Hörmander functional calculus on UMD lattice valued $L^p$ spaces under generalised Gaussian estimates

Abstract

We consider self-adjoint semigroups $T_t=\exp (-tA)$ acting on $L^2(\Omega )$ and satisfying (generalised) Gaussian estimates, where $\Omega$ is a metric measure space of homogeneous type of dimension $d$. The aim of the article is to show that $A\otimes {\mathrm{Id}}_Y$ admits a Hörmander type ${\mathcal{H}}_2^{\beta }$ functional calculus on $L^p(\Omega ;Y)$ where $Y$ is a UMD lattice, thus extending the well-known Hörmander calculus of $A$ on $L^p(\Omega )$. We show that if $T_t$ is lattice positive (or merely admits an $H^{\infty }$ calculus on $L^p(\Omega ;Y)$) then this is indeed the case. Here the derivation exponent has to satisfy $\beta >\alpha \cdot d+\frac{1}{2}$, where $\alpha \in (0,1)$ depends on $p$, and on convexity and concavity exponents of $Y$. A part of the proof is the new result that the Hardy-Littlewood maximal operator is bounded on $L^p(\Omega ;Y)$. Moreover, our spectral multipliers satisfy square function estimates in $L^p(\Omega ;Y)$. In a variant, we show that if $e^{itA}$ satisfies a dispersive $L^1(\Omega )\to L^{\infty }(\Omega )$ estimate, then $\beta >\frac{d+1}{2}$ above is admissible independent of convexity and concavity of $Y$. Finally, we illustrate these results in a variety of examples.

accepted for publication in Journal d'Analyse Mathématique