Gevrey regularity for integro-differential operators

TL;DR

This paper establishes Gevrey regularity for viscosity solutions of certain integro-differential equations with singular kernels, extending known regularity results to a broader class of operators including the fractional Laplacian.

Contribution

It proves that solutions belong to Gevrey classes under conditions on the kernel and right-hand side, generalizing regularity results for fractional Laplacian equations.

Findings

Viscosity solutions are Gevrey regular when the data is Gevrey.

Includes fractional Laplacian as a special case.

Extends regularity theory for integro-differential equations.

Abstract

We prove for some singular kernels $K(x,y)$ that viscosity solutions of the integro-differential equation $\int_{\mathbb{R}^n} \left[u(x+y)+u(x-y)-2u(x)\right]\,K(x,y)dy=f(x)$ locally belong to some Gevrey class if so does $f$. The fractional Laplacian equation is included in this framework as a special case.