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- W2131555137 abstract "The Bochner-Martinelli formula gives the connection between complex and harmonic analysis in C n . This becomes especially apparent in the solution of the @-Neumann problem: any function that is orthogonal to the holomorphic functions is the @-normal derivative of a harmonic function. considered the boundary values of the Bochner-Martinelli integral. However, in C n 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. Interest in the Bochner-Martinelli representation grew in the 1970's in connection with the in- creased attention to integral methods in multidimensional complex analysis. Moreover, it turned out that the very general Cauchy-Fantappie representation found by Leray is easily obtained from the Bochner-Martinelli representation (Khenkin). Koppelman's representation for exterior differential forms, which has the Bochner-Martinelli representation as a special case, appeared at the same time. The Cauchy-Fantappie and Koppelman representations found significant applications in mul- tidimensional complex analysis: constructing good integral representations for holomorphic func- tions, an explicit solution of the ∂-equation and estimates of this solution, uniform approximation of holomorphic functions on compact sets, etc. In sum, one may say that the Bochner-Martinelli formula gives the connection between com- plex and harmonic analysis in C n . This becomes especially apparent in the solution of the ∂- Neumann problem: any function that is orthogonal to the holomorphic functions is the ∂-normal derivative of a harmonic function." @default.
- W2131555137 created "2016-06-24" @default.
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- W2131555137 date "2011-01-01" @default.
- W2131555137 modified "2023-09-28" @default.
- W2131555137 title "Some Application of the Bochner-Martinelli Integral" @default.
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