Regularity of harmonic functions for anisotropic fractional Laplacian

TL;DR

This paper establishes regularity properties of harmonic functions associated with anisotropic fractional Laplacians, showing they are Hölder continuous and, under stronger conditions, differentiable up to third order.

Contribution

It proves Hölder continuity and higher differentiability of harmonic functions for anisotropic fractional Laplacians under various regularity assumptions.

Findings

Harmonic functions are Hölder continuous under mild conditions.

Green function and Poisson kernel exhibit higher regularity with stronger assumptions.

Harmonic functions can be differentiable up to third order in certain cases.

Abstract

We prove that bounded harmonic functions of anisotropic fractional Laplacians are H\"older continuous under mild regularity assumptions on the corresponding L\'evy measure. Under some stronger assumptions the Green function, Poisson kernel and the harmonic functions are even differentiable of order up to three.