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

Abstract

We study solutions to $L u=f$ in $\Omega \subset \mathbb{R}^n$, being $L$ the generator of any, possibly nonsymmetric, stable Lévy process. On the one hand, we study the regularity of solutions to $L u=f$ in $\Omega, u=0$ in $\Omega^c$, in $C^{1, \alpha}$ domains $\Omega$. We show that solutions $u$ satisfy $u / d^\gamma \in C^{\varepsilon_0}(\bar{\Omega})$, where $d$ is the distance to $\partial \Omega$, and $\gamma=\gamma(L, \nu)$ is an explicit exponent that depends on the Fourier symbol of operator $L$ and on the unit normal $v$ to the boundary $\partial \Omega$. 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 Hölder regularity of $u / d^\gamma$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.