DISPERSIVE ESTIMATES WITH LOSS OF DERIVATIVES VIA THE HEAT SEMIGROUP AND THE WAVE OPERATOR

Abstract. This paper aims to give a general (possibly compact or noncompact) analog of Strichartz inequalities with loss of derivatives, obtained by Burq, Gérard, 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 Schrödinger group (both related to a same self-adjoint operator). One of the novelty is to forget the endpoint L1 − L ∞ dispersive estimates and to look for a weaker H1 − 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 L2 − L2 dispersive properties. We also use the link between the wave propagator and the unitary Schrödinger group to prove how short time dispersion for waves implies dispersion for the Schrödinger group