Decay estimates for a class of semigroups related to self-adjoint operators on metric measure spaces

Abstract

Assume that $(X,d,\hat{I}¼)$ is a metric space endowed with a non-negative Borel measure $\hat{I}¼$ satisfying the doubling condition and the additional condition that $\hat{I}¼(B(x,r))\gtrsim r^n$ for any $x\in X,r>0$ and some $n\ge 1$. Let $L$ be a non-negative self-adjoint operator on $L^2(X,\hat{I}¼)$. We assume that $e^{-tL}$ satisfies a Gaussian upper bound and the Schrödinger operator $e^{itL}$ satisfies an $L^1\to L^{\infty }$ decay estimate of the form \begin{equation*} \|e^{itL}\|_{L^1\to L^\infty} \lesssim |t|^{-\frac{n}{2}}. \end{equation*} Then for a general class of dispersive semigroup $e^{it\ddot{I}•(L)}$, where $\ddot{I}•:{\mathbb{R}}^{+}\to \mathbb{R}$ is smooth, we establish a similar $L^1\to L^{\infty }$ decay estimate by a suitable subordination formula connecting it with the Schrödinger operator $e^{itL}$. As applications, we derive new Strichartz estimates for several dispersive equations related to Hermite operators, twisted Laplacians and Laguerre operators.