Riemann problem in the weighted spaces L1(ρ)

Abstract

In the unit disc bounded by the circle T = {z, |z| = 1} we consider the Riemann boundary value problem in the weighted space L¹(ρ), where

\(\rho \left( t \right) = {\prod\nolimits_{k = 1}^m {\left| {t - {t_k}} \right|} ^{{\alpha _k}}}\)

, t_k ∈ T, k = 1, 2,..., m, and α_k, k = 1, 2,..., m are real numbers. The question of interest is to determine an analytic outside the circle T function ϕ(z), ϕ(∞) = 0 to satisfy

\({\lim _{r \to 1 - 0}}||{\Phi ^ + }\left( {rt} \right) - a\left( t \right){\Phi ^ - }\left( {{r^{ - 1}}t} \right) - f\left( t \right)|{|_{{L^1}\left( {{\rho _r}} \right)}} = 0\)

, where f ∈ L¹(ρ), a(t) ∈ Cδ(T), δ>0, and ρ_r are some continuations of function ρ inside the circle. The normal solvability of this problem is established.