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The Bochner-Martinelli integral and its applications. (Integral Bokhnera- Martinelli i ego primeneniya.) (Russian. English summary) Zbl 0756.32001
Novosibirsk: Nauka. 240 p. (1992).
The theme of this monograph is the connection between harmonic and complex analysis in \(\mathbb{C}^ n\). This connection is exemplified by the Bochner-Martinelli integral, which represents holomorphic functions via a harmonic kernel. The book discusses both direct applications of the Bochner-Martinelli integral, such as the Hartogs extension theorem, and illustrations of the general theme, such multiplication of distributions represented by harmonic functions.
The book will be useful to researchers in the function theory of several complex variables, for it brings together many interesting results that until now have been available only in scattered journal articles. To some extent the book updates and overlaps with the earlier monograph of L. A. Ajzenberg and Sh. A. Dautov [Differential forms orthogonal to holomorphic functions or forms, and their properties (Transl. Math. Monographs 56, Amer. Math. Soc., Providence) (1983; Zbl 0511.32002)]. There is a bibliography of 212 items.
The book is divided into five chapters: 1. The Bochner-Martinelli integral; 2. CR-functions given on a hypersurface; 3. Distributions given on a hypersurface; 4. The \(\overline\partial\)- Neumann problem for smooth functions and distributions; 5. Some applications and unsolved problems. These are subdivided into the following sections: 1. the Bochner- Martinelli integral representation; 2. boundary behavior of the Bochner- Martinelli integral; 3. jump theorems for the Bochner-Martinelli integral; 4. boundary behavior of derivatives of the Bochner-Martinelli integral; 5. the Bochner-Martinelli integral in the ball; 6. analytic representation of CR-functions; 7. the Hartogs-Bochner extension theorem; 8. holomorphic extension from a part of the boundary of a domain; 9. removable singularities of CR-functions; 10. the analogue of Riemann’s theorem for CR-functions; 11. harmonic representation of distributions; 12. multiplication of distributions; 13. the generalized Fourier transform; 14. statement of the \(\overline\partial\)-Neumann problem; 15. holomorphic functions representable by the Bochner-Martinelli integral; 16. iterates of the Bochner-Martinelli integral; 17. a uniqueness theorem for the \(\overline\partial\)-Neumann problem; 18. solvability of the \(\overline\partial\)-Neumann problem; 19. an integral representation for the solution of the \(\overline\partial\)-Neumann problem in the ball; 20. multidimensional logarithmic residues; 21. multidimensional analogues of Carleman’s formula; 22. the Poincaré-Bertrand formula; 23. problems connected with the possibility of holomorphic extension.

MSC:
32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.)
43A99 Abstract harmonic analysis
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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