DIRICHLET BOUNDARY VALUE PROBLEM IN THE WEIGHTED SPACES $L^{1}(\rho)$

Abstract

The Dirichlet boundary value problem in the weighted spaces $L^{1}(\rho)$ on the unit circle $T=\{z: |z|=1\}$ is investigated, where $\rho(t)={|t-t_{k}|}^{\alpha_{k}}$,~~$k=1,\dots,m$, \lb $t_{k}\in T$ and $\alpha_{k}$ are arbitrary real numbers. The problem is to determine a function $\Phi(z)$ analytic in unit disc such that: $ \lim_{r\rightarrow 1-0}\|Re\Phi(rt)-f(t)\|_{L^{1}(\rho_{r})}=0, $ where $f\in L^{1}(\rho)$. In the paper necessary and sufficient conditions for solvability of the problem are given and the general solution is written in the explicit form.