Square function and heat flow estimates on domains

TL;DR

This paper proves square function estimates on Lp spaces using heat kernel bounds and provides simplified proofs for related dispersive PDE estimates, along with new Lp bounds for heat flow derivatives in Hilbert spaces.

Contribution

It offers a direct proof of the classical square function estimate via a Mikhlin multiplier theorem and introduces simplified methods for dispersive PDE applications and heat flow derivative bounds.

Findings

Square function estimate proven via Gaussian bounds

Simplified proof for dispersive PDE applications

Lp bounds for heat flow derivatives in Hilbert spaces

Abstract

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.