Table Of ContentLecture Notes in Mathematics 2320
Yinqin Li
Dachun Yang
Long Huang
Real-Variable
Theory of Hardy
Spaces Associated
with Generalized
Herz Spaces
of Rafeiro and Samko
Lecture Notes in Mathematics
Volume 2320
Editors-in-Chief
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Yinqin Li (cid:129) Dachun Yang (cid:129) Long Huang
Real-Variable Theory
of Hardy Spaces Associated
with Generalized Herz
Spaces of Rafeiro and Samko
YinqinLi DachunYang
LaboratoryofMathematicsandComplex (CorrespondingAuthor)
Systems(MinistryofEducationofChina) LaboratoryofMathematicsandComplex
SchoolofMathematicalSciences Systems(MinistryofEducationofChina)
BeijingNormalUniversity SchoolofMathematicalSciences
Beijing,Beijing,China BeijingNormalUniversity
Beijing,Beijing,China
LongHuang
SchoolofMathematicsandInformation
Science
KeyLaboratoryofMathematicsand
InterdisciplinarySciencesofthe
GuangdongHigherEducationInstitute
GuangzhouUniversity
Guangzhou,Guangdong,China
This work was supported by National Key Research and Development Program of
China (2020YFA0712900), National Natural Science Foundation of China (11971058)
(12071197), and Basic and Applied Basic Research Foundation of Guangdong Province
(2021A1515110905)
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-981-19-6787-0 ISBN978-981-19-6788-7 (eBook)
https://doi.org/10.1007/978-981-19-6788-7
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Preface
It is well known that Herz spaces certainly play an important role in harmonic
analysisandpartialdifferentialequationsandhavebeensystematicallystudiedand
developedsofar;see,forinstance,[79,146,150,174,275]forclassicalHerzspaces,
[83,239–241]forweightedHerzspaces,[132,196,202,235–238,276]forvariable
Herz spaces, [61, 68, 69, 93, 179, 262, 263] for Herz-type Hardy spaces, [59, 62,
64,67,257–259]forHerz-typeBesovspaces,and[57,60,212,256,260,261,264]
for Herz-type Triebel–Lizorkinspaces. Observe that the classical Herz space was
originally introduced by Herz [104] in 1968 to study the Bernstein theorem on
absolutely convergent Fourier transforms, while the research on Herz spaces can
be traced back to the work of Beurling [13]. Indeed, in 1964, to study some
convolutionalgebras,Beurling [13] first introduceda special Herz space Ap(Rn),
with p ∈ (1,∞)[see Remark 7.1.2(iv)forits definition],whichis also calledthe
Beurlingalgebra.Afterthat,greatprogresshasbeenmadeonHerzspacesandtheir
applications. For instance, in 1985, Baernstein and Sawyer [9] generalized these
Herzspacesandgavemanyapplicationsinboththe embeddingandthemultiplier
theoremsforclassicalHardyspacesonthen-dimensionalEuclideanspace;in1984,
tostudytheWienerthirdTauberiantheoremforthen-dimensionalEuclideanspace,
Feichtinger[79]introducedanothernormofAp(Rn),whichisobviouslyequivalent
tothenormdefinedbyBeurling[13].
Moreover,Herzspacesplayacrucialroleintheconvergenceandthesummability
problems of both Fourier transforms and Fourier series. Recall that the study of
summability meanswas originally motivatedby the famousconvergenceproblem
ofDirichletintegraloperators.Asone ofthe deepestresultsin harmonicanalysis,
in the celebrated works of both Carleson [28] and Hunt [129], they showed that
Dirichletintegraloperatorsconvergealmosteverywhereinone-dimensionalcase.In
recentdecades,viareplacingDirichletintegraloperatorsbysomeothersummability
means,thesummabilityofFouriertransformswassystematicallystudiedbyButzer
andNessel[19],TrigubandBelinsky[230],andFeichtingerandWeisz[80–82]as
wellasWeisz[246–253].Particularly,Weisz[254,255]establishedtheconnection
v
vi Preface
between the Herz spaces and the summability of both Fourier series and Fourier
transforms.Letusgivemoredetailsonthelattercase.Tothisend,let
θ ∈L1(Rn)∩C (Rn),
0
where C (Rn) denotes the set of all the continuous functions f on Rn satisfying
0
that
lim |f(x)|=0.
|x|→∞
Recallthat,foranygivenp ∈ [1,2]andT := (T ,...,T ) ∈ (0,∞)n andforany
1 n
f ∈Lp(Rn),theθ-meanoff isdefinedbysetting,foranyx ∈Rn,
(cid:2) (cid:3) (cid:4)
σθf(x):= θ −ξ1,··· ,−ξn f(cid:5)(ξ)e2πix·ξdξ,
T Rn T1 Tn
√
wheref(cid:5)denotestheFouriertransformoff,i := −1,and
(cid:6)n
x·ξ := x ξ
j j
j=1
for any x := (x ,··· ,x ),ξ := (ξ ,...,ξ ) ∈ Rn (see, for instance, [82,
1 n 1 n
(3·1)]). Then, in the recent book [254], Weisz showed that the θ-means of some
functions converge to these functions themselves at all their Lebesgue points of
thesefunctionsunderconsiderationifandonlyiftheFouriertransformofθ belongs
to a suitable Herz space. This means that those Herz spaces are the best choice
in the study of the summability of Fourier transforms. Furthermore, Herz spaces
also prove important in the recent article [207] of Sawano et al. Indeed, in 2017,
Sawanoetal.[207]introducedboththeballquasi-BanachfunctionspaceXandthe
associated Hardy space H (Rn) via the grand maximal function. As was pointed
X
outin[207],Sawanoetal.useda certaininhomogeneousHerzspacetoovercome
the essential difficulty appearing in the proof of the convergence of the atomic
decompositionofH (Rn).Also,Herz-typespacesproveusefulinthestudyrelated
X
to partial differential equations. For instance, Scapellato [210] showed that the
variable Herz spaces are the key tools in the study of the regularity of solutions
to elliptic equations. Drihem [65, 66] studied semilinear parabolic equations with
initial data in Herz spaces or Herz-type Triebel–Lizorkin spaces. In addition, the
Fourier–Herz space has been proved to be one of the most suitable spaces to
investigate the global stability for fractional Navier–Stokes equations; see, for
instance,[31,41,159,190].FormoreprogressonapplicationsofHerzspaces,we
referthereaderto[74,112,114,116,126,180,184,189,195,274].
Ontheotherhand,asagoodsubstituteofHerzspaces,Herz-typeHardyspaces
are also usefulin manymathematicalfields such as harmonicanalysis and partial
Preface vii
differentialequations,whichhavebeensystematicallystudiedanddevelopedsofar;
see, for instance, [32, 40, 63, 86, 120, 158, 279]. Recall that the study of Hardy
spaces can be traced back to the works of Hardy and Littlewood via tools from
complex analysis (see [97, 98, 152]). Then, based on the real-variable methods
introducedbyCalderónandZygmundinthe1950s(see[25,26,281]),theclassical
real Hardy space Hp(Rn) was originally initiated by Stein and Weiss [215] and
thensystematicallydevelopedbyFeffermanandStein[77].Lateron,Coifman[44]
establishedtheatomiccharacterizationoftheHardyspaceHp(R);Strömberg[216]
showed that an orthonormalwavelet basis is always an unconditionalbasis of the
HardyspaceH1(Rn),andthisremarkablediscoverypreludedthewaveletanalysis.
Moreover, Calderón et al. [21–23] investigated the Cauchy integral on Lipschitz
curvesusingweightedHardyspaces,andKenig[143,144] studiedthe(weighted)
Hardy spaces on Lipschitz domains. For more developmentsof Hardy spaces, we
referthereaderto[105,106,148,160,186,187]forthereal-variabletheoryofHardy
spaces as well as [46, 71, 183, 211, 213, 243, 244] for the applications of Hardy
spaces to both harmonic analysis and partial differential equations. Furthermore,
as a variant of the classical real Hardy spaces, the Hardy spaces associated with
the Beurling algebras on the real line were first introduced by Chen and Lau
[40] in 1989, in which they studied the dual spaces and the maximal function
characterizations of these Herz–Hardy spaces. Later, García-Cuerva [86] in 1989
generalized the results of Chen and Lau [40] to higher-dimensional case, and
García-CuervaandHerrero[87]in 1994furtherstudiedthemaximalfunction,the
atomic, and the Littlewood–Paley function characterizationsof these Herz–Hardy
spaces. Also, since 1990, Lu and Yang made a series of studies on the Hardy
spaces associated with the Beurling algebras or with the Herz spaces (see, for
instance, the monograph [175] and its references). In particular, in 1992, Lu and
Yang [162] first studied both the Littlewood–Paley function and the φ-transform
characterizationsof the Herz–Hardy space HK (Rn). Moreover,in 1995,Lu and
2
Yang[163]establishedtheatomicandthemolecularcharacterizationsoftheHerz–
Hardy space with general indices and then, in 1997, they further gave various
maximalfunctioncharacterizationsofweightedHerz–Hardyspacesin[172].
Meanwhile, Lu and Yang [164] proved that some oscillatory singular integral
operators are bounded from Herz–Hardy spaces to Herz spaces and showed that
their boundednessfails on Herz–Hardy spaces via a counterexample. Recall that,
as was pointed out by Pan [193], the oscillatory singular integral operators may
not be bounded from the classical Hardy space H1(Rn) to the Lebesgue space
L1(Rn). Thus, in some sense, the results of [164] showed that the Herz–Hardy
spaceisapropersubstitutionofH1(Rn)inthestudyonoscillatorysingularintegral
operators.In addition,the commutatorsand the multiplier theoremson the Herz–
Hardyspaces were investigatedby Lu and Yang,respectively,in [170] and [173];
boththeinterpolationofgeneralizedHerzspacesanditsapplicationsweregivenin
HernándezandYang[102,103];weightedHerzspacesandtheirapplicationswere
considered by Lu and Yang in [166]; embedding theorems for Herz spaces were
studied in [167]. Furthermore,Hernándezet al. [101] established the φ-transform
and the wavelet characterizations for some Herz and Herz-type Hardy spaces by
viii Preface
meansofalocalversionofthediscretetentspacesattheorigin.Formoreprogress
on the Herz–Hardy spaces on Rn and their applications, we refer the reader to
[111,119,134,165,169,171,175,198,245].Moreover,Herz-typespacesonlocal
fieldsandVilenkingroupswerealsoinvestigatedbyFan,Lu,andYang,respectively,
in[168]and[73,174,268].
Nowadays, more and more new function spaces continually spring up to meet
the increasing requirements arising in harmonic analysis and partial differential
equations;see,forinstance,[1,15,65,185,197,199,270].Particularly,extending
theclassicalHerzspacestosomemoregeneralsettingshasalsoattractedconsider-
ableattentionrecently.Forinstance,RafeiroandSamko[197]creativelyintroduced
local and global generalized Herz spaces recently, which are the generalization
of classical homogeneousHerz spaces and connect with generalized Morrey type
spaces. To be precise, Rafeiro and Samko [197] showed that the scale of these
Herz spaces include Morrey type spaces and complementary Morrey type spaces
and, as applications, they also obtained the boundedness of a class of sublinear
operators on these generalized Herz spaces. Note that Morrey type spaces have
been studied in [2, 3, 191, 203, 204]. Moreover, as a generalization of classical
variable Herz spaces introduced by Izuki [130], Rafeiro and Samko [199] further
extendedthegeneralizedHerzspacestothevariableexponentsetting.Furthermore,
in [199], Rafeiro and Samko showed that the generalized variable exponentHerz
spaces coincide with the generalized variable exponent Morrey type spaces, and
alsoestablishedsomeboundednessofsublinearoperatorsonthesespaces.
Observe that Herz-type Hardy spaces have found lots of applications in many
branchesof mathematicsand that both the real-variable theory of function spaces
and their applications are always one of the central topics of harmonic analysis.
Thenitisanaturalandmeaningfultopictointroduceanddevelopthereal-variable
theory of Hardy spaces associated with local and global generalized Herz spaces
of Rafeiro and Samko [197], which is the subjectof the presentbook. To achieve
this,wefirstinvestigatesomebasicpropertiesofthesegeneralizedHerzspacesand
realize that the Hardy spaces associated with generalized Herz spaces completely
fallintotheframeworkofHardyspacesassociatedwithballquasi-Banachfunction
spaces which were first studied by Sawano et al. [207] as was mentioned above.
Indeed,aboutthe real-variabletheory of Hardy spaces associated with ball quasi-
Banachfunctionspaces,Sawanoetal.[207]triedtoanswerthefollowingimportant
issue:Characterizethequasi-BanachspaceXforwhichtheassociatedHardyspace
H (Rn) can be described by various maximal functions, atomic decompositions,
X
moleculardecompositions,Littlewood–Paleyfunctions,andsoon.Tobeprecise,let
Xbeaballquasi-BanachfunctionspaceonRn(see[207]orDefinition1.2.13below
foritsdefinition).Recallthat,ifthepoweredHardy–Littlewoodmaximaloperators
areboundedonbothXanditsassociatespace,andifXsupportsaFefferman–Stein
vector-valuedinequality,thenSawanoetal.[207]establishedvariousreal-variable
characterizations of the Hardy space H (Rn) associated with X, respectively, in
X
termsofatoms,molecules,andtheLusinareafunction.AftertheworkofSawano
et al. [207], the real-variabletheoryof functionspaces associated with ball quasi-
