On the flows associated to selfadjoint operators on metric measure spaces

Let X be a metric space with a doubling measure satisfying μ(B)≳rBn for any ball B with any radius rB> 0. Let L be a non negative selfadjoint operator on L2(X). We assume that e-tL satisfies a Gaussian upper bound and that the flow eitL satisfies a typical L1- L∞ dispersive estimate of the form ‖eitL‖L1→L∞≲|t|-n/2. Then we prove a similar L1- L∞ dispersive estimate for a general class of flows eitϕ(L), with φ(r) of power type near 0 and near ∞. In the case of fractional powers φ(L) = Lν, ν∈ (0 , 1) , we deduce dispersive estimates for eitLν with data in Sobolev, Besov or Hardy spaces HLp with p∈ (0 , 1] , associated to the operator L.