Discontinuous Riemann boundary problem in weighted spaces (Ռիմանի խզվող եզրային խնդիրը կշռային տարածություններում)

Abstract

The Riemann boundary problem in weighted spaces L 1 (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: lim r→1−0 kΦ+(rt)−a(t)Φ−(r −1 t)− f(t)kL 1(ρ) = 0, where f ∈ L 1 (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.