Radially weighted harmonic functions in the unit disc

Let ω be a radial positive continuous function on the unit disk D. In this paper, we consider the weighted harmonic equation ∂zω-1∂¯zu=0, with Dirichlet boundary conditions u=f, where f is a distribution on T=∂D. We show if ω is integrable, then the Dirichlet problem has a unique solution. Furthermore, if ω satisfies some extra conditions, we prove the existence of the solution given by the convolution of the ω-Poisson kernel and the boundary data f. These results extend the case of the standard weight given by ωα(z)=(1-|z|2)α with α>-1, studied by Olofsson and Wittsten [20, 21]. We also prove a sharp estimate of the differential at zero of an ω-harmonic function, given as an ω-Poisson extension of some boundary function f∈Lp(T).