The Cauchy-Green formula with rational approximation on the sets with a finite perimeter in the complex plane
type: Item
instance of: scholarly article
title: The Cauchy-Green formula with rational approximation on the sets with a finite perimeter in the complex plane
zbMATH Open document ID: 0571.30035
DOI: 10.1016/0022-1236(85)90083-7
published in: Journal of Functional Analysis
publication date: 1985-01-01T00:00:00Z
review text: The paper deals with a generalization of the classical Cauchy-Green theorem to compact sets with finite perimeter on the complex plane. The representation formula obtained here, allows to reconstruct Lipschitz functions defined on a set with finite perimeter by its values on the ''reduced boundary'' of the set and the values of its \({\bar \partial}\)- derivative. As a corollary, a natural generalization of the F. and M. Riesz theorem for nowhere dense sets of finite perimeter is obtained. The notion of ''boundary values'' for functions in generalized \(H^{\infty}\) (i.e. \(L^{\infty}\)-weak (*) Closure of rational functions) of such sets is developed and the analog of Cauchy integral formula is proved.
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 3910737
zbMATH Keywords: sets with finite perimeter
zbMATH Keywords: reduced boundary
zbMATH Keywords: rational functions
author: Dmitry Khavinson
MaRDI profile type: Publication
full work available at URL: 0022-1236(85)90083-7
OpenAlex ID: W2006532527
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cites work: Nuovi teoremi relativi alle misure \((r - 1)\)-dimensionali in uno spazio ad \(r\) dimensioni
cites work: Q5618011
cites work: The Gauss-Green Theorem
cites work: Q5565773
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cites work: Functions whose partial derivatives are measures
cites work: Analytic capacity and measure
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description: scientific article; zbMATH DE number 3910737