Regularity theory for general stable operators

TL;DR

This paper establishes sharp regularity estimates for solutions to nonlocal stable operators, covering interior and boundary behavior, and demonstrates the optimality of these results with counterexamples.

Contribution

It provides the first comprehensive regularity theory for solutions to general stable operators with arbitrary spectral measures, including boundary regularity and sharpness of estimates.

Findings

Solutions are $C^{eta}$ with $eta= ext{min}( ext{interior regularity},$ boundary regularity)

Solutions exhibit $C^{2s}$ regularity when $f$ is bounded and $s eq1/2$

Results are proven to be sharp through counterexamples.

Abstract

We establish sharp regularity estimates for solutions to $Lu=f$ in $\Omega\subset\mathbb R^n$, being $L$ the generator of any stable and symmetric L\'evy process. Such nonlocal operators $L$ depend on a finite measure on $S^{n-1}$, called the spectral measure. First, we study the interior regularity of solutions to $Lu=f$ in $B_1$. We prove that if $f$ is $C^\alpha$ then $u$ belong to $C^{\alpha+2s}$ whenever $\alpha+2s$ is not an integer. In case $f\in L^\infty$, we show that the solution $u$ is $C^{2s}$ when $s\neq1/2$, and $C^{2s-\epsilon}$ for all $\epsilon>0$ when $s=1/2$. Then, we study the boundary regularity of solutions to $Lu=f$ in $\Omega$, $u=0$ in $\mathbb R^n\setminus\Omega$, in $C^{1,1}$ domains $\Omega$. We show that solutions $u$ satisfy $u/d^s\in C^{s-\epsilon}(\overline\Omega)$ for all $\epsilon>0$, where $d$ is the distance to $\partial\Omega$. Finally, we show that our results are sharp by constructing two counterexamples.