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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

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