On Maximal L^p-regularity

TL;DR

This paper establishes maximal L^q regularity for certain Cauchy problems under weak assumptions by leveraging off-diagonal estimates and Hardy space techniques, extending results to weighted cases.

Contribution

It introduces a novel approach using Hardy space methods and off-diagonal estimates to prove maximal regularity with minimal assumptions.

Findings

Maximal L^q regularity is achieved under weak off-diagonal estimates.

Weighted maximal regularity results are also obtained.

The approach simplifies previous methods by using Hardy space interpolation.

Abstract

The aim of this paper is to propose weak assumptions to prove maximal L^q regularity for Cauchy problem: du/dt - Lu(t)=f(t). Mainly we only require "off-diagonal" estimates on the real semigroup (e^{tL})_{t>0} to obtain maximal L^q regularity. The main idea is to use a one kind of Hardy space H^1 adapted to this problem and then use interpolation results. These techniques permit us to prove weighted maximal regularity too.