Hölder regularity and Harnack inequality for the logarithmic Laplacian

Abstract

In this article, we establish Schauder-type estimates for the logarithmic Laplacian. We show that for a $\kappa$-Hölder inhomogeneous term, the solution is also $\kappa$-Hölder in the interior. In fact, the interior regularity slightly exceeds $\kappa$-Hölder smoothness up to a logarithmic correction. Additionally, we prove a Harnack inequality for non-negative solutions.

22 pages