Variable Lebesgue Spaces: Foundations and Harmonic Analysis

315 Pages·2013·3.011 MB·English

by  David V. Cruz-Uribe, Alberto Fiorenza (auth.)

Applied and Numerical Harmonic Analysis
SeriesEditor
JohnJ.Benedetto
UniversityofMaryland
CollegePark,MD,USA
EditorialAdvisoryBoard
AkramAldroubi JelenaKovacˇevic´
VanderbiltUniversity CarnegieMellonUniversity
Nashville,TN,USA Pittsburgh,PA,USA
AndreaBertozzi GittaKutyniok
UniversityofCalifornia TechnischeUniversita¨tBerlin
LosAngeles,CA,USA Berlin,Germany
DouglasCochran MauroMaggioni
ArizonaStateUniversity DukeUniversity
Phoenix,AZ,USA Durham,NC,USA
HansG.Feichtinger ZuoweiShen
UniversityofVienna NationalUniversityofSingapore
Vienna,Austria Singapore,Singapore
ChristopherHeil ThomasStrohmer
GeorgiaInstituteofTechnology UniversityofCalifornia
Atlanta,GA,USA Davis,CA,USA
Ste´phaneJaffard YangWang
UniversityofParisXII MichiganStateUniversity
Paris,France EastLansing,MI,USA
Forfurthervolumes:
http://www.springer.com/series/4968
David V. Cruz-Uribe Alberto Fiorenza
(cid:2)
Variable Lebesgue Spaces
Foundations and Harmonic Analysis
DavidV.Cruz-Uribe AlbertoFiorenza
DepartmentofMathematics DipartimentodiArchitettura
TrinityCollege Universita`diNapoli
Hartford “FedericoII”
Connecticut Napoli
USA Italy
ISBN978-3-0348-0547-6 ISBN978-3-0348-0548-3(eBook)
DOI10.1007/978-3-0348-0548-3
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013930225
MathematicalSubjectClassification(2010):42B20,42B25,42B35,46A19,46B25,46E30,46E35
(cid:2)c SpringerBasel2013
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection
with reviews or scholarly analysis or material supplied specifically for the purpose of being entered
and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of
this publication or parts thereof is permitted only under the provisions of the Copyright Law of the
Publisher’slocation,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.
PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violations
areliabletoprosecutionundertherespectiveCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
ThisbookrepresentsthefruitsofourcollaborationonthevariableLebesguespaces.
Our work in this area stretches back over a decade. Its genesis is memorable: it
began in Naples during an exceptionally cold January in 2002, shortly after the
introductionoftheeuro.Weworkedthroughthedetailsofapreprintwehadreceived
from our colleagueLars Diening which gaveconditionsfor the maximaloperator
tobeboundedonvariableLebesguespacesdefinedonboundedsets.Ourfirsttask
was to try to understand these previously unknown spaces. The more we worked
with them, the more intrigued we became. Subsequently, we began to study the
maximal operator on unbounded domains; a search for applications led us to the
otherclassical operatorsof harmonicanalysisandthe interplaybetweenweighted
norminequalitiesandvariableLebesguespaces.
In 2007, the first author was invited to teach a graduate course on variable
Lebesgue spaces at the University of Naples, Federico II, an invitation he gladly
accepted. The notes for that course became the basis for this book. One problem,
however,wasthatourknowledgeofthefieldcontinuedtoevolveevenaswetried
toconvertthosenotesintoa finalmanuscript.Insteadofwritingwe wouldstopto
prove new theorems, leading to repeated revisions and expansions of the text. At
thispoint,however,wethinkwehavereachedareasonableplacetostop:wehave
written(webelieve)anintroductiontovariableLebesguespacesthatwillbeuseful
forawideaudience.Simplyput,wethinkwehavefinallygottenitright.
Manyindividualshavecontributeddirectlyandindirectlytothisbook.Wewant
toacknowledgethelateChristophNeugebauer,whocollaboratedwithusonourfirst
paperonvariableLebesguespacesandprovidedkeyinsights.Wewanttothankour
colleaguesLarsDiening,PeterHa¨sto¨,AlesˇNekvindaandStefanSamko,whofreely
shared with us preprints of their work. Their generosity kept us abreast of a very
rapidlyevolvingfield.WewanttothankJeanMichelRakotosonforhiscollegiality
and for sharing with us his ideas and questions on variable Lebesgue spaces. We
also want to thank our colleague Claudia Capone and the students who attended
the variableLebesguespace coursefortheir patience aswe tried forthe first time
to shapeourknowledgeintoa coherentwhole.We especiallywanttothankCarlo
Sbordone,who first broughtus together and has providedcontinuingsupport and
v
vi Preface
encouragementforourjointlabors.Andfinally,wewanttothankourwivesandour
childrenforpatientlybearingwithusasthisbookbecameareality.
Hartford,CT,USA DavidV.Cruz-Uribe,SFO
Napoli,Italy AlbertoFiorenza
Contents
1 Introduction .................................................................. 1
1.1 AnOverviewofVariableLebesgueSpaces .......................... 2
1.2 ABriefHistoryofVariableLebesgueSpaces........................ 4
1.3 TheOrganizationofthisBook........................................ 8
1.4 PrerequisitesandNotation ............................................ 11
2 StructureofVariableLebesgueSpaces.................................... 13
2.1 ExponentFunctions ................................................... 13
2.2 TheModular........................................................... 17
2.3 TheSpaceLp.(cid:3)/.(cid:2)/.................................................... 18
2.4 Ho¨lder’sInequalityandtheAssociateNorm......................... 26
2.5 EmbeddingTheorems................................................. 35
2.6 ConvergenceinLp.(cid:3)/.(cid:2)/.............................................. 43
2.7 CompletenessandDenseSubsetsofLp.(cid:3)/.(cid:2)/ ....................... 54
2.8 TheDualSpaceofaVariableLebesgueSpace....................... 62
2.9 TheLebesgueDifferentiationTheorem .............................. 66
2.10 NotesandFurtherResults............................................. 68
2.10.1 References.................................................... 68
2.10.2 Musielak-OrliczSpacesandModularSpaces.............. 70
2.10.3 BanachFunctionSpaces..................................... 72
2.10.4 AlternativeDefinitionsoftheModular ..................... 74
2.10.5 VariableLebesgueSpacesandOrliczSpaces .............. 75
2.10.6 MoreonConvergence ....................................... 75
2.10.7 VariableSequenceSpaces................................... 77
3 TheHardy-LittlewoodMaximalOperator................................ 79
3.1 BasicProperties........................................................ 79
3.2 TheCaldero´n-ZygmundDecomposition ............................. 82
3.3 TheMaximalOperatoronVariableLebesgueSpaces............... 88
3.4 TheProofofTheorem3.16 ........................................... 93
3.5 ModularInequalities................................................... 107
3.6 InterpolationandConvexity........................................... 113
vii
viii Contents
3.7 NotesandFurtherRemarks........................................... 117
3.7.1 References.................................................... 117
3.7.2 MoreonModularInequalities............................... 118
3.7.3 LlogLInequalitiesinVariableLebesgueSpaces.......... 119
3.7.4 TheFractionalMaximalOperator........................... 120
3.7.5 HardyOperatorsonVariableLebesgueSpaces ............ 122
3.7.6 OtherMaximalOperators ................................... 124
3.7.7 DecreasingRearrangements................................. 126
3.7.8 RealandComplexInterpolation ............................ 127
4 BeyondLog-Ho¨lderContinuity ............................................ 129
4.1 ControlatInfinity:TheN Condition............................... 130
1
4.2 AUsefulTool:MuckenhouptA Weights ........................... 142
p
4.3 ApplicationsofWeightstotheMaximalOperator................... 152
4.4 LocalControl:TheK Condition..................................... 160
0
4.5 ANecessaryandSufficientCondition................................ 177
4.6 NotesandFurtherResults............................................. 180
4.6.1 References.................................................... 180
4.6.2 MoreontheK Condition................................... 181
0
4.6.3 DiscontinuousExponents.................................... 183
4.6.4 PerturbationofExponents................................... 184
4.6.5 WeightedVariableLebesgueSpaces........................ 185
5 ExtrapolationintheVariableLebesgueSpaces .......................... 191
5.1 BasicPropertiesofConvolutions..................................... 191
5.2 ApproximateIdentitiesonVariableLebesgueSpaces............... 197
5.3 TheFailureofYoung’sInequality .................................... 202
5.4 RubiodeFranciaExtrapolation....................................... 205
5.5 ApplicationsofExtrapolation......................................... 213
5.6 NotesandFurtherResults............................................. 227
5.6.1 References.................................................... 227
5.6.2 PointwiseEstimates.......................................... 228
5.6.3 MoreonApproximateIdentities ............................ 229
5.6.4 ApplicationsofExtrapolation............................... 230
5.6.5 SharpMaximalOperatorEstimates......................... 230
5.6.6 LocaltoGlobalEstimates ................................... 231
5.6.7 TheVariableRieszPotential ................................ 233
5.6.8 Vector-ValuedMaximalOperators.......................... 234
5.6.9 TwoClassicalPDEs.......................................... 235
5.6.10 TheFourierTransform....................................... 236
6 BasicPropertiesofVariableSobolevSpaces.............................. 239
6.1 TheSpaceWk;p.(cid:3)/.(cid:2)/ ................................................. 239
6.2 DensityofSmoothFunctions......................................... 243
6.3 ThePoincare´Inequalities ............................................. 249
6.4 SobolevEmbeddingTheorems ....................................... 252
Contents ix
6.5 NotesandFurtherResults............................................. 260
6.5.1 References.................................................... 260
6.5.2 AnAlternativeDefinitionoftheNorm ..................... 262
6.5.3 BoundaryRegularity......................................... 262
6.5.4 ExtensionTheorems......................................... 263
6.5.5 MoreontheDensityofSmoothFunctions ................. 264
6.5.6 MoreonthePoincare´Inequalities........................... 265
6.5.7 MoreontheSobolevEmbeddingTheorem................. 266
6.5.8 CompactEmbeddings ....................................... 268
6.5.9 MeanContinuity............................................. 269
6.5.10 Gagliardo-NirenbergInequalities ........................... 270
A Appendix:OpenProblems.................................................. 271
Bibliography...................................................................... 279
SymbolIndex..................................................................... 295
AuthorIndex...................................................................... 301
SubjectIndex..................................................................... 305
Chapter 1
Introduction
The variable Lebesgue spaces, as their name implies, are a generalization of
the classical Lebesgue spaces, replacing the constant exponent p with a variable
exponent function p.(cid:3)/. The resulting Banach function spaces Lp.(cid:3)/ have many
properties similar to the Lp spaces, but they also differ in surprising and subtle
ways. For this reason the variable Lebesgue spaces have an intrinsic interest, but
they are also very important for their applications to partial differentialequations
and variational integrals with non-standard growth conditions. The past 20 years,
andespeciallythepastdecade,havewitnessedanexplosivegrowthinthestudyof
theseandrelatedspaces.
The goal of this book is to provide an introduction to the variable Lebesgue
spaces. We first establish their structure and function space properties, paying
special attention to the differences between bounded and unbounded exponents.
Next,wedevelopthemachineryofharmonicanalysisonvariableLebesguespaces.
We first concentrate on the Hardy-Littlewood maximal operator, and then extend
theRubiodeFranciatheoryofextrapolationtothissetting.Todosoweintroduce
thetheoryofMuckenhouptA weightsandweightednorminequalities.Withthese
p
toolswecanthenstudyotheroperators,particularlyconvolutionoperators,singular
integraloperatorsandRieszpotentials.Finally,asanapplicationoftheseresultswe
givetheessentialpropertiesofthevariableSobolevspaces.
In writing this book we had two differentaudiencesin mind. First, we wanted
towriteanintroductionsuitableforresearchersandstudentsinterestedinlearning
aboutthevariableLebesguespaces.Atthesametime,wehopedtocreateauseful
referenceformathematiciansalreadyactiveinthearea.Forbothaudienceswehave
provided a coherent treatment of the material—in terms of notation, hypotheses
and overall point of view—and thereby united results by many authors from a
rapidlyevolvingfield.Wehavealsoincludedaconciseintroductionofweightsand
weighted norm inequalities. These have become veryimportanttools in the study
ofthe variableLebesguespaces, andwe havegivena carefultreatmentofthe key
ideasneededtousethem.
We have not, however, merely summarized existing work. We have included
manynewandpreviouslyunpublishedresultsandnewproofsofknownresults.Our
D.V.Cruz-UribeandA.Fiorenza,VariableLebesgueSpaces,AppliedandNumerical 1
HarmonicAnalysis,DOI10.1007/978-3-0348-0548-3 1,©SpringerBasel2013