Global Calder\'on-Zygmund theory for fractional Laplacian type equations

TL;DR

This paper develops advanced boundary regularity results for solutions to fractional Laplacian equations, providing sharp estimates based on kernel regularity, and enhances understanding of nonlocal boundary behaviors.

Contribution

It introduces new boundary regularity results and sharp Calderón-Zygmund estimates for fractional Laplacian equations with various kernel regularities.

Findings

Established boundary regularity results for fractional Laplacian solutions.

Derived sharp Calderón-Zygmund estimates depending on kernel regularity.

Analyzed point-wise behaviors of maximal functions of solutions.

Abstract

We establish several fine boundary regularity results of weak solutions to non-homogeneous $s$-fractional Laplacian type equations. In particular, we prove sharp Calder\'on-Zygmund type estimates of $u/d^s$ depending on the regularity assumptions on the associated kernel coefficient including VMO, Dini continuity or the H\"older continuity, where $u$ is a weak solution to such a nonlocal problem and $d$ is the distance to the boundary function of a given domain. Our analysis is based on point-wise behaviors of maximal functions of $u/d^s$.