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

TL;DR

This paper proves interior Calderon-Zygmund estimates in Sobolev spaces for a broad class of nonlocal equations with continuous coefficients, extending classical theory to non-convolution and manifold settings.

Contribution

It establishes new interior regularity estimates for nonlocal equations with continuous kernels, applicable to non-convolution operators and fractional p-Laplace equations.

Findings

Proves interior $W^{t,p}$ estimates for nonlocal equations.

Extends Calderon-Zygmund theory to non-convolution kernels.

Applicable to equations on manifolds and fractional p-Laplace equations.

Abstract

Given $2\leq p<\infty$, $s\in (0, 1)$ and $t\in (1, 2s)$, we establish interior $W^{t,p}$ Calderon-Zygmund estimates for solutions of nonlocal equations of the form \[ \int_{\Omega} \int_{\Omega} K\left (x,|x-y|,\frac{x-y}{|x-y|}\right ) \frac{(u(x)-u(y))(\varphi(x)-\varphi(y))}{|x-y|^{n+2s}} dx dy = g[\varphi], \quad \forall \phi\in C_c^{\infty}(\Omega) \] 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 $|\nabla u| \neq 0$.