DISCONTINUOUS RIEMANN BOUNDARY PROBLEM IN WEIGHTED SPACES - Petrosyan, V.G. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2017)
The Riemann boundary problem in weighted spaces $L^1(ρ) $ on $T = {t; |t| = 1}, $ where $ρ(t) =|t -t_0|^α, t_0 \in T$ and $α > -1$, is investigated. The problem is to find analytic functions $Φ^+(z)$ and $Φ^-(z)$, $Φ^-(∞)= 0$ defined on the interior and exterior domains of $T$ respectively, such that: $\lim_\limits{ r→ 1-0} ||Φ^+(rt)-a(t)Φ^-(r^1t)- f (t)||_{L^1(ρ) }= 0,$ where $f\in L^1(ρ), a(t) \in H_0(T;t_1, t_2,...,t_m)$. The article gives necessary and sufficient conditions for solvability of the problem and with explicit form of thr solutions.
Keywords: Riemann boundary problem, weighted spaces, Cauchy type integral, Holder classes