The Schrödinger equation in $L^p$ spaces for operators with heat kernel satisfying Poisson type bounds

Abstract: Let $L$ be a non-negative self-adjoint operator acting on $L^2(X)$ where $X$ is a space of homogeneous type with a dimension $n$. In this paper, we study sharp endpoint $L^p$-Sobolev estimates for the solution of the initial value problem for the Schr\"odinger equation, $i \partial_t u + L u=0 $ and show that for all $f\in L^p(X), 1<p<\infty,$ \begin{eqnarray*} \left| e^{itL} (I+L)^{-{\sigma n}} f\right|{p} \leq C(1+|t|)^{\sigma n} |f|{p}, \ \ \ t\in{\mathbb R}, \ \ \ \sigma\geq \big|{1\over 2}-{1\over p}\big|, \end{eqnarray*} where the semigroup $e^{-tL}$ generated by $L$ satisfies a Poisson type upper bound. This extends the previous result in \cite{CDLY1} in which the semigroup $e^{-tL}$ generated by $L$ satisfies the exponential decay.