$W^{\sigma,\epsilon}$-estimates for nonlocal elliptic equations

TL;DR

This paper establishes epsilon-integrability of fractional derivatives for solutions to certain nonlocal fully nonlinear elliptic equations, extending classical second-order results to the fractional setting.

Contribution

It introduces a novel approach combining potential estimates with ABP-type estimates to analyze fractional derivatives in nonlocal elliptic equations.

Findings

Fractional derivatives are epsilon-integrable for solutions.

Potential estimates are successfully adapted to the nonlocal setting.

ABP-type estimates help control superlevel sets of fractional derivatives.

Abstract

We prove that the fractional derivatives of solutions to a class of nonlocal fully nonlinear elliptic equations are epsilon-integrable. We follow Fanghua Lin's original approach to the analogous problem for second order equations, by first proving a potential estimate, then combining this estimate with the ABP-type estimate by N. Guillen and R. Schwab to control the size of the superlevel sets of the fractional derivatives of solutions.