On $L_p$-estimates for a class of non-local elliptic equations

TL;DR

This paper proves the continuity and solvability of a class of non-local elliptic equations with variable kernels in $L_p$ spaces, using analytic methods, and establishes interior estimates and uniqueness for related martingale problems.

Contribution

It introduces a novel analytic approach to establish $L_p$-estimates and solvability for non-local elliptic equations with non-homogeneous, irregular kernels.

Findings

Proves continuity of non-local operators from $H^\sigma_p$ to $L_p$

Establishes unique strong solvability in $L_p$ spaces

Provides interior $L_p$-estimates for solutions

Abstract

We consider non-local elliptic operators with kernel $K(y)=a(y)/|y|^{d+\sigma}$, where $0 < \sigma < 2$ is a constant and $a$ is a bounded measurable function. By using a purely analytic method, we prove the continuity of the non-local operator $L$ from the Bessel potential space $H^\sigma_p$ to $L_p$, and the unique strong solvability of the corresponding non-local elliptic equations in $L_p$ spaces. As a byproduct, we also obtain interior $L_p$-estimates. The novelty of our results is that the function $a$ is not necessarily to be homogeneous, regular, or symmetric. An application of our result is the uniqueness for the martingale problem associated to the operator $L$.