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This paper aims to give a general (possibly compact or noncompact) analog ofStrichartz inequalities with loss of derivatives, obtained by Burq, G\'erard,and Tzvetkov [19] and Staffilani and Tataru [51]. Moreover we present a newapproach, relying only on the heat semigroup in order to understand theanalytic connexion between the heat semigroup and the unitary Schr\"odingergroup (both related to a same self-adjoint operator). One of the novelty is toforget the endpoint $L^1-L^\infty$ dispersive estimates and to look for aweaker $H^1-BMO$ estimates (Hardy and BMO spaces both adapted to the heatsemigroup). This new point of view allows us to give a general framework(infinite metric spaces, Riemannian manifolds with rough metric, manifolds withboundary,...) where Strichartz inequalities with loss of derivatives can bereduced to microlocalized $L^2-L^2$ dispersive properties. We also use the linkbetween the wave propagator and the unitary Schr\"odinger group to prove howshort time dispersion for waves implies dispersion for the Schr\"odinger group.

Dispersive estimates with loss of derivatives via the heat semigroup and the wave operator