$L_p$-estimates for nonlocal equations with general L\'evy measures

TL;DR

This paper establishes $L_p$-estimates and solvability results for nonlocal parabolic equations involving general Le9vy measures, allowing for singular measures and no regularity assumptions in time.

Contribution

It provides the first comprehensive $L_p$-theory for nonlocal operators with very general, potentially singular Le9vy measures, including existence, uniqueness, and regularity results.

Findings

Proves continuity of nonlocal operators with general Le9vy measures.

Establishes unique strong solvability of nonlocal parabolic equations in $L_p$ spaces.

Shows the applicability of the theory to weighted mixed-norm spaces depending on parameters.

Abstract

We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(\sigma)} \right) \nu_t(dy), \end{equation*} where $\nu_t$ is a general L\'evy measure of order $\sigma \in(0,2)$. We allow this class of L\'evy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $\sigma$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.