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# The Bochner-Martinelli Integral and Its Applications

## byAlexander M. Kytmanov

### Paperback | October 8, 2011

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The Bochner-Martinelli integral representation for holomorphic functions or'sev eral complex variables (which has already become classical) appeared in the works of Martinelli and Bochner at the beginning of the 1940's. It was the first essen tially multidimensional representation in which the integration takes place over the whole boundary of the domain. This integral representation has a universal 1 kernel (not depending on the form of the domain), like the Cauchy kernel in e . However, in en when n > 1, the Bochner-Martinelli kernel is harmonic, but not holomorphic. For a long time, this circumstance prevented the wide application of the Bochner-Martinelli integral in multidimensional complex analysis. Martinelli and Bochner used their representation to prove the theorem of Hartogs (Osgood Brown) on removability of compact singularities of holomorphic functions in en when n > 1. In the 1950's and 1960's, only isolated works appeared that studied the boundary behavior of Bochner-Martinelli (type) integrals by analogy with Cauchy (type) integrals. This study was based on the Bochner-Martinelli integral being the sum of a double-layer potential and the tangential derivative of a single-layer potential. Therefore the Bochner-Martinelli integral has a jump that agrees with the integrand, but it behaves like the Cauchy integral under approach to the boundary, that is, somewhat worse than the double-layer potential. Thus, the Bochner-Martinelli integral combines properties of the Cauchy integral and the double-layer potential.

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Title:The Bochner-Martinelli Integral and Its ApplicationsFormat:PaperbackDimensions:308 pagesPublished:October 8, 2011Publisher:Springer-Verlag/Sci-Tech/TradeLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:3034899041

ISBN - 13:9783034899048

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Table of Contents

1 The Bochner-Martinelli Integral.- 1 The Bochner-Martinelli integral representation.- 1.1 Green's formula in complex form.- 1.2 The Bochner-Martinelli formula for smooth functions.- 1.3 The Bochner-Martinelli representation for holomorphic functions.- 1.4 Some integral representations.- 2 Boundary behavior.- 2.1 The Sokhotski?-Plemelj formula for functions satisfying a Hölder condition.- 2.2 Analogue of Privalov's theorem for integrable functions.- 2.3 Further results.- 3 Jump theorems.- 3.1 Integrable and continuous functions.- 3.2 Functions of class Lp.- 3.3 Distributions.- 3.4 Differential forms.- 4 Boundary behavior of derivatives.- 4.1 Formulas for finding derivatives.- 4.2 Jump theorem for derivatives.- 4.3 Jump theorem for the "normal" derivative.- 5 The Bochner-Martinelli integral in the ball.- 5.1 The spectrum of the Bochner-Martinelli operator.- 5.2 Computation of the Bochner-Martinelli integral in the ball.- 5.3 Some applications.- 5.4 Characterization of the ball using the Bochner-Martinelli operator.- 2 CR-Functions Given on a Hypersurface.- 6 Analytic representation of CR-functions.- 6.1 Currents.- 6.2 The problem of analytic representation.- 6.3 The theorem on analytic representation.- 6.4 Some corollaries.- 6.5 Further results and generalizations.- 7 The Hartogs-Bochner extension theorem.- 7.1 The Hartogs-Bochner theorem.- 7.2 Weinstock's extension theorem.- 7.3 The theorem of Harvey and Lawson.- 8 Holomorphic extension from a part of the boundary.- 8.1 Statement of the problem.- 8.2 Lupacciolu's theorem.- 8.3 The$$\bar \partial $$-problem for the Bochner-Martinelli kernel.- 8.4 Proof of Lupacciolu's theorem.- 8.5 Extension of the class of compact sets.- 8.6 The case of a hypersurface.- 8.7 Further results and generalizations.- 9 Removable singularities of CR-functions.- 9.1 Bounded CR-functions.- 9.2 Integrable CR-functions.- 9.3 Further results.- 10 Analogue of Riemann's theorem for CR-functions.- 10.1 Statement of the problem and results.- 10.2 Auxiliary results.- 10.3 Analogue of Smirnov's theorem.- 10.4 Proof of the main result.- 10.5 Further results.- 3 Distributions Given on a Hypersurface.- 11 Harmonic representation of distributions.- 11.1 Statement of the problem.- 11.2 Boundary values of harmonic functions of finite order of growth.- 11.3 Corollaries.- 11.4 Theorems on harmonic extension.- 12 Multiplication of distributions.- 12.1 Different approaches to multiplication of distributions.- 12.2 Definition of the product of distributions using harmonic representations.- 12.3 Properties of the product of distributions given on a hypersurface.- 12.4 Properties of products of distributions in D?(Rn).- 12.5 Multiplication of hyperfunctions with compact support.- 12.6 Multiplication in the sense of Mikusi?ski.- 12.7 Multipliable distributions.- 12.8 Boundary values of polyharmonic functions of finite order of growth.- 12.9 The class of homogeneous multipliable distributions.- 12.10 Further results.- 13 The generalized Fourier transform.- 13.1 Functions of slow growth.- 13.2 Distributions of slow growth.- 13.3 The inversion formula.- 13.4 Analogue of Vladimirov's theorem.- 13.5 Determination of the Fourier transform of some distributions.- 4 The$$\bar \partial $$-Neumann Problem.- 14 Statement of the$$\bar \partial $$-Neumann problem.- 14.1 The Hodge operator.- 14.2 Statement of the problem.- 14.3 The homogeneous$$\bar \partial $$-Neumann problem.- 15 Functions represented by Bochner-Martinelli.- 15.1 Smooth functions.- 15.2 Continuous functions.- 15.3 Functions with the one-dimensional holomorphic extension property.- 15.4 Generalizations for differential forms.- 16 Iterates of the Bochner-Martinelli integral.- 16.1 The theorem on iterates.- 16.2 Auxiliary results.- 16.3 Proof of the theorem on iterates and some corollaries.- 17 Uniqueness theorem for the $$\bar \partial $$-Neumann problem.- 17.1 Proof of the theorem.- 17.2 Corollaries of the uniqueness theorem.- 18 Solvability of the $$\bar \partial $$-Neumann problem.- 18.1 The tangential $$\bar \partial $$?-equation.- 18.2 The $$\bar \partial $$-Neumann problem for smooth functions.- 18.3. The $$\bar \partial $$-Neumann problem for distributions.- 18.4 Generalization to differential forms.- 19 Integral representation in the ball.- 19.1 The $$\bar \partial $$-Neumann problem in the ball.- 19.2 Auxiliary results.- 19.3 Proof of the main theorem.- 5 Some Applications and Open Problems.- 20 Multidimensional logarithmic residues.- 20.1 The residue formula for smooth functions.- 20.2 The formula for logarithmic residues.- 20.3 The singular Bochner-Martinelli integral.- 20.4 The formula for logarithmic residues with singularities on the boundary.- 21 Multidimensional analogues of Carleman's formula.- 21.1 The classical Carleman-Goluzin-Krylov formula.- 21.2 Holomorphic extension from a part of the boundary.- 21.3 Yarmukhamedov's formula.- 21.4 A?zenberg's formula.- 22 The Poincaré-Bertrand formula.- 22.1 The singular Bochner-Martinelli integral depending on a parameter.- 22.2 Estimates of some integrals.- 22.3 Composition of the singular Bochner-Martinelli integral and an integral with a weak singularity.- 22.4 The Poincaré-Bertrand formula.- 23 Problems on holomorphic extension.- 23.1 Functions representable by the Cauchy-Fantappiè formula.- 23.2 Differential criteria for holomorphicity of functions.- 23.3 The generalized $$\bar \partial $$-Neumann problem.- 23.4 The general form of integral representations in C2.- 6 Holomorphic Extension of Functions.- 24 Holomorphic extension of hyperfunctions.- 24.1 Hyperfunctions as boundary values of harmonic functions.- 24.2 Holomorphic extension of hyperfunctions into a domain.- 25 Holomorphic extension of functions.- 25.1 Holomorphic extension using the Bochner-Martinelli integral.- 25.2 Holomorphic extension using Cauchy-Fantappiè integrals.- 26 The Cauchy problem for holomorphic functions.- 26.1 Statement of the problem.- 26.2 Some additional information on the Bochner-Martinelli integral.- 26.3 Weak boundary values of holomorphic functions of class Lq(D).- 26.4 Doubly orthogonal bases in spaces of harmonic functions.- 26.5 Criteria for solvability of Problem 1.