Calderon-Zygmund theory for non-convolution type nonlocal equations with continuous coefficient

Abstract

Given $2\le p<\infty$, $s\in (0,1)$ and $t\in (1,2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form \[ \int_Ω \int_Ω K\left (x,|x-y|,\frac{x-y}{|x-y|}\right ) \frac{(u(x)-u(y))(φ(x)-φ(y))}{|x-y|^{n+2s}} dx dy = g[φ], \quad \forall ϕ\in C_c^{\infty}(Ω) \] where $\Omega \subset {\mathbb{R}}^n$ is an open set. Here we assume $K$ is bounded, nonnegative and continuous in the first entry -- and ellipticity is ensured by assuming that $K$ is strictly positive in a cone. The setup is chosen so that it is applicable for nonlocal equations on manifolds, but the structure of the equation is general enough that it also applies to the certain fractional $p$-Laplace equations around points where $u\in C^1$ and $|\nabla u|\ne 0$.