Multidimensional Integral Representations: Problems of Analytic Continuation

236 Pages·2015·2.116 MB·English

by  Alexander M. Kytmanov, Simona G. Myslivets (auth.)

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
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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.
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