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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.

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