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arXiv · 1407.4086

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

Frederic Bernicot·Valentin Samoyeau
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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.

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Frederic Bernicot, Valentin Samoyeau. 2014-07-15. Dispersive estimates with loss of derivatives via the heat semigroup and the wave operator. https://arxiv.org/abs/1407.4086

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