Global-in-time maximal regularity for the Cauchy problem of the heat equation in BMO and applications

TL;DR

This paper proves global-in-time maximal regularity for the heat equation and Schrödinger operators in BMO spaces, extending local results and employing advanced heat kernel estimates and oscillation techniques.

Contribution

It introduces a new method for establishing global BMO-maximal regularity for heat and Schrödinger equations, improving upon previous local-in-time results.

Findings

Established global-in-time maximal regularity in BMO spaces for heat equation.

Extended results to Schrödinger operators with potentials in RH_q.

Developed novel techniques using heat kernel estimates and oscillation analysis.

Abstract

In this article, we establish global-in-time maximal regularity for the Cauchy problem of the classical heat equation $\partial_t u(x,t)-\Delta u(x,t)=f(x,t)$ with $u(x,0)=0$ in a certain $\rm BMO$ setting, which improves the local-in-time result initially proposed by Ogawa and Shimizu in \cite{OS, OS2}. In further developing our method originally formulated for the heat equation, we obtain analogous global ${\rm BMO}$-maximal regularity associated to the Schr\"odinger operator $\mathcal L=-\Delta+V$, where the nonnegative potential $V$ belongs to the reverse H\"older class ${\rm RH}_q$ for some $q> n/2$. This extension includes several inhomogeneous estimates as ingredients, such as Carleson-type estimates for the external forces. Our new methodology is to exploit elaborate heat kernel estimates, along with matched space-time decomposition on the involving integral-type structure of maximal operators, as well as some global techniques such as those from de Simon's work and Schur's lemma. One crucial trick is to utilize the mean oscillation therein to contribute a higher and necessary decay order for global-in-time estimates.