Regularity for fully nonlinear integro-differential operators with kernels of variable orders

TL;DR

This paper establishes regularity results such as Harnack inequality and Hölder estimates for fully nonlinear integro-differential operators with variable order kernels, extending classical fractional Laplacian theory.

Contribution

It introduces a new approach using a constant involving the kernel's variable order and proves regularity results for solutions of these generalized operators.

Findings

Proves uniform Harnack inequality for solutions.

Establishes Hölder continuity of viscosity solutions.

Extends regularity theory to variable order kernels.

Abstract

We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of the kernel is not characterized by a single number, we use the constant \begin{align*} C_\varphi = \left( \int_{\mathbb{R}^n} \frac{1-\cos y_1}{\vert y \vert^n \varphi (\vert y \vert)} \, dy \right)^{-1} \end{align*} instead of $2-\sigma$, where $\varphi$ satisfies a weak scaling condition. We obtain the uniform Harnack inequality and H\"older estimates of viscosity solutions to the nonlinear integro-differential equations.