Regularity theory for general stable operators

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 Lévy 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 $s\ne 1/2$, 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.

arXiv admin note: text overlap with arXiv:1404.1197