Let \(\omega \) be a radial positive continuous function on the unit disk \({{\mathbb {D}}}\) . In this paper, we consider the weighted harmonic equation \(\partial _z \omega ^{-1} {\overline{\partial }}_z u=0\) , with Dirichlet boundary conditions \(u=f\) , where f is a distribution on \({{\mathbb {T}}}= \partial {{\mathbb {D}}}\) . We show if \(\omega \) is integrable, then the Dirichlet problem has a unique solution. Furthermore, if \(\omega \) satisfies some extra conditions, we prove the existence of the solution given by the convolution of the \(\omega \) -Poisson kernel and the boundary data f. These results extend the case of the standard weight given by \(\omega _\alpha (z)=(1-|z|^2)^\alpha \) with \(\alpha >-1\) , studied by Olofsson and Wittsten [20, 21]. We also prove a sharp estimate of the differential at zero of an \(\omega \) -harmonic function, given as an \(\omega \) -Poisson extension of some boundary function \(f \in L^p({{\mathbb {T}}})\) .
