Square function and heat flow estimates on domains
Résumé
The first purpose of this note is to provide a proof of the usual square function estimate on Lp (Ω). 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 (Ω; H) bounds for the derivatives of the heat flow with values in a given Hilbert space H.
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