Nonlocal elliptic and parabolic equations with general stable operators in weighted Sobolev spaces

Abstract

We study nonlocal elliptic and parabolic equations on $C^{1,\ddot{I}„}$ open sets in weighted Sobolev spaces, where $\ddot{I}„\in (0,1)$. The operators we consider are infinitesimal generators of symmetric stable Lévy processes, whose Lévy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable.

39 pages, minor revision, to appear in SIAM J. Math. Anal