Regularity for fully nonlinear integro-differential operators with regularly varying kernels

TL;DR

This paper extends regularity results for nonlinear integro-differential operators with symmetric, regularly varying kernels, establishing uniform Harnack inequalities and Hölder estimates that remain stable as the order approaches 2.

Contribution

It generalizes known regularity results to a broader class of kernels with logarithmic variations, ensuring estimates are uniform near the critical order 2.

Findings

Established uniform Harnack inequality for the class of kernels.

Proved Hölder regularity estimates that do not blow up as the order approaches 2.

Extended fractional Laplacian regularity results to kernels with logarithmic variations.

Abstract

In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre \cite{CS1} are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and H\"older estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{\sigma, \beta}$ satisfying $$ K_{\sigma,\beta}(y)\asymp \frac{ 2-\sigma}{|y|^{n+\sigma}}\left( \log\frac{2}{|y|^2}\right)^{\beta(2-\sigma)}\quad \mbox{near zero} $$ with respect to $\sigma\in(0,2)$ close to $2$ (for a given $\beta\in\mathbb R$), where the regularity estimates do not blow up as the order $ \sigma\in(0,2)$ tends to $2.$