FriedrichHaslinger ComplexAnalysis DeGruyterGraduate Also of Interest LinearAlgebra.ACourseforPhysicistsandEngineers ArakMathai,HansJ.Haubold,2017 ISBN978-3-11-056235-4,e-ISBN(PDF)978-3-11-056250-7, e-ISBN(EPUB)978-3-11-056259-0 FunctionalAnalysis.ATerseIntroduction GerardoChacón,HumbertoRafeiro,JuanCamiloVallejo,2016 ISBN978-3-11-044191-8,e-ISBN(PDF)978-3-11-044192-5, e-ISBN(EPUB)978-3-11-043364-7 IntervalAnalysisandAutomaticResultVerification GünterMayer,2017 ISBN978-3-11-050063-9,e-ISBN(PDF)978-3-11-049946-9, e-ISBN(EPUB)978-3-11-049805-9 Thed-barNeumannProblemandSchrödingerOperators FriedrichHaslinger,2014 ISBN978-3-11-031530-1,e-ISBN(PDF)978-3-11-031535-6, e-ISBN(EPUB)978-3-11-037783-5 InvariantDistancesandMetricsinComplexAnalysis MarekJarnicki,2013 ISBN978-3-11-025043-5,e-ISBN(PDF)978-3-11-025386-3 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 LibraryofCongressCataloging-in-PublicationData ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress. BibliographicinformationpublishedbytheDeutscheNationalbibliothek TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie; detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de. ©2018WalterdeGruyterGmbH,Berlin/Boston Typesetting:VTeXUAB,Lithuania Printingandbinding:CPIbooksGmbH,Leck Coverimage:FriedrichHaslinger ♾Printedonacid-freepaper PrintedinGermany www.degruyter.com 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