Non-symmetric stable operators: regularity theory and integration by parts

We study solutions to $Lu=f$ in $Omegasubsetmathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable Levy process. On the one hand, we study the regularity of solutions to $Lu=f$ in $Omega$, $u=0$ in $Omega^c$, in $C^{1,alpha}$ domains~$Omega$. We show that solutions $u$ satisfy $u/d^gammain C^{varepsilon_circ}big(overlineOmegabig)$, where $d$ is the distance to $partialOmega$, and $gamma=gamma(L, u)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $ u$ to the boundary $partialOmega$. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded $C^{1,alpha}$ domains. We do it via a new efficient approximation argument, which exploits the Holder regularity of $u/d^gamma$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.