  In this Short Note we complement the intriguing harmonic analytic perspective
due to P. Auscher and A. Axelsson for the abstract evolution equations. This
concerns a unified approach to temporally weighted estimates for the forward
and backward maximal regularity operators in presence of quadratic estimates
and functional calculi. In particular we provide several invariance properties
for the maximal regularity operators either in evolution form or in balayage
form.
