On Schrödinger Groups of Operators Satisfying Sub-Gaussian Estimates

Abstract

Let X be a space of homogeneous type. Assume that L is a non negative, selfadjoint operator on $L^{2}(X)$ satisfying the sub-Gaussian upper bounds. In this paper, we prove that

$$\begin{aligned} \big \Vert (I+L)^{-n/2}e^{i\tau L}f\big \Vert _{1,\infty } \le C (1 + |\tau |)^{n/2}\Vert f\Vert _{1}, \quad \forall \tau \in \mathbb {R}. \end{aligned}$$

By interpolation, we obtain

$$\begin{aligned} \big \Vert (I+L)^{-n|1/p-1/2|}e^{i\tau L}f\big \Vert _{p} \le C (1 + |\tau |)^{n|1/p-1/2|}\Vert f\Vert _{p}, \quad \forall \tau \in \mathbb {R}, \ \ 1<p<\infty . \end{aligned}$$