Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator

Samoyeau, Valentin

Confluentes Mathematici, Tome 11 (2019) no. 1, pp. 59-78.

This paper can be considered as a sequel of [4] by Bernicot and Samoyeau, where the authors have proposed a general way of deriving Strichartz estimates for the Schrödinger equation from a dispersive property of the wave propagator. It goes through a reduction of $H^1-\mathrm{BMO}$ dispersive estimates for the Schrödinger propagator to $L^2-L^2$ microlocalized (in space and in frequency) dispersion inequalities for the wave operator. This paper aims to contribute in enlightening our comprehension of how dispersion for waves implies dispersion for the Schrödinger equation. More precisely, the hypothesis of our main theorem encodes dispersion for the wave equation in an uniform way, with respect to the light cone. In many situations the phenomena that arise near the boundary of the light cone are the more complicated ones. The method we present allows to forget those phenomena we do not understand very well yet. The second main step shows the Strichartz estimates with loss of derivatives we can obtain under those assumptions. The setting we work with is general enough to recover a large variety of frameworks (infinite metric spaces, Riemannian manifolds with rough metric, some groups, ...) where the lack of knowledge of the wave propagator is an obstacle to our understanding of the dispersion phenomena.

Classification : 35B30, 42B37, 47D03, 47D06
Mots clés : dispersive inequalities; Strichartz inequalities; heat semigroup; Schrödinger group; wave operator

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