DISCONTINUOUS RIEMANN BOUNDARY PROBLEM IN WEIGHTED SPACES

The Riemann boundary problem in weighted spaces L1(ρ) on T=t;|t|=1, where ρ(t)=|t−t0|α,t0∈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: limr→1−0||Φ+(rt)−a(t)Φ−(r1t)−f(t)||L1(ρ)=0, where f∈L1(ρ),a(t)∈H0(T;t1,t2,...,tm). The article gives necessary and sufficient conditions for solvability of the problem and with explicit form of thr solutions.