Maximal gamma-regularity

TL;DR

This paper establishes new maximal regularity estimates in square function spaces, extending results to stochastic equations with $1<p<2$ and allowing rough initial data, advancing harmonic analysis and stochastic PDE theory.

Contribution

It introduces a novel class of maximal regularity results in square function spaces applicable to both deterministic and stochastic equations, including cases with rough initial data.

Findings

Maximal regularity estimates in square function spaces are proven.

New results for stochastic equations with $1<p<2$ are established.

Initial values with roughness comparable to $L^2$ are accommodated.

Abstract

In this paper we prove maximal regularity estimates in "square function spaces" which are commonly used in harmonic analysis, spectral theory, and stochastic analysis. In particular, they lead to a new class of maximal regularity results for both deterministic and stochastic equations in $L^p$-spaces with $1<p<\infty$. For stochastic equations, the case $1<p<2$ was not covered in the literature so far. Moreover, the "square function spaces" allow initial values with the same roughness as in the $L^2$-setting.