Sharpness of the Brascamp–Lieb inequality in Lorentz spaces

Neal Bez; Sanghyuk Lee; Shohei Nakamura; Yoshihiro Sawano

doi:10.3934/era.2017.24.006

Article Contents

2017, Volume 24: 53-63. Doi: 10.3934/era.2017.24.006

This volume Previous Article Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations Next Article A note on parallelizable dynamical systems

Sharpness of the Brascamp–Lieb inequality in Lorentz spaces

1.
Department of Mathematics, Graduate School of Science and Engineering, Saitama University, Saitama 338-8570, Japan
2.
Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Korea
3.
Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo, 192-0397, Japan

Received: January 12, 2017

Revised: April 16, 2017

Published: August 2017

This work was supported by JSPS Grant-in-Aid for Young Scientists (A) no. 16H05995 (Bez), NRF (Republic of Korea) Grant no. 2015R1A2A2A05000956 (Lee) and partially supported by JSPS Grand-in-Aid for Scientific Research (C) no. 16K05209 (Sawano). The first author would like to thank Jon Bennett for helpful discussions.

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Abstract

We provide necessary conditions for the refined version of the Brascamp-Lieb inequality where the input functions are allowed to belong to Lorentz spaces, thereby establishing the sharpness of the range of Lorentz exponents in the subcritical case. Using similar considerations, some sharp refinements of the Strichartz estimates for the kinetic transport equation are established.

Keywords:

Brascamp-Lieb inequality,
Lorentz spaces.

Mathematics Subject Classification: 35B45(Primary); 35P10, 35B65(Secondary).

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