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 2 s -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 → 1 - which allows us to recover the local theory.