Remarks on maximal regularity

TL;DR

This paper establishes weighted estimates for the maximal regularity operator, explores weak solutions to the abstract Cauchy problem, and provides a new proof of maximal regularity for certain operators using advanced mathematical techniques.

Contribution

It introduces new weighted estimates for maximal regularity operators and offers a novel proof approach for maximal regularity of accretive operators.

Findings

Weighted estimates for maximal regularity operator established

New proof of maximal regularity for accretive operators provided

Enhanced understanding of weak solutions to the abstract Cauchy problem

Abstract

We prove weighted estimates for the maximal regularity operator. Such estimates were motivated by boundary value problems. We take this opportunity to study a class of weak solutions to the abstract Cauchy problem. We also give a new proof of maximal regularity for closed and maximal accretive operators following from Kato's inequality for fractional powers and almost orthogonality arguments.