Generalization of I.Vekua's integral representations of holomorphic functions and their application to the Riemann–Hilbert–Poincaré problem

Abstract

I. Vekua’s integral representations of holomorphic functions, whose m-th derivative (m≥0) is Hӧlder-continuous in a closed domain bounded by the Lyapunov curve, are generalized for analytic functions whose m-th derivative is representable by a Cauchy type integral whose density is from variable exponent Lebesgue space Lp(⋅)(Γ;ω) with power weight. An integration curve is taken from a wide class of piecewise-smooth curves admitting cusp points for certain p and ω. This makes it possible to obtain analogues of I. Vekua’s results to the Riemann–Hilbert–Poincaré problem under new general assumptions about the desired and the given elements of the problem. It is established that the solvability essentially depends on the geometry of a boundary, a weight function ω(t) and a function p(t).