Homogeneous Sobolev global-in-time maximal regularity and related trace estimates

TL;DR

This paper establishes global-in-time maximal regularity and trace estimates for a class of sectorial operators on UMD Banach spaces, extending the understanding of abstract Cauchy problems and their solutions.

Contribution

It proves new maximal regularity and trace estimates for non-invertible sectorial operators, broadening the scope of abstract Cauchy problem analysis.

Findings

Proves global-in-time -maximal regularity for certain sectorial operators.

Establishes trace estimates ensuring solution continuity in trace spaces.

Connects maximal regularity with recent advances in operator and interpolation theory.

Abstract

In this paper, we prove global-in-time $\dot{\mathrm{H}}^{\alpha,q}$-maximal regularity for a class of injective, but not invertible, sectorial operators on a UMD Banach space X , provided $q\in(1,+\infty) , $\alpha\in(-1+1/q,1/q)$. We also prove the corresponding trace estimate, so that the solution to the canonical abstract Cauchy problem is continuous with values in a not necessarily complete trace space.In order to put our result in perspective, we also provide a short review on L q-maximal regularity which includes some recent advances such as the revisited homogeneous operator and interpolation theory by Danchin, Hieber, Mucha and Tolksdorf. This theory will be used to build the appropriate trace space, from which we want to choose the initial data, and the solution of our abstract Cauchy problem to fall in.