Sharp $L^p$ estimates for Schrödinger groups on spaces of homogeneous type

Abstract

We prove an $L^p$ estimate $$\Vert e^{-itL}\phi (L)f{\Vert }_p\lesssim (1+|t|{)}^s\Vert f{\Vert }_p,t\in \mathbb{R},s=n\left|\frac{1}{2}-\frac{1}{p}\right|$$ for the Schrödinger group generated by a semibounded, selfadjoint operator $L$ on a metric measure space $\mathcal{X}$ of homogeneous type (where $n$ is the doubling dimension of $\mathcal{X}$). The assumptions on $L$ are a mild $L^{p_0}\to L^{p_0^{\prime }}$ smoothing estimate and a mild $L^2\to L^2$ off--diagonal estimate for the corresponding heat kernel $e^{-tL}$. The estimate is uniform for $\phi$ varying in bounded sets of $\mathcal{S}(\mathbb{R})$, or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on $L$, that the estimate extends to $$\Vert e^{-itL}\phi (\theta L)f{\Vert }_p\lesssim (1+{\theta }^{-1}|t|{)}^s\Vert f{\Vert }_p,\theta >0,t\in \mathbb{R},$$ with uniformity also for $\theta$ varying in bounded subsets of $(0,+\infty )$. For nonnegative operators uniformity holds for all $\theta >0$.

26 pages