Dirichlet boundary value problem in the weighted spaces L¹(ρ) (Դիրիխլեի եզրային խնդիրը L¹(ρ) կշռային տարածությունների մեջ)

Abstract

The Dirichlet boundary value problem in the weighted spaces L 1 (ρ) on the unit circle T = {z: |z| = 1} is investigated, where ρ(t) = |t −tk | αk , k = 1,...,m, tk ∈ T and αk are arbitrary real numbers. The problem is to determine a function Φ(z) analytic in unit disc such that: limr→1−0 kReΦ(rt)− f(t)kL 1(ρr) = 0, where f ∈ L 1 (ρ). In the paper necessary and sufficient conditions for solvability of the problem are given and the general solution is written in the explicit form.