Square function and heat flow estimates on domains

The first purpose of this note is to provide a proof of the usual square function estimate on L-p(). It turns out to follow directly from a generic Mikhlin multiplier theorem obtained by Alexopoulos, and we provide a sketch of its proof in the Appendix for the reader's convenience. We also relate such bounds to a weaker version of the square function estimate which is enough in most instances involving dispersive PDEs and relies on Gaussian bounds on the heat kernel (such bounds are the key to Alexopoulos'result as well). Moreover, we obtain several useful L-p(;H) bounds for (the derivatives of) the heat flow with values in a given Hilbert space H.