Global Calderón-Zygmund theory for fractional Laplacian type equations

Abstract

We establish several fine boundary regularity results of weak solutions to non-homogeneous s-fractional Laplacian type equations. In particular, we prove sharp Calderón-Zygmund type estimates of $u/d^s$ depending on the regularity assumptions on the associated kernel coefficient including VMO, Dini continuity or the Hölder 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 $u/d^s$.