  The first purpose of this note is to provide a proof of the usual square
function estimate on Lp (?). It turns out to follow directly from a generic
Mikhlin multiplier theorem obtained by Alexopoulos, which mostly relies on
Gaussian bounds on the heat kernel. We also provide a simple proof of a weaker
version of the square function estimate, which is enough in most instances
involving dispersive PDEs. Moreover, we obtain, by a relatively simple
integration by parts, several useful Lp (?; H) bounds for the derivatives of
the heat ?ow with values in a given Hilbert space H.
