Smirnov classes of analytic functions with variable exponent in multiply connected domains

Abstract

Let be a multiply connected domain with boundary , where , , are simple closed rectifiable curves such that lie outside one another, but all of them lie inside . The paper introduces the Smirnov classes with variable exponent , where , , are given positive measurable functions on . The properties of functions from these classes are established, in particular: an expansion theorem, representability by a Cauchy integral, and generalizations of Smirnov’s and Tumarkin’s theorems related to simply connected domains for multiply connected domains. Also, the question of belonging of Cauchy-type integrals with a density from to the class is investigated.