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 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: _ r→ 1-0 ||Φ^+(rt)-a(t)Φ^-(r^1t)- f (t)||_L^1(ρ) = 0, where fL^1(ρ), a(t) 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.