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