DISCONTINUOUS RIEMANN BOUNDARY PROBLEM IN WEIGHTED SPACES

Abstract

The Riemann boundary problem in weighted spaces $L^1(\rho) $ on $T = {t; |t| = 1}, $ where $\rho(t) =|t -t_0|^\alpha, t_0 \in T$ and $\alpha > -1$, is investigated. The problem is to find analytic functions $\Phi^+(z)$ and $\Phi^-(z)$, $\Phi^-(\infty)= 0$ defined on the interior and exterior domains of $T$ respectively, such that: $\lim_\limits{ r\rightarrow 1-0} ||\Phi^+(rt)-a(t)\Phi^-(r^1t)- f (t)||_{L^1(\rho) }= 0,$ where $f\in L^1(\rho),  a(t) \in H_0(T;t_1, t_2,...,t_m)$. The article gives necessary and sufficient conditions for solvability of the problem and with explicit form of thr solutions.