Maximal regularity and Hardy spaces

TL;DR

This paper establishes the boundedness of maximal regularity operators on Hardy spaces for various elliptic operators on manifolds, Lie groups, and domains, leading to new insights into maximal regularity in harmonic analysis.

Contribution

It introduces Hardy spaces tailored for maximal regularity analysis and proves boundedness of associated operators, extending classical results to broader geometric settings.

Findings

Boundedness of maximal regularity operators on Hardy spaces

Reobtaining maximal $L^q$ regularity on $L^p$ spaces

Extension of maximal regularity results to manifolds and Lie groups

Abstract

In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.