	34	]	]	X
ma-381	317	35	y.	y.	PROPN
ma-381	317	36	liang	liang	PROPN
ma-381	317	37	,	,	PUNCT
ma-381	317	38	y.	y.	PROPN
ma-381	317	39	sawano	sawano	PROPN
ma-381	317	40	,	,	PUNCT
ma-381	317	41	t.	t.	PROPN
ma-381	317	42	ullrich	ullrich	PROPN
ma-381	317	43	,	,	PUNCT
ma-381	317	44	d.	d.	PROPN
ma-381	317	45	yang	yang	PROPN
ma-381	317	46	and	and	CCONJ
ma-381	317	47	w.	w.	PROPN
ma-381	317	48	yuan	yuan	PROPN
ma-381	317	49	,	,	PUNCT
ma-381	317	50	a	a	DET
ma-381	317	51	new	new	ADJ
ma-381	317	52	framework	framework	NOUN
ma-381	317	53	for	for	ADP
ma-381	317	54	generalized	generalized	ADJ
ma-381	317	55	besov	besov	NOUN
ma-381	317	56	-	-	PUNCT
ma-381	317	57	type	type	NOUN
ma-381	317	58	and	and	CCONJ
ma-381	317	59	triebel	triebel	NOUN
ma-381	317	60	-	-	PUNCT
ma-381	317	61	lizorkin	lizorkin	NOUN
ma-381	317	62	-	-	PUNCT
ma-381	317	63	type	type	NOUN
ma-381	317	64	spaces	space	NOUN
ma-381	317	65	,	,	PUNCT
ma-381	317	66	diss	diss	PROPN
ma-381	317	67	.	.	PROPN
ma-381	317	68	math	math	PROPN
ma-381	317	69	.	.	PUNCT
ma-381	318	1	(	(	PUNCT
ma-381	318	2	rozprawy	rozprawy	PROPN
ma-381	318	3	mat	mat	NOUN
ma-381	318	4	.	.	PUNCT
ma-381	318	5	)	)	PUNCT
ma-381	319	1	489	489	NUM
ma-381	319	2	(	(	PUNCT
ma-381	319	3	2013	2013	NUM
ma-381	319	4	)	)	PUNCT
ma-381	319	5	,	,	PUNCT
ma-381	319	6	1–114.[9	1–114.[9	NUM
ma-381	319	7	]	]	X
ma-381	320	1	l.	l.	PROPN
ma-381	320	2	grafakos	grafakos	PROPN
ma-381	320	3	,	,	PUNCT
ma-381	320	4	modern	modern	ADJ
ma-381	320	5	fourier	fourier	NOUN
ma-381	320	6	analysis	analysis	NOUN
ma-381	320	7	,	,	PUNCT
ma-381	320	8	2nd	2nd	ADJ
ma-381	320	9	ed	ed	NOUN
ma-381	320	10	.	.	PROPN
ma-381	320	11	,	,	PUNCT
ma-381	320	12	grad	grad	PROPN
ma-381	320	13	.	.	PUNCT
ma-381	321	1	texts	text	NOUN
ma-381	321	2	math	math	PROPN
ma-381	321	3	.	.	PUNCT
ma-381	321	4	,	,	PUNCT
ma-381	321	5	250	250	NUM
ma-381	321	6	,	,	PUNCT
ma-381	321	7	springer	springer	NOUN
ma-381	321	8	,	,	PUNCT
ma-381	321	9	new	new	PROPN
ma-381	321	10	york	york	PROPN
ma-381	321	11	,	,	PUNCT
ma-381	321	12	2009	2009	NUM
ma-381	321	13	.	.	PUNCT
ma-381	322	1	https://doi.org/10.28924/ada/ma.5.21	https://doi.org/10.28924/ada/ma.5.21	PROPN
ma-381	322	2	1	1	NUM
ma-381	322	3	.	.	PUNCT
ma-381	322	4	introduction	introduction	NOUN
ma-381	322	5	2	2	NUM
ma-381	322	6	.	.	PUNCT
ma-381	322	7	prerequisites	prerequisite	NOUN
ma-381	322	8	for	for	ADP
ma-381	322	9	hardy	hardy	ADJ
ma-381	322	10	-	-	PUNCT
ma-381	322	11	fofana	fofana	NOUN
ma-381	322	12	spaces	space	NOUN
ma-381	322	13	3	3	NUM
ma-381	322	14	.	.	X
ma-381	322	15	cauchy	cauchy	PROPN
ma-381	322	16	-	-	PUNCT
ma-381	322	17	riemann	riemann	PROPN
ma-381	322	18	equations	equation	NOUN
ma-381	322	19	,	,	PUNCT
ma-381	322	20	riesz	riesz	NOUN
ma-381	322	21	transforms	transform	VERB
ma-381	322	22	and	and	CCONJ
ma-381	322	23	hardy	hardy	ADJ
ma-381	322	24	-	-	PUNCT
ma-381	322	25	fofana	fofana	NOUN
ma-381	322	26	spaces	space	NOUN
ma-381	322	27	4	4	NUM
ma-381	322	28	.	.	PUNCT
ma-381	322	29	temperature	temperature	NOUN
ma-381	322	30	cauchy	cauchy	PROPN
ma-381	322	31	-	-	PUNCT
ma-381	322	32	riemann	riemann	PROPN
ma-381	322	33	equations	equation	NOUN
ma-381	322	34	and	and	CCONJ
ma-381	322	35	hardy	hardy	ADJ
ma-381	322	36	-	-	PUNCT
ma-381	322	37	fofana	fofana	NOUN
ma-381	322	38	spaces	space	NOUN
ma-381	322	39	references	reference	NOUN