Table Of ContentFriedrichHaslinger
ComplexAnalysis
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Friedrich Haslinger
Complex Analysis
|
A Functional Analytic Approach
MathematicsSubjectClassification2010
30-01,30H20,32-01,32A36,32W05,35J10
Author
Prof.DrFriedrichHaslinger
FakultätfürMathematik
UniversitätWien
Oskar-Morgenstern-Platz1
A-1090Wien
Austria
[email protected];
http://www.mat.univie.ac.at/˜has/
ISBN978-3-11-041723-4
e-ISBN(PDF)978-3-11-041724-1
e-ISBN(EPUB)978-3-11-042615-1
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Preface
Thisbookprovidesathoroughintroductiontocomplexanalysisinonevariableand
featuresspecialtopicsofseveralvariablesconnectedwiththeCauchy–Riemanndif-
ferentialequationfromafunctionalanalysispointofview.Chapters1and2arebasic
andcovertheclassicalmaterialofthecomplexplaneandofholomorphicfunctions,
theirpowerseriesandtheCauchyintegralformulawiththeconsequencesforthebe-
haviorofzeroesandthemaximumprinciple.Thehomologyandthehomotopyver-
sionofCauchy’stheoremarerelatedtoimportanttopologicalconceptsandwillbe
crucialforthecharacterizationofsimplyconnecteddomainsinChapter4.Thecalcu-
lusofresiduesofmeromorphicfunctionsisusedtoevaluaterealintegralsandFourier
transformswithoutcomputingantiderivatives.Thereareonly2sectionswhichdonot
belongtoastandardtreatment:holomorphicparameterintegrals,wheretheknowl-
edgeofbasicpropertiesoftheLebesgueintegralisassumed,andtheinhomogeneous
Cauchyformula,whereStokes’integraltheoremfromrealanalysisisused.Thisisthe
firstplacewherewerefertorealmethodsincomplexanalysis.
Chapter 3 is devoted to analytic continuation and the monodromy theorem. In
Chapter 4 we continue with real methods such as the 𝒞∞-partition of unity and
the inhomogeneous Cauchy–Riemann equations. This is also true for the general
treatmentofRunge’sapproximationtheoremusingtheHahn–Banachtheorem.Now
we are able to prove Mittag-Leffler’s theorem on the existence of global meromor-
phicfunctionswithprescribedsingularitiesand,bysolutionsoftheinhomogeneous
Cauchy–Riemannequations,thecohomologyversionofMittag-Leffler’stheorem.In-
finiteproductsareintroducedinordertoobtainWeierstraß’factorizationtheoremon
theexistenceofholomorphicfunctionswithprescribedzeroes.Thisgivesthetoolsto
showthateachdomaininthecomplexplaneisadomainofholomorphy,i.e.hasa
holomorphicfunctionwhichcannotbeanalyticallyextendedbeyondanypointofthe
boundaryofthedomain.Inaddition,normalfamiliesofholomorphicfunctionsare
studied,aconceptwhichreferstocompactnessinspacesofholomorphicfunctions
and provides the main idea in the proof of the Riemann mapping theorem, which
statesthateachsimplyconnecteddomainnotequaltoℂisbiholomorphicequivalent
totheunitdisc.Chapter4endswithacharacterizationofsimplyconnecteddomains
by means of properties of holomorphic functions. In Chapter 5 we study harmonic
functions–aconceptofrealanalysis,solvetheDirichletproblemconstructingacon-
tinuous function on the closure of a domain which is harmonic in the interior and
coincideswithagivencontinuousfunctionontheboundaryofthedomain.Further-
more,wecollectthebasicsofsubharmonicfunctions.
We would like to emphasize that this presentation of the classical theory for
one complex variable was inspired by the first 30 pages of Lars Hörmander’s book
“AnIntroductiontoComplexAnalysisinSeveralVariables”[41],which,aftermorethan
50years,isstillsingularinitseleganceandimportance.
https://doi.org/10.1515/9783110417241-201
VI | Preface
Chapter6describesthemaindifferencesbetweentheunivariateandmultivariate
theories with emphasis on the inhomogeneous Cauchy–Riemann differential equa-
tions.Weexplaintheconceptofpseudoconvexitywithoutgivingafullproofofthe
characterizationofdomainsofholomorphybypseudoconvexity.
In the following chapters we treat basic functional analytic results on Hilbert
spacesandspectraltheoryofoperatorsonHilbertspacesforboundedandunbounded
operatorsinordertoprovidethetoolsforBergmanspacesofholomorphicfunctions
(Hilbertspaceswiththereproducingproperty)andforathoroughdiscussionofthe
canonical solution operator to 𝜕. In Chapter 8 the solution operator to 𝜕 restricted
to holomorphic L2-functions in one complex variable is investigated, pointing out
that the Bergman kernel of the associated Hilbert space of holomorphic functions
plays an important role. We investigate operator properties like compactness and
Schatten-class membership (nuclear and Hilbert–Schmidt operators). At this place
wehavetheprerequisitestostudythespaceofallholomorphicfunctionsonadomain
endowedwiththetopologyofuniformconvergenceonallcompactsubsetsofthedo-
main.Chapter9isdevotedtothestudyofthesespaces,whichturnouttobecomplete
metricspaceswiththeMontelproperty(nuclearFréchetspaces).Inaddition,wede-
scribethedualspacesoftheseFréchetspacesagainascertainspacesofholomorphic
functionsandrepresentthemassequencespaces(Köthesequencespaces).
InChapter10weconsiderthegeneral𝜕-complexandderivepropertiesofthecom-
plexLaplacianonL2-spacesofboundedpseudoconvexdomains.Forthispurposewe
firstconcentrateonbasicresultsaboutdistributions,Sobolevspaces,andunbounded
operatorsonHilbertspaces.ThekeyresultistheKohn–Morreyformula,whichispre-
sentedindifferentversions.Usingthisformulathebasicpropertiesofthe𝜕-Neumann
operator–theboundedinverseofthecomplexLaplacian–areproved.Itturnsoutto
beusefultoinvestigateanevenmoregeneralsituation,namelythetwisted𝜕-complex,
where𝜕iscomposedwithapositivetwistfactor.Inthiswayoneobtainsarathergen-
eralbasicestimate,fromwhichonegetsHörmander’sL2-estimatesforthesolutionof
theCauchy–Riemannequationtogetherwithresultsonrelatedweightedspacesofen-
tirefunctions.Thelastchaptercontainsanaccountoftheapplicationofthe𝜕-methods
to Schrödinger operators, Pauli and Dirac operators and to Witten–Laplacians. We
use the 𝜕-methods and some spectral theory to settle the question whether certain
Schrödingeroperatorswithmagneticfieldonℝ2havecompactresolvent.
Mostofthematerialinthisbookisself-containedwiththeexceptionofsomeparts
inChapters6and11.Eachsinglechaptercontainsexercisesenhancingandreinforc-
ingthematerialdiscussedinthetext.Notesattheendofeachchapterrefertothe
literatureandmoreadvancedresults.Theprerequisitesforreadingthisbookare:real
analysis,basicmeasuretheoryandpointsettopology.
IwaspartiallysupportedbytheFWF-grantP28154oftheAustrianScienceFund.
Vienna FriedrichHaslinger
February2017
Contents
Preface|V
1 Complexnumbersandfunctions|1
1.1 Complexnumbers|1
1.2 Sometopologicalconceptsinℝn|3
1.3 Holomorphicfunctions|6
1.4 TheCauchy–Riemannequations|7
1.5 Ageometricinterpretationofthecomplexderivative|13
1.6 Uniformconvergence|15
1.7 Powerseries|18
1.8 Lineintegrals|21
1.9 Primitivefunctions|24
1.10 Elementaryfunctions|28
1.11 Exercises|33
1.12 Notes|38
2 Cauchy’sTheoremandCauchy’sformula|39
2.1 Windingnumbers|39
2.2 ThetheoremofCauchy–GoursatandCauchy’sformula|43
2.3 ImportantconsequencesofCauchy’sTheorem|50
2.4 Isolatedsingularities|53
2.5 ThemaximumprincipleandCauchy’sestimates|56
2.6 Openmappings|61
2.7 Holomorphicparameterintegrals|64
2.8 Complexdifferentialforms|66
2.9 TheinhomogeneousCauchyformula|68
2.10 GeneralversionsofCauchy’sTheoremandCauchy’sformula|70
2.11 Laurentseriesandmeromorphicfunctions|80
2.12 Theresiduetheorem|84
2.13 Exercises|95
2.14 Notes|102
3 Analyticcontinuation|103
3.1 Regularandsingularpoints|103
3.2 Analyticcontinuationalongacurve|105
3.3 TheMonodromyTheorem|107
3.4 Exercises|110
3.5 Notes|110
VIII | Contents
4 Constructionandapproximationofholomorphicfunctions|111
4.1 Apartitionofunity|111
4.2 TheinhomogeneousCauchy–Riemanndifferentialequations|113
4.3 TheHahn–BanachTheorem|117
4.4 Runge’sapproximationtheorems|120
4.5 Mittag-Leffler’sTheorem|130
4.6 TheWeierstraßFactorizationTheorem|136
4.7 SomeapplicationsoftheMittag-LefflerandWeierstraß
Theorems|142
4.8 Normalfamilies|144
4.9 TheRiemannMappingTheorem|146
4.10 Characterizationofsimplyconnecteddomains|151
4.11 Exercises|152
4.12 Notes|154
5 Harmonicfunctions|155
5.1 Definitionandimportantproperties|155
5.2 TheDirichletproblem|157
5.3 Jensen’sformula|161
5.4 Subharmonicfunctions|165
5.5 Exercises|167
5.6 Notes|168
6 Severalcomplexvariables|169
6.1 Complexdifferentialformsandholomorphicfunctions|169
6.2 TheinhomogeneousCRequations|174
6.3 Domainsofholomorphy|180
6.4 Exercises|187
6.5 Notes|188
7 Bergmanspaces|191
7.1 Elementaryproperties|191
7.2 Examples|199
7.3 Biholomorphicmappings|203
7.4 Exercises|206
7.5 Notes|207
8 Thecanonicalsolutionoperatorto𝜕|209
8.1 CompactoperatorsonHilbertspaces|209
8.2 Thecanonicalsolutionoperatorto𝜕restrictedtoA2(𝔻)|222
8.3 (0,1)-formswithholomorphiccoefficients|226
8.4 Exercises|228
Contents | IX
8.5 Notes|229
9 NuclearFréchetspacesofholomorphicfunctions|231
9.1 GeneralpropertiesofFréchetspaces|231
9.2 Thespaceℋ(DR(0))anditsdualspace|232
9.3 Exercises|237
9.4 Notes|238
10 The𝜕-complex|239
10.1 UnboundedoperatorsonHilbertspaces|239
10.2 DistributionsandSobolevspaces|255
10.3 Friedrichs’lemma|261
10.4 Afinitedimensionalanalog|266
10.5 The𝜕-Neumannoperator|267
10.6 Densityinthegraphnorm|275
10.7 Propertiesofthe𝜕-Neumannoperator|280
10.8 Exercises|287
10.9 Notes|288
11 Thetwisted𝜕-complexandSchrödingeroperators|291
11.1 Anexactsequenceofunboundedoperators|291
11.2 Thetwistedbasicestimates|292
11.3 Hörmander’sL2-estimates|295
11.4 The𝜕-Neumannoperatoronweighted(0,q)-forms|297
11.5 Weightedspacesofentirefunctions|305
11.6 Spectralanalysisofself-adjointoperators|310
11.7 Realdifferentialoperators|315
11.8 DiracandPaulioperators|322
11.9 Compactresolvents|323
11.10 Exercises|327
11.11 Notes|329
Bibliography|331
Index|335
