  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.
