$ W^{\sigma,p}$ A Priori Estimates for Fully Nonlinear Integro-Differential Equations

TL;DR

This paper develops $W^{\sigma,p}$ a priori estimates for fully nonlinear integro-differential equations of order $\sigma$, extending classical $W^{2,p}$ estimates and improving maximum principles based on $L^p$ norms.

Contribution

It introduces new $W^{\sigma,p}$ estimates for nonlinear integro-differential equations and enhances Aleksandrov-Bakelman-Pucci maximum principles with $L^p$ dependence.

Findings

Established $W^{\sigma,p}$ estimates for a class of equations.

Improved maximum principles depending only on $L^p$ norms.

Extended classical estimates to fractional order integro-differential equations.

Abstract

$W^{\sigma,p}$ estimates are studied for a class of fully nonlinear integro-differential equations of order $\sigma$, which are analogues of $W^{2,p}$ estimates by Caffarelli. We also present Aleksandrov-Bakelman-Pucci maximum principles, which are improvements of estimates proved by Guillen-Schwab, depending only on $L^p$ norms of inhomogeneous terms.