Maximal regularity for evolution equations with critical singular perturbations

TL;DR

This paper extends maximal regularity results for evolution equations to include critical singular perturbations, providing new endpoint and weighted regularity theories relevant for stochastic PDEs.

Contribution

It generalizes maximal regularity to critical unbounded perturbations and develops a weighted theory for applications in stochastic PDEs.

Findings

Proved maximal regularity at the critical endpoint case.

Developed a weighted maximal regularity theory for mixed-scale perturbations.

Applied results to linearized skeleton equations in large deviations for stochastic PDEs.

Abstract

Assuming $A$ has maximal $L^p$-regularity, this paper investigates perturbations of $A$ by time-dependent operators $B$ that are unbounded and satisfy a critical $L^q$-integrability condition in time. We establish two main results. The first proves maximal $L^p$-regularity for the critical endpoint case, generalizing previous work by Pr\"uss and Schnaubelt (2001). The second develops a weighted maximal regularity theory for mixed-scale perturbations, motivated by the linearized skeleton equations appearing in large deviations theory for stochastic PDEs.