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

TL;DR

This paper develops a general framework for dispersive estimates with derivative loss using heat semigroup techniques, applicable to various geometric settings, and links wave and Schrödinger dispersions.

Contribution

Introduces a novel heat semigroup-based approach to dispersive estimates with derivative loss, bypassing endpoint estimates and extending to complex geometric contexts.

Findings

Establishes $H^1-BMO$ estimates as a weaker alternative to $L^1-L^$ dispersive estimates.

Provides a unified framework for Strichartz inequalities with loss of derivatives on diverse spaces.

Shows short-time wave dispersion implies Schrf6dinger dispersion.

Abstract

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.