Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels : Subcritical Case

TL;DR

This paper establishes regularity results for a new class of fully nonlinear integro-differential operators with nonsymmetric kernels in the subcritical case, extending classical PDE regularity theory to nonlocal operators.

Contribution

It introduces a novel class of nonsymmetric kernels and proves key regularity estimates, including comparison principles and Harnack inequalities, for subcritical integro-differential equations.

Findings

Comparison principle established

Harnack inequality proved

Interior C^{1,α}-regularity obtained

Abstract

We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump L\`evy processes. If the index of the operator $\sigma$ is in $ (1,2)$ (subcritical case), then we obtain a comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci estimate, a Harnack inequality, a H\"older regularity, and an interior $\rm C^{1,\alpha}$-regularity for fully nonlinear integro-differential equations associated with such a class. Moreover, our estimates remain uniform as the index $\sigma$ of the operator is getting close to two, so that they can be regarded as a natural extension of regularity results for elliptic partial differential equations.