Sharp Lp estimates for Schrödinger groups on spaces of homogeneous type

We prove an Lp estimate (Equation Presented) for the Schrödinger group generated by a semibounded, self-adjoint operator L on a metric measure space X of homogeneous type (where n is the doubling dimension of X). The assumptions on L are a mild Lp0 → Lp' 0 smoothing estimate and a mild L2 → L2 off-diagonal estimate for the corresponding heat kernel e -tL. The estimate is uniform for φ varying in bounded sets of S(R), or more generally of a suitable weighted Sobolev space. We also prove, under slightly stronger assumptions on L, that the estimate extends to (Equation Presented) with uniformity also for θ varying in bounded subsets of (0,+∞). For nonnegative operators uniformity holds for all θ > 0.