The Dirichlet problem in weighted norm

Abstract

We study the classical Dirichlet problem in the disc with the weighted uniform norm for the weight function $${w(x) = v(x)\prod_{j=1}^s|\sin(\frac{x-x_j}{2})|^{\lambda_j}}$$ , $${\{\lambda_{j}\}_{j=1}^{s}}$$ are positive numbers and v is a strictly positive continuous function on the circle. Remarkably the problem has solution if and only if none of the numbers $${\{\lambda_{j}\}_{j=1}^{s}}$$ is natural.