Sharp regularity for the anisotropic fractional Laplacian with Morrey data

Abstract

We obtain sharp regularity for the anisotropic fractional Laplace equation with a right-hand side belonging to a Morrey space. We derive local $C^{σ+λ-n}-$regularity of solutions when $σ+λ-n\neq1$, where $σ\in(0,2)$ is the order of the operator and $λ\in(n-σ,n)$ is the Morrey decay rate of the right-hand side. In the case $σ+λ-n=1$, we show that solutions are locally in the Zygmund class, and hence are Log-Lipschitz continuous. Explicit examples attest to the sharpness of the conditions.