The Riemann–Hilbert Problem in Weighted Classes of Cauchy Type Integrals with Density from LP( · )(Γ)

We consider the Riemann–Hilbert problem in the following setting: find a function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi} \in K^{p(\centerdot)}(D;\omega)$$\end{document} whose boundary values ϕ+(t) satisfy the condition\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm Re}[(a(t) + ib(t))\phi^{+}(t)] = c(t)$$\end{document} a.e. on Γ. Here D is a simply connected domain bounded by a simple closed curve Γ, and Kp( · )(D;ω) is the set of functions ϕ(z) representable in the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi(z) = \omega^{-1} (z) (K_{\Gamma\varphi})(z)$$\end{document}, where ω(z) is a weight function and (KΓφ)(z) is a Cauchy type integral whose density φ is integrable with a variable exponent p(t). It is assumed that Γ is a piecewise-Lyapunov curve without zero angles, ω(z) is an arbitrary power function and p(t) satisfies the Log-Hölder condition. The solvability conditions are established and solutions are constructed. These solutions largely depend on the coefficients a, b, c, the weight ω, on the values of p(t) at the angular points of Γ and on the values of angles at these points.