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

TL;DR

This paper investigates nonlocal elliptic and parabolic equations involving general stable operators within weighted Sobolev spaces, addressing singular measures and extending analysis to less regular time-dependent measures.

Contribution

It introduces a framework for analyzing nonlocal equations with highly singular stable operators in weighted Sobolev spaces, including time-measurable measures for parabolic cases.

Findings

Established existence and regularity results for solutions.

Extended the class of admissible Le9vy measures to very singular ones.

Provided new analytical tools for nonlocal equations in weighted Sobolev spaces.

Abstract

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