The Dirichlet problem for Lévy-stable operators with \(L^2\)-data

We prove Sobolev regularity for distributional solutions to the Dirichlet problem for generators of 2s-stable processes and exterior data, inhomogeneity in weighted \(L^2\) $L^2$ -spaces. This class of operators includes the fractional Laplacian. For these rough exterior data the theory of weak variational solutions is not applicable. Our regularity estimate is robust in the limit \(s\rightarrow 1-\) $s\to 1-$ which allows us to recover the local theory.