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

Abstract

We study solutions to $Lu=f$ in $\Omega \subset {\mathbb{R}}^n$, being $L$ the generator of any, possibly non-symmetric, stable Lévy 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 $u/d^{\gamma }\in C^{{\epsilon }_{\circ }}\left(\overline{\Omega }\right)$, 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 $\nu$ 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.