Martingale Hardy Spaces and Summability of One-Dimensional Vilenkin-Fourier Series

633 Pages·2022·6.371 MB·English

by  Lars-Erik Persson, George Tephnadze, Ferenc Weisz

Lars-Erik Persson
George Tephnadze
Ferenc Weisz
Martingale Hardy
Spaces and
Summability of
One-Dimensional
Vilenkin-Fourier
Series
Martingale Hardy Spaces and Summability
of One-Dimensional Vilenkin-Fourier Series
Lars-Erik Persson • George Tephnadze (cid:129)
Ferenc Weisz
Martingale Hardy Spaces
and Summability
of One-Dimensional
Vilenkin-Fourier Series
Lars-ErikPersson GeorgeTephnadze
UiTTheArticUniversityofNorway SchoolofScienceandTechnology
Narvik,Norway UniversityofGeorgia
Tbilisi,Georgia
KarlstadUniversity
Sweden
FerencWeisz
DepartmentofNumericalAnalysis
EötvösLorándUniversity
Budapest,Hungary
ISBN978-3-031-14458-5 ISBN978-3-031-14459-2 (eBook)
https://doi.org/10.1007/978-3-031-14459-2
Mathematics Subject Classification: 40F05, 42A38, 42B25, 42B05, 42B08, 42B30, 42C10, 43A75,
60G42
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Preface
The classical Fourier Analysis has been developedin an almost unbelievableway
from the first fundamental discoveries by J. J. Fourier (1768–1830). Especially
a number of wonderful results have been proved, and new directions of such
researchhavebeendeveloped,e.g.,concerningwavelettheory,Gabortheory,time-
frequency analysis, fast Fourier transform, abstract harmonic analysis, etc. One
importantreasonforthisisthatthisdevelopmentisnotonlyimportantforimproving
the “state of the art”, but also for its importance in other areas of mathematics
and also for several applications(e.g. theory of signal transmission, multiplexing,
filtering, image enhancement,coding theory, digital signal processing and pattern
recognition).
TheclassicaltheoryofFourierseriesdealswithdecompositionofafunctioninto
sinusoidal waves. Unlike these continuous waves, the Vilenkin (Walsh) functions
arerectangularwaves.ThedevelopmentofthetheoryofVilenkin-Fourierserieshas
beenstronglyinfluencedbytheclassicaltheoryoftrigonometricseries.Becauseof
this,itisinevitabletocompareresultsofVilenkinseriestothoseontrigonometric
series.Therearemanysimilaritiesbetweenthesetheories,butthereexistdifferences
also.Muchofthesecanbeexplainedbymodernabstractharmonicanalysis,which
studiesorthonormalsystemsfromthepointofviewofthestructureofatopological
group.
The classical theory of Lebesgue Lp spaces started to be developed around
1910. The weak-Lp spaces are function spaces which are closely related to Lp
spaces.Oneofthemostimportantearlyapplicationsofthesespacesweremadeby
J. Marcinkiewiczin 1939,where he provedMarcinkiewicz interpolationtheorem.
Concerningtheimportanceofthistheorem,thecorrespondingso-calledweak-type
estimates and further developmentto which nowadays is called real interpolation
methodwereferto[30].
The Hardy spaces H were introduced by Riesz [281] in 1923, who named
p
them after G. H. Hardy, because of the paper [156] from 1915. Such classical
investigationsofHardyspacesbyG.H.Hardy,F.andM.Riesz,J.E.Littlewoodand
others of H employed complex methods and showed that, for several problems
p
v
vi Preface
in Fourier Analysis, the use of the H (0<p ≤1) scale of spaces is preferable
p
over that of the Lp(0<p ≤1) scale. During four decades, a powerfultheory of
H (0<p ≤1) spaces, especially in the n-dimensionalcase, has been developed
p
by means of real methods, and various new applications to Fourier Analysis and
singularoperatorshavebeengiven,seee.g.CoifmanandWeiss[70],Feffermanand
Stein[93],TaiblesonandWeiss[339]andthereferencesgiventherein.
The history of martingaletheorygoesback to the earlyfifties when Doob[85]
pointedouttheconnectionbetweenmartingalesandanalyticfunctions.Onthebasis
of Burkholder’s scientific achievements (see [59–61]), the theory of continuous
parametermartingalescanperfectlywellbeappliedincomplexanalysisandinthe
theoryofclassicalHardyspaces.ThisconnectionisthemainpointofDurrett’sbook
[87]. Attention was also drawn to martingale Hardy spaces when Burkholder and
Gundy[151,152]provedtheinequalitynamedafterthemeversince.Thisinequality
states that the Lp normsof the maximalfunction and the quadratic variation of a
one parameter martingale are equivalent for 1 < p < ∞. Some years later, this
resultwasextendedtop = 1byDavis[75].In1973thedualoftheone-parameter
martingale Hardy space generated by the maximal function was characterized by
Garsia[105]andHerz[161]asthespaceoffunctionsofboundedmeanoscillation
(BMO).Next,wementionthatsomeimportantstepsintheearlydevelopmentcan
be found in the book by Schipp et al. [295] from 1990. The research continued
intensivelyalsoafterthis.Someofthemostimportantstepsinthesedevelopments
are presentedin the two books[400] and [423] by F. Weisz from 1994and 2002,
respectively.
Theaim ofthisbookistodiscuss, developandapplythenewestdevelopments
of this fascinating theory connected to modern harmonic analysis. In particular,
we present and prove new estimations of the Vilenkin-Fourier coefficients and
prove some new results concerning boundedness of maximal operators of partial
sums. Moreover, we derive necessary and sufficient conditions for the modulus
of continuity so that norm convergence of the partial sums is valid and develop
new methods to prove Hardy type inequalities for the partial sums with respect
to the Vilenkin systems. We also do the similar investigationfor the Fejér means.
Furthermore,we investigatesome Nörlundmeans but only in the case when their
coefficientsaremonotone.Somewell-knownexamplesofNörlundmeansareFejér
means, Cesàro means and Nörlund logarithmic means. In addition, we consider
Riesz logarithmic means, which are not examples of Nörlund means. It is also
provedthattheseresultsarethebestpossibleinaspecialsense.Asapplicationsboth
some well-knownand new results are pointed out. Finally, we want to pronounce
that we prove and discuss the analogy of the famous result of Carleson-Hunt
concerningalmosteverywhereconvergenceofpartialsumsforFourierseriesinLp
spaces,p >1inthecaseofVilenkinsystems.
The bookcontainsnine chaptersand oneappendix,which containssome basic
factsconcerningWalshandKaczmarzsystems.Onereasonforthisisthatitwillbe
more convenient for the reader to compare with the classical theory, and another
reason is that it gives us a possibility to raise new open questions. It is maybe
surprisingthatsomeoftheseopenquestionsconcerntheclassicalsituationbutare
Preface vii
motivatedbytheresultsweprovedinthisnewsituation.Infact,wehaveespecially
pointed out a great number of open questions in this book. We hope that this can
stimulate the further development of this fascinating area. We now continue by
describingthemaincontentofeachofthechapters.
In Chap.1, we first define Vilenkin groups and functions and study basic
properties. We include some necessary preliminaries, in particular some classical
inequalities,andwe evenpresentproofsoftheseinequalitiesbyusingaconvexity
approach,whichwethinkisnotpresentedinthisforminsomebookbefore.After
that,weinvestigatetheclassicaltheoryofLebesguespacesandweak-Lpspacesand
westateandproveMarcinkiewiczinterpolationtheorem.WestudyDirichletkernels
and Lebesgue constants and prove two-sided estimates for them, which are very
crucial to prove important results in Chap.6. We study some classical conditions
in the space of integrable functions, which provide pointwise convergence and
convergenceinL1 normofpartialsums.We also givesomeequivalentdefinitions
of the modulus of continuity of Lp functions and give necessary and sufficient
conditionsconcerningnormconvergenceofpartialsumsinL1.
In Chap.2, we define martingales, and by using technique of martingale the-
ory, we prove boundedness of maximal functions on Lebesgue spaces, from
which it follows almost everywhereconvergenceof subsequencesof partial sums
with respect to Vilenkin systems. We also give the proof of Calderon-Zygmund
decomposition theorem. Moreover, we prove an analogy of the Carleson-Hunt
theorem with respect to Vilenkin systems. Finally, we prove an analogy of the
KolmogorovtheoremandconstructtheintegrablefunctionwhoseVilenkin-Fourier
seriesdivergeseverywhere.Itisalso provedthatforany1 ≤ p ≤ ∞ andforany
setwithmeasure0,thereexistsafunctionf ∈ Lp,whoseVilenkin-Fourierseries
divergesonthisset.
In Chap.3, we first define Fejér means. We investigateFejér kernelsand study
two-sidedestimatesforthem.Afterthatweprovealmosteverywhereconvergence
of Fejér means. Moreover, we define Lebesgue and Vilenkin-Lebesgue points of
integrablefunctionsandproveconvergenceofFejérmeansofintegrablefunctions
with respect to Vilenkin systems in Vilenkin-Lebesgue points, which are almost
everywhere points for any integrable function. We also study convergence in Lp
normofFejérmeans.Moreover,someimportantupperandlowerestimatesofFejér
kernels are studied, which are very crucial to prove important results in Chap.7.
Moreover, we define what an approximate identity is and study some conditions,
which provide convergenceof convolution operators by approximate identity and
Lp functions.
InChap.4,wefirstdefinesomesummabilitymethods,whicharecalledNörlund
andT meansandderivenecessaryandsufficientconditionswhichprovideregularity
of these summability methods. We prove new estimates for the kernels of these
summability methods, which are very important to prove our main results in
Chap.8. Moreover, we study some conditions for Nörlund and T means from
whichitfollowsconvergenceinLp norms,whenp ≥ 1.Finally,westudyalmost
everywhereconvergenceofNörlundandT meansinLp spaces,whenp ≥1.
viii Preface
InChap.5,wefirstdefinemartingalesandgivesomebasicresults,whichweuse
intheproofsofourmainresults.Moreover,wedefinemartingaleHardyspacesand
provethattheyareisomorphictoLebesguespacesforp >1.Inthecasewhen0<
p ≤ 1weprovesomeimportanttheoremsforthistheoryandalsoacrucialatomic
decompositionresult for these spaces, which, in particular,simplify our proofsof
boundednessofsome classicaloperatorsinFourieranalysis.Finally,we construct
concrete martingales, which help us to provesharpness of our main results in the
laterchapters.Thistechniquetoprovesharpnessisfairlynewandhopefullyuseful
alsofarbeyondthemainscopeofthisbook.
Chapter 6 is devoted to present and prove some new and known results about
Vilenkin-FouriercoefficientsandpartialsumsofmartingalesinHardyspaces.First,
we show that the Fourier coefficients of martingales f ∈ H are not uniformly
p
bounded when 0 < p < 1. By applying these results, we can prove some
known Hardy and Paley type inequalities with a new more simple method. After
that, we investigate partial sums with respect to the Vilenkin system and prove
boundednessofweighedmaximaloperatorsofpartialsumsonthemartingaleHardy
spaces.Moreover,wederivenecessaryandsufficientconditionsforthemodulusof
continuityforwhichnormconvergenceofthe partialsumshold,andwe presenta
new proof of a Hardy type inequality for it. We also investigate convergenceand
divergencerateofsubsequencesofpartialsumswithrespecttoVilenkinsystemsin
themartingaleHardyspacesfor0 < p < 1.Finally,weprovestrongconvergence
resultsofpartialsumswithrespecttoVilenkinsystems.
InChap.7,weinvestigatesomeanalogousproblemsconcerningthepartialsums
of Fejér means. First, we consider maximal operator Fejér means. Moreover, we
investigate some weighted maximal operators of Fejér means and prove some
boundednessresults for them. After that, we apply these results to find necessary
andsufficientconditionsforthemodulusofcontinuityforwhichnormconvergence
ofFejérmeansholds.Finally,weprovesomenewHardytypeinequalitiesforFejér
means, which are also called strong convergenceresults of Fejér means. We also
provesharpnessofallourmainresultsinthisChapter.
In Chap.8, we consider boundedness of maximal operators and weighted
maximal operators of Nörlund and T means. After that, we prove some strong
convergencetheorems for these summability methods. Since Fejér, Cesàro, Riesz
andNörlundlogarithmicmeansareexamplesofNörlundandT means,bothsome
well-known and new results can be pointed out. We also investigate Riesz and
NörlundlogarithmicmeanssimultaneouslyattheendofthisChapter.
In Chap.9, we consider Hardy spaces with variable exponents. Let p(·) be
a measurable function defined on [0,1) satisfying p− =: infx∈(cid:2)p(x) > 0,
supx∈(cid:2)p(x) =: p+ < ∞andthelog-Höldercontinuitycondition.We investigate
themartingaleHardyspacesHp(·)andprovetheiratomicdecompositions.Similarly
tothespaceswithaconstantp,weobtainthattheHardyspacesHp(·)areequivalent
to the Lebesgue spaces Lp(·) if p− > 1. We generalize the classical results and
show that the partial sums of the Vilenkin-Fourierseries convergeto the function
innormiff ∈ Lp(·) andp− > 1.TheboundednessofthemaximalFejéroperator
Preface ix
on Hp(·) is proved whenever p− > 1/2 and the condition p1− − p1+ < 1 hold.
Oneofthekeypointsoftheproofisthatweintroducetwonewmaximaloperators
andprovetheir boundednesson Lp(·) with p− > 1.As a consequence,we obtain
theoremsaboutalmosteverywhereandnormconvergenceoftheFejérmeansofthe
Vilenkin-Fourierseries.
For the readers’ convenience, we have also included an Appendix (Chap.10)
withsomebasicinformationaboutthedyadicgroupandtheWalsh andKaczmarz
systems.Moreover,weprovesomeinterestingresultsconcerningsummabilitywith
respect to the Walsh system, which are not known for Vilenkin systems. We also
giveaproofconcerningboundednessofthemaximaloperatorofFejérmeanswith
respecttotheKaczmarzsystemandsharpnessofthisresult.Inparticular,thisfairly
newmethodofproof,evenofsharpness,inthecaseofKaczmarzsystemcanbevery
usefulfor researcherslookingfor the results for similar Walsh system and related
systems.
HowtoRead the Book?
Each chapter is divided into sections, similarly as the numbers of the formulas.
Hence, for example 4.5 means the fifth Section of Chap.4 and (4.3.6) means the
sixth formula in Sect.4.3 of Chap.4. For the convenience of the reader, we have
alsoaddedalistofsymbolsandnotationsattheendofthebook.
Thethreefirstchaptersarebasicincludingmanydefinitions,basictheorems,etc.
This part can be used as an introduction, e.g., in a course for PhD students and
researchers with interest to broaden the knowledge in Fourier Analysis with new
problemsandthinking.Moreover,thispartisbasicandthebestistoreaditbefore
toreadtheothersixchapters,which,intheirturn,canbereadindependentofeach
other.However,itisalsopossibletostarttoreadanychapterindependentlybyjust
on some places going back to limited and well-described information from some
previouschapters.
Attheendofeachchapter,thereisasectioncalled“FinalCommentsandOpen
Questions”,wherewehavecollectedsomehistoricalandotherremarks,pointedout
some relations to the theory of classical Fourier Analysis and raised a number of
openquestions.
Especially,we hopethatthis interplaybetweenclassical and“modern”Fourier
Analysis and some corresponding open questions will be very useful for a broad
audienceofreadersandserveasasourceofinspirationforfurtherresearchinthis
fascinatingarea.
Luleå,Sweden Lars-ErikPersson
May2022 GeorgeTephnadze
FerencWeisz
Acknowledgements
ThesecondauthorwantstothankShotaRustaveliNationalScienceFoundationfor
the financial support within the frame of project no. FR-19-676. The third author
was supported by the Hungarian National Research Development and Innovation
Office—NKFIH,KH130426.
We are very grateful to several researchers around the world (co-authors,
colleagues, etc.) for various kinds of contributions and support, which essentially
have contributed to improve the quality of this book. As typical examples of
researchers in this “supporting team”, we want to mention Roland Duduchava,
Hans-GeorgFeichtinger,NatashaSamko,FerencSchipp,ZurabVashakidze,Georgi
TutberidzeandDavitBaramidze.
WealsothankDr.HanaTurcinováforpermittingustouseherphotoofthemagic
Nordiclightwechosenasasymbolandwhichistakenexactlyattheplace(“Hotel
Infinity”)wherewefinalizedthebook.
Finally,andmostimportant,ourmostcordialthanksgotoourwonderfulfamilies
for their patience, encouragement,support and love during all our work with this
book.
xi