The Bochner-Martinelli Integral and Its Applications

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.

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 Holder 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 Poincare 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 Poincare-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 Poincare-Bertrand formula.- 23 Problems on holomorphic extension.- 23.1 Functions representable by the Cauchy-Fantappie 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-Fantappie 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.