  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.
