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

We prove an L^p estimate for the Schrodinger group e^{-itL} generated by a semibounded, selfadjoint operator L on a metric measure space X of homogeneous type. 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}