Sharp endpoint $L^p$ estimates for Schr\"odinger groups

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 $n$. Suppose that the heat operator $e^{-tL}$ satisfies the generalized Gaussian $(p_0, p'_0)$-estimates of order $m$ for some $1\leq p_0 < 2$. In this paper we prove {\it sharp} endpoint $L^p$-Sobolev bound for the Schr\"odinger group $e^{itL}$, that is for every $p\in (p_0, p'_0)$ there exists a constant $C=C(n,p)>0$ independent of $t$ such that \begin{eqnarray*} \left\| (I+L)^{-{s}}e^{itL} f\right\|_{p} \leq C(1+|t|)^{s}\|f\|_{p}, \ \ \ t\in{\mathbb R}, \ \ \ s\geq n\big|{1\over 2}-{1\over p}\big|. \end{eqnarray*} As a consequence, the above estimate holds for all $1<p<\infty$ when the heat kernel of $L$ satisfies a Gaussian upper bound. This extends classical results due to Feffermann and Stein, and Miyachi for the Laplacian on the Euclidean spaces ${\mathbb R}^n$. We also give an application to obtain an endpoint estimate for $L^p$-boundedness of the Riesz means of the solutions of the Schr\"odinger equations.