Solving Poisson's equation for the standard weighted Laplacian in the unit disc

Abstract

We will find Green's function for the standard weighted Laplacian and use the corresponding Green's potential to solve Poisson's equation in the unit disc with zero boundary values, in the sense of radial $L^1$-means, for complex Borel measures $\mu$ satisfying the condition $\int_\mathbb{D} (1-|w|^2)^{\alpha+1} d|\mu|(w) < \infty$ for $\alpha>-1$.