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Regularity theory for general stable operators

We establish sharp regularity estimates for solutions to $Lu=f$ in $Ω\subset\mathbb R^n$, 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 $C^α$ then $u$ belong to $C^{α+2s}$ whenever $α+2s$ is not an integer. In case $f\in L^\infty$, we show that the solution $u$ is $C^{2s}$ when $s\neq1/2$, and $C^{2s-ε}$ for all $ε>0$ when $s=1/2$. Then, we study the boundary regularity of solutions to $Lu=f$ in $Ω$, $u=0$ in $\mathbb R^n\setminusΩ$, in $C^{1,1}$ domains $Ω$. We show that solutions $u$ satisfy $u/d^s\in C^{s-ε}(\overlineΩ)$ for all $ε>0$, where $d$ is the distance to $\partialΩ$. Finally, we show that our results are sharp by constructing two counterexamples.