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

TL;DR

This paper investigates the regularity of solutions and establishes new integration by parts identities for non-symmetric stable operators, extending known results for the fractional Laplacian to more general settings.

Contribution

It provides the first regularity results for solutions to non-symmetric stable operators in $C^{1,eta}$ domains and introduces new integration by parts formulas in half spaces and bounded domains.

Findings

Solutions satisfy $u/d^ heta o C^{eta}$ regularity near boundary.

New integration by parts identities extend fractional Laplacian results.

Approximation techniques exploit boundary regularity for general domains.

Abstract

We study solutions to $Lu=f$ in $\Omega\subset\mathbb R^n$, being $L$ the generator of any, possibly non-symmetric, stable L\'evy 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 $u/d^\gamma\in C^{\varepsilon_\circ}\big(\overline\Omega\big)$, 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 H\"older regularity of $u/d^\gamma$. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.