The Cauchy-Green formula with rational approximation on the sets with a finite perimeter in the complex plane
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.
scientific article; zbMATH DE number 3910737
scientific article