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Svetlin G. Georgiev Theory of Distributions Second Edition Theory of Distributions Svetlin G. Georgiev Theory of Distributions Second Edition SvetlinG.Georgiev DepartmentofDifferentialEquations FacultyofMathematicsandInformatics UniversityofSofia“St.KlimentOhridski” Sofia,Bulgaria ISBN978-3-030-81264-5 ISBN978-3-030-81265-2 (eBook) https://doi.org/10.1007/978-3-030-81265-2 1stedition:©SpringerInternationalPublishing2015 2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewhole orpart ofthematerial isconcerned, specifically therights oftranslation, reprinting, reuse ofillustrations, recitation, broadcasting, reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface to the Second Edition In the 5 yearssince the first edition of this book was published,I have receiveda lot of messages and letters from readers commentingon the book and suggesting howitcouldbeimproved.Withtheaimofthisinformation,Ihaverevisedthefirst editionofthebook.Thechangesitssecondeditionareasfollows. In Chap. 1, a section titled “Lp Spaces” has been added. In this section, Lp spaces for p ≥ 1 are introduced. They are deduced from the Hölder, Young, Minkowski inequalities and an interpolation inequality. In this section, certain criteriaforuniformintegrabilityofsomeclassesoffunctionsaregiven.TheRiesz- FischertheoremandtheLp dominatedconvergencetheoremarestudied.Also,the conceptionforseparableanddualspacesisintroducedandtheRieszrepresentation theorem for bounded linear functionals on Lp spaces is proven. A new section titled "Change of Variables" has been added to Chap. 2. Here, the change of variables for some classes of distributions is investigated and a representation of the Dirac delta function is provided. In Chap. 3, a new section titled “The Local StructureofDistributions”hasbeenadded.Inthissection,acriterionforlinearand continuous extension of a distribution with compact support is provided and that anydistributionwithcompactsupporthasafiniteorderisproven. Anewsectioncalled“NotesandReferences”hasbeenintroducedinallchapters. In this section, some additional materials are provided for each chapter. Some problemsareprovidedwithdetailedproofs.Thebook’sindexhasbeenupdated. Theaimofthesecondeditionistopresentaclearandwell-organizedtreatment of the concept behind the development of mathematics and solution techniques. The material of this book is presented in a highly readable, mathematically solid format.Manypracticalproblemsareillustrated,displayingthescopeofthetheory ofdistributions. Paris,France SvetlinG.Georgiev February2021 v Preface to the First Edition The theoryof partialdifferentialequationsis withouta doubtoneof the branches ofanalysisinwhichideasandmethodsofdifferentfieldsofmathematicsmanifest themselves and are interlaced—from functional and harmonic analysis to differ- ential geometry and topology. Because of that, the study of this topic represents a constant endeavour and requires undertaking several challenges. The main aim of this bookis to explainmanyof the fundamentalideasunderlyingthe theoryof distributions. The book consists of ten chapters. The first chapter deals with the well-known classicaltheoryregardingthespaceC∞,theSchwartzspaceandtheconvolutionof locallyintegrablefunctions.Itmayalsoserveasanintroductiontotypicalquestions related to cones in Rn. Chapter 2 collects the definitions of distributions, their order,sequences,supportandsingularsupport,andmultiplicationbyC∞functions. In Chaps. 3 and 4 we introduce differentiation and homogeneous distributions. The notion of direct multiplication of distributions is developed in Chap. 5. The followingtwoChaps.6and7,dealwithspecificproblemsaboutconvolutionsand tempereddistributions.InChaps.8and9wecollectedbasicmaterialandproblems regardingintegraltransforms.Sobolevspacesarediscussedinthetenth,andfinal, chapter. The volume is aimed at graduate students and mathematicians seeking an accessible introduction to some aspects of the theory of distributions, and is well suitedforaone-semesterlecturecourse. ItisapleasuretoacknowledgethegreathelpIreceivedfromProfessorMokhtar Kirane, University of La Rochelle, La Rochelle, France, who made valuable suggestionsthathavebeenincorporatedinthetext. Iexpressmygratitudeinadvancetoanybodywhowillinformmeaboutmistakes, misprints,orexpresscriticismorothercomments,bywritingtothee-mailaddresses [email protected],[email protected]. Paris,France SvetlinG.Georgiev January2015 vii Contents 1 Introduction................................................................. 1 1.1 TheSpacesC∞andS .............................................. 1 0 1.2 TheLp Spaces........................................................ 12 1.2.1 Definition..................................................... 12 1.2.2 TheInequalitiesofHölderandMinkowski................. 14 1.2.3 SomeProperties.............................................. 18 1.2.4 TheRiesz–FischerTheorem................................. 20 1.2.5 Separability................................................... 27 1.2.6 Duality........................................................ 29 1.2.7 GeneralLp Spaces........................................... 44 1.3 TheConvolutionofLocallyIntegrableFunctions.................. 48 1.4 ConesinRn........................................................... 55 1.5 AdvancedPracticalProblems........................................ 56 1.6 NotesandReferences................................................ 76 2 GeneralitiesonDistributions.............................................. 77 2.1 Definitions............................................................ 77 2.2 OrderofaDistribution............................................... 84 2.3 ChangeofVariables.................................................. 85 2.4 SequencesandSeries................................................. 88 2.5 Support................................................................ 92 2.6 SingularSupport ..................................................... 98 2.7 Measures.............................................................. 100 2.8 MultiplyingDistributionsbyC∞ Functions........................ 102 2.9 AdvancedPracticalProblems........................................ 103 2.10 NotesandReferences................................................ 112 3 Differentiation .............................................................. 113 3.1 Derivatives............................................................ 113 3.2 TheLocalStructureofDistributions................................ 118 3.3 ThePrimitiveofaDistribution ...................................... 124 3.4 SimpleandDoubleLayersonSurfaces............................. 127 ix x Contents 3.5 AdvancedPracticalProblems........................................ 131 3.6 NotesandReferences................................................ 142 4 HomogeneousDistributions............................................... 143 4.1 Definition............................................................. 143 4.2 Properties............................................................. 144 4.3 AdvancedPracticalProblems........................................ 151 4.4 NotesandReferences................................................ 154 5 TheDirectProductofDistributions...................................... 155 5.1 Definition............................................................. 155 5.2 Properties............................................................. 157 5.3 AdvancedPracticalProblems........................................ 160 5.4 NotesandReferences................................................ 164 6 Convolutions ................................................................ 165 6.1 Definition............................................................. 165 6.2 Properties............................................................. 167 6.3 Existence.............................................................. 169 6.4 TheConvolutionAlgebrasD(cid:4)(Γ+)andD(cid:4)(Γ).................... 170 6.5 RegularizationofDistributions...................................... 171 6.6 FractionalDifferentiationandIntegration........................... 172 6.7 AdvancedPracticalProblems........................................ 176 6.8 NotesandReferences................................................ 194 7 TemperedDistributions.................................................... 195 7.1 Definition............................................................. 195 7.2 DirectProduct........................................................ 197 7.3 Convolution........................................................... 198 7.4 AdvancedPracticalProblems........................................ 200 7.5 NotesandReferences................................................ 203 8 IntegralTransforms........................................................ 205 8.1 TheFourierTransforminS(Rn)................................... 205 8.2 TheFourierTransforminS(cid:4)(Rn) .................................. 206 8.3 PropertiesoftheFourierTransforminS(cid:4)(Rn) .................... 208 8.4 TheFourierTransformofDistributionswithCompactSupport... 209 8.5 TheFourierTransformofConvolutions............................. 210 8.6 TheLaplaceTransform .............................................. 211 8.6.1 Definition..................................................... 211 8.6.2 Properties..................................................... 212 8.7 AdvancedPracticalProblems........................................ 214 8.8 NotesandReferences................................................ 218 9 FundamentalSolutions .................................................... 219 9.1 DefinitionandProperties............................................. 219 9.2 FundamentalSolutionsofOrdinaryDifferentialOperators........ 223 9.3 FundamentalSolutionoftheHeatOperator ........................ 226 Contents xi 9.4 FundamentalSolutionoftheLaplaceOperator..................... 227 9.5 AdvancedPracticalProblems........................................ 227 9.6 NotesandReferences................................................ 230 10 SobolevSpaces.............................................................. 231 10.1 Definitions............................................................ 231 10.2 ElementaryProperties................................................ 232 10.3 ApproximationbySmoothFunctions............................... 236 10.4 Extensions............................................................ 241 10.5 Traces................................................................. 244 10.6 SobolevInequalities.................................................. 247 10.7 TheSpaceH−s....................................................... 255 10.8 AdvancedPracticalProblems........................................ 256 10.9 NotesandReferences................................................ 258 References......................................................................... 259 Index............................................................................... 261 Chapter 1 Introduction 1.1 TheSpacesC∞ andS 0 WithNn∪{0}wedenotethespaceofmulti-indicesα =(α ,α ,...,α ),α ∈N∪ 1 2 n k {0},k = 1,2,...,n.Forα = (α ,α ,...,α ),β = (β ,β ,...,β ) ∈ Nn∪{0}, 1 2 n 1 2 n we will write α ≤ β if α ≤ β , k = 1,2,...,n. Set D = (D ,D ,...,D ), k k 1 2 n ∂ ∂|α| D = ,k =1,2,...,n,Dα = . k ∂x ∂α1∂α2...∂αn k x1 x2 xn LetX ⊂Rnbeanopenset.IfK ⊂XisacompactsetweshallwriteK ⊂⊂X. Thefollowingconventionswillalso be usedthroughoutthebook:U(x ,R) isthe 0 open ball of radius R with centre at the point x , S(x ,R) = ∂U(x ,R) is the 0 0 0 sphereofradiusRwithcentreatx ,U(x ,R)=U(x ,R)∪S(x ,R)istheclosed 0 0 0 0 ball of radius R with centre at the point x and U = U(0,R), S = S(0,R), 0 R R U =U(0,R). R IfAandB aresetsinRn,byd(A,B)ordist(A,B)weshalldenotethedistance betweenthesetsAandB,thatis d(A,B)=dist(A,B)=infx∈A,y∈B|x−y|. We shall use A(cid:6) to denotethe (cid:6)-neighbourhoodof a set A, i.e., A(cid:6) = A+U . If (cid:6) Aisanopenset,thenA willdesignatethesetofpointsinAthataremorethan(cid:6) (cid:6) awayfromtheboundary∂A,i.e.,A ={x :x ∈A,dist(x,∂A)>(cid:6)}. (cid:6) We use intA to denote the set of interior points of the set A. With κ we will A denotethe characteristicfunctionofA, i.e.,κ (x) = 1 forx ∈ A andκ (x) = 0 A A forx (cid:9)∈A. Definition1.1 ThesetAiscalledconvexifforanypointsxandyinAthesegment λx+(1−λ)y, λ∈[0,1], liesentirelyinA. ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 1 S.G.Georgiev,TheoryofDistributions, https://doi.org/10.1007/978-3-030-81265-2_1