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 dispersive estimates for the Schrödinger propagator to 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.
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DOI: 10.5802/cml.56
Mots-clés : dispersive inequalities; Strichartz inequalities; heat semigroup; Schrödinger group; wave operator
Valentin Samoyeau 1
@article{CML_2019__11_1_59_0, author = {Valentin Samoyeau}, title = {Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator}, journal = {Confluentes Mathematici}, pages = {59--78}, publisher = {Institut Camille Jordan}, volume = {11}, number = {1}, year = {2019}, doi = {10.5802/cml.56}, mrnumber = {4002394}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.56/} }
TY - JOUR AU - Valentin Samoyeau TI - Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator JO - Confluentes Mathematici PY - 2019 SP - 59 EP - 78 VL - 11 IS - 1 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.56/ DO - 10.5802/cml.56 LA - en ID - CML_2019__11_1_59_0 ER -
%0 Journal Article %A Valentin Samoyeau %T Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator %J Confluentes Mathematici %D 2019 %P 59-78 %V 11 %N 1 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.56/ %R 10.5802/cml.56 %G en %F CML_2019__11_1_59_0
Valentin Samoyeau. Strichartz estimates with loss of derivatives under a weak dispersion property for the wave operator. Confluentes Mathematici, Volume 11 (2019) no. 1, pp. 59-78. doi : 10.5802/cml.56. https://cml.centre-mersenne.org/articles/10.5802/cml.56/
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