Table Of ContentSvetlin G. Georgiev
Theory of
Distributions
Theory of Distributions
Svetlin G. Georgiev
Theory of Distributions
123
SvetlinG.Georgiev
DepartmentofDifferentialEquations
UniversityofSofia“St.KlimentOhridski”
Sofia,Bulgaria
ISBN978-3-319-19526-1 ISBN978-3-319-19527-8 (eBook)
DOI10.1007/978-3-319-19527-8
LibraryofCongressControlNumber:2015944329
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2015
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Preface
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.
Thisbookconsistsoftenchapters.Chapter1dealswiththewell-knownclassical
theoryregardingthespaceC1,theSchwartzspaceandtheconvolutionoflocally
integrable functions. It may also serve as an introduction to typical questions
related to cones in Rn. Chapter 2 collects the definitions of distributions, their
order,sequences,supportandsingularsupport,andmultiplicationbyC1functions.
In Chaps.3 and 4 we introduce differentiation and homogeneous distributions.
The notion of direct multiplication of distributions is developed in Chap.5. The
followingtwochapters,6and7,dealwithspecificproblemsaboutconvolutionsand
tempereddistributions.InChaps.8and9wecollectedbasicmaterialandproblems
regarding integral transforms. Sobolev spaces are discussed in Chap.10, the final
chapter.
This 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
v
Contents
1 Introduction ................................................................ 1
1.1 TheSpacesC1andS .............................................. 1
0
1.2 ConvolutionofLocallyIntegrableFunctions....................... 5
1.3 ConesinRn........................................................... 8
1.4 Exercises.............................................................. 10
2 GeneralitiesonDistributions ............................................. 27
2.1 Definition............................................................. 27
2.2 OrderofaDistribution............................................... 31
2.3 Sequences............................................................. 32
2.4 Support................................................................ 34
2.5 SingularSupport ..................................................... 36
2.6 Measures.............................................................. 38
2.7 MultiplyingDistributionsbyC1 Functions........................ 39
2.8 Exercises.............................................................. 40
3 Differentiation .............................................................. 65
3.1 Derivatives............................................................ 65
3.2 ThePrimitiveofaDistribution ...................................... 68
3.3 DoubleLayersonSurfaces .......................................... 71
3.4 Exercises.............................................................. 71
4 HomogeneousDistributions .............................................. 87
4.1 DefinitionandProperties............................................. 87
4.2 Exercises.............................................................. 88
5 DirectProductofDistributions ........................................... 99
5.1 Definition............................................................. 99
5.2 Properties............................................................. 101
5.3 Exercises.............................................................. 104
vii
viii Contents
6 Convolutions ................................................................ 109
6.1 Definition............................................................. 109
6.2 Properties............................................................. 111
6.3 Existence.............................................................. 113
6.4 TheConvolutionAlgebrasD0.(cid:2)C/andD0.(cid:2)/.................... 115
6.5 RegularizationofDistributions...................................... 116
6.6 FractionalDifferentiationandIntegration........................... 117
6.7 Exercises.............................................................. 121
7 TemperedDistributions ................................................... 151
7.1 Definition............................................................. 151
7.2 DirectProduct........................................................ 153
7.3 Convolution........................................................... 154
7.4 Exercises.............................................................. 156
8 IntegralTransforms ....................................................... 161
8.1 FourierTransforminS.Rn/ ........................................ 161
8.2 FourierTransforminS0.Rn/........................................ 162
8.3 PropertiesoftheFourierTransforminS0.Rn/..................... 164
8.4 FourierTransformofDistributionswithCompactSupport........ 165
8.5 FourierTransformofConvolutions ................................. 166
8.6 LaplaceTransform ................................................... 167
8.6.1 Definition..................................................... 167
8.6.2 Properties..................................................... 168
8.7 Exercises.............................................................. 170
9 FundamentalSolutions .................................................... 179
9.1 DefinitionandProperties............................................. 179
9.2 Exercises.............................................................. 182
10 SobolevSpaces ............................................................. 187
10.1 Definitions............................................................ 187
10.2 ElementaryProperties................................................ 188
10.3 ApproximationbySmoothFunctions............................... 191
10.4 Extensions............................................................ 196
10.5 Traces................................................................. 199
10.6 SobolevInequalities.................................................. 201
10.7 TheSpaceH(cid:2)s ....................................................... 210
10.8 Exercises.............................................................. 211
References......................................................................... 215
Index............................................................................... 217
Chapter 1
Introduction
1.1 TheSpacesC1 and S
0
LetX (cid:2)Rnbeanopenset.
Definition1.1 We call space of basic functions the space C1.X/ of smooth
0
functionswithcompactsupportdefinedonX.
With Nn [f0g we denote the space of multi-indices ˛ D .˛1;˛2;:::;˛n/, ˛k 2
N[f0g, k D 1;2;:::;n. Set D D .D1;D2;:::;Dn/, Dk D @@xk, k D 1;2;:::;n,
D˛ D @˛1@@˛j2˛:j::@˛n.IfK (cid:2) X isacompactsetweshallwriteK (cid:2)(cid:2)X.Thefollowing
x1 x2 xn
conventions will also be used throughout the book: U.x0;R/ is the open ball of
radiusRwithcentreatthepointx0,S.x0;R/ D @U.x0;R/isthesphereofradiusR
withcentreatx0,andUR DU.0;R/,SR DS.0;R/.
IfAandBaresetsinRn,bydist.A;B/weshalldenotethedistancebetweenthesets
AandB,thatis
dist.A;B/Dinfx2A;y2Bjx(cid:3)yj:
WeshalluseA(cid:3) todenotethe(cid:3)-neighbourhoodofasetA,i.e.A(cid:3) DACU(cid:3).IfAis
anopensetA(cid:3) willdesignatethesetofpointsinAthataremorethan(cid:3) awayfrom
theboundary@A,i.e.A(cid:3) DfxWx2A;dist.x;@A/>(cid:3)g.
WeuseintAtodenotethesetofinteriorpointsofthesetA.
Definition1.2 ThesetAiscalledconvexifforanypointsxandyinAthesegment
(cid:4)xC.1(cid:3)(cid:4)/y; (cid:4)2Œ0;1(cid:5);
liesentirelyinA.
WewillwritechAtodenotetheconvexhullofasetA.
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S.G.Georgiev,TheoryofDistributions,DOI10.1007/978-3-319-19527-8_1
