On the flows associated to selfadjoint operators on metric measure spaces

Abstract

Let X be a metric space with a doubling measure satisfying $\mu (B)\gtrsim r_{B}^n$ for any ball B with any radius $r_{B}>0$. Let L be a non negative selfadjoint operator on $L^{2}(X)$. We assume that $e^{-tL}$ satisfies a Gaussian upper bound and that the flow $e^{itL}$ satisfies a typical $L^{1}-L^{\infty }$ dispersive estimate of the form

$$\begin{aligned} \Vert e^{itL}\Vert _{L^{1}\rightarrow L^{\infty }}\lesssim |t|^{-\frac{n}{2}}. \end{aligned}$$

Then we prove a similar $L^{1}-L^{\infty }$ dispersive estimate for a general class of flows $e^{it \phi (L)}$, with $\phi (r)$ of power type near 0 and near $\infty$. In the case of fractional powers $\phi (L)=L^{\nu }$, $\nu \in (0,1)$, we deduce dispersive estimates for $e^{itL^{\nu }}$ with data in Sobolev, Besov or Hardy spaces $H^{p}_{L}$ with $p\in (0,1]$, associated to the operator L.