Regularity results for stable-like operators

TL;DR

This paper investigates regularity properties of stable-like operators, showing that solutions to certain integro-differential equations exhibit Hölder continuity improvements under specific conditions.

Contribution

It establishes new regularity results for solutions of stable-like operators with variable coefficients, extending known theory to broader classes of non-local operators.

Findings

Solutions gain Hölder regularity when the right-hand side is Hölder continuous.

Regularity results depend on the order  of the operator and the smoothness of coefficients.

The paper provides conditions on the coefficient function A(x,h) for regularity enhancement.

Abstract

For $\alpha\in [1,2)$ we consider operators of the form $$L f(x)=\int_{R^d} [f(x+h)-f(x)-1_{(|h|\leq 1)} \nabla f(x)\cdot h] \frac{A(x,h)}{|h|^{d+\alpha}}$$ and for $\alpha\in (0,1)$ we consider the same operator but where the $\nabla f$ term is omitted. We prove, under appropriate conditions on $A(x,h)$, that the solution $u$ to $L u=f$ will be in $C^{\alpha+\beta}$ if $f\in C^\beta$.