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 $L^1-L^{\infty }$ 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 Schrödinger group to prove how short time dispersion for waves implies dispersion for the Schrödinger group.

arXiv comment48 pages