Multidimensional Integral Representations: Problems of Analytic Continuation PDF
Preview Multidimensional Integral Representations: Problems of Analytic Continuation
Alexander M. Kytmanov Simona G. Myslivets Multidimensional Integral Representations Problems of Analytic Continuation Multidimensional Integral Representations Alexander M. Kytmanov • Simona G. Myslivets Multidimensional Integral Representations Problems of Analytic Continuation 123 AlexanderM.Kytmanov SimonaG.Myslivets InstituteofMathematicsandComputer InstituteofMathematicsandComputer Science Science SiberianFederalUniversity SiberianFederalUniversity Krasnoyarsk,Russia Krasnoyarsk,Russia MultidimensionalIntegralRepresentations ISBN978-3-319-21658-4 ISBN978-3-319-21659-1 (eBook) DOI10.1007/978-3-319-21659-1 LibraryofCongressControlNumber:2015949242 MathematicsSubjectClassification(2010):32A25,32A26,32A40,32A50 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com) Foreword This monograph is devoted to integral representations for holomorphic functions in several complex variables such as: Bochner–Martinelli, Cauchy–Fantappiè, Koppelman, etc. and their applications to analytic continuation functions with a one-dimensional property of holomorphic extension. This book also contains multidimensionalboundaryanaloguesoftheMoreratheorem. Tel-Aviv,Israel LevAizenberg June2015 v Preface TheBochner–Martinelliintegralrepresentationforholomorphicfunctionsofseveral complexvariablesappearedintheworksofMartinelli(1938)andBochner(1943). It was the first essentially multidimensionalrepresentationwith integrationtaking place over the whole boundary of the domain. This integral representation has a universalkernel(notdependingontheformofthedomain),liketheCauchykernel inC1.However,inCn whenn >1,theBochner–Martinellikernelisharmonic,but notholomorphic.Foralongtime,thiscircumstancehinderedthewideapplication oftheBochner–Martinelliintegralinmultidimensionalcomplexanalysis. InterestintheBochner–Martinellirepresentationgrewinthe1970sinconnection withtheincreasedattentiontointegralmethodsinmultidimensionalcomplexanal- ysis.Moreover,itturnedoutthattheverygeneralCauchy–Fantappièrepresentation suggested by Leray is easily obtained from the Bochner–Martinellirepresentation (Khenkin). Koppelman’s representation for exterior differential forms, which has theBochner–Martinellirepresentationasaspecialcase,emergedatthesametime. The Cauchy–Fantappiè and Koppelman representations were extensively used in multidimensional complex analysis: yielding good integral representations for holomorphic functions, explicit solution of the @N-equation and estimates of this solution,uniformapproximationofholomorphicfunctionsoncompactsets,etc. In the early 1970s, it was shown that, notwithstanding the non-holomorphicity of the kernel, the Bochner–Martinelli representation holds only for holomorphic functions. In 1975, Harvey and Lawson obtained a result for odd-dimensional manifolds on spanning by complex chains; the Bochner–Martinelli formula lies at its foundation. In the 1980s and 1990s, the Bochner–Martinelli formula was successfully exploited in the theory of function of several complex variables: in multidimensionalresidues,incomplex(algebraic)geometry,inquestionsofrigidity ofholomorphicmappings,infindinganaloguesofCarleman’sformula,etc. TheschoolofmultidimensionalcomplexanalysispromotedbyL.A.Aizenberg and A.P. Yuzhakov in Krasnoyarsk in the 1960s last century was involved in the development of the theory of integral representations and residues and their applications. A series of monographs on integral representations and residues by L.A. Aizenberg, Sh.A. Dautov, A.P. Yuzhakov, A.K. Tsih, A.M. Kytmanov, and vii viii Preface N.N. Tarkhanov were published in the 1980s and 1990s. Over the 20 years since then,newresultshavebeenobtainedandnewareasofresearchexplored. Ourmonographsummarizestheresultsobtainedbytheauthorsinrecentyears, includinginparticularthestudiesondifferentfamiliesofcomplexlinesandcurves sufficient for analytic continuation of functions from the boundary of a bounded domain,multidimensionalboundaryanaloguesoftheMoreratheorem. In a sense, this monographis a sequel to an earlier book of one of the authors [45].Inanycase,thefirsttwochaptersofourbookarealmostentirelytakenfrom [45]. The results of the monograph were delivered as part of specialized courses at the Institute of Mathematics and Computer Science of the Siberian Federal Universitybetween1995and2015. Chaptersarenumberedthroughoutthemonograph,sectionsarenumberedthrou- ghoutthe chapters. All statements, comments, formulas, and examplesare tied to thenumberoftherespectivesection. Krasnoyarsk,Russia AlexanderKytmanov June2015 SimonaMyslivets Acknowledgements The authors used the financial support of RFBR, grant 14-01-00544, and grant 14.Y26.31.0006ofthe Russian Governmenttosupportresearchschoolsunderthe supervisionofleadingscientistsintheSiberianFederalUniversity. ix
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