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

Abstract

We prove an L^{p} estimate \| e^{-itL} \varphi(L)f \|_{p} \lesssim (1+|t|)^s \|f\|_p, \quad t\in \mathbb{R}, \quad s=n\Big|\frac{1}{2}-\frac{1}{p}\Big| for the Schrödinger group generated by a semibounded, self-adjoint 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}'} 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 \varphi varying in bounded sets of \mathscr{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 \|e^{-itL} \varphi(\theta L) f\|_{p} \lesssim (1+\theta^{-1}|t|)^s \|f\|_p, \quad \theta > 0, \quad 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 .