On Carleman formulas and on the class of holomorphic functions representable by them

Abstract

Carleman formulas, unlike the Cauchy formula, restore a function holomorphic in a domain D by its values on a part M of the boundary ∂D, provided that M is of positive Lebesgue measure. An extensive survey of Carleman formulas is found in [AIZ]. In the present paper new Carleman formulas are obtained for domains in ℂn and the question about a description of the class of holomorphic functions that are represented by Carleman formula is investigated. In [AIZTV] we considered the simplest Carleman formulas in one and several complex variables on particular simply connected domains. It was shown there that a necessary and sufficient condition for a holomorphic function f to be represented by Carleman formula over the set M is that f must belong to "the Hardy class ℋ1 near the set M". In the present paper we look at the case of Fok-Kuni integral representation formula. This is a particular form of abstract Carleman formula, but it involves simply connected domains, not covered by previous results. Furthermore we initiate the study of the same questions for non-simply connected domains. We obtain the description of the class of functions representable by Carleman formula on annulii in ℂ and their generalizations, the Reinhardt domains in ℂn.