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Regularized
Brascamp–Lieb inequalities
Neal Bez and Shohei Nakamura |
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Vol. 18 (2025), No. 7, 1567–1613
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Abstract |
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Given any (forward) Brascamp–Lieb inequality on euclidean space, a famous theorem of Lieb guarantees that gaussian near-maximizers always exist. Recently, Barthe and Wolff used mass transportation techniques to establish a counterpart to Lieb’s theorem for all nondegenerate cases of the inverse Brascamp–Lieb inequality. Here we build on work of Chen, Dafnis and Paouris and employ heat-flow techniques to understand the inverse Brascamp–Lieb inequality for certain regularized input functions, in particular extending the Barthe–Wolff theorem to such a setting. Inspiration arose from work of Bennett, Carbery, Christ and Tao for the forward inequality, and we recover their generalized Lieb’s theorem using a clever limiting argument of Wolff. In fact, we use Wolff’s idea to deduce regularized inequalities in the broader framework of the forward-reverse Brascamp–Lieb inequality, in particular allowing us to recover the gaussian saturation property in this framework first obtained by Courtade, Cuff, Liu and Verdú. |
Keywords
Brascamp–Lieb inequality, heat flow
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Mathematical Subject Classification
Primary: 42B37
Secondary: 44A12, 52A40
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Milestones
Received: 8 November 2021
Revised: 22 August 2024
Accepted: 20 September 2024
Published: 13 June 2025
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Authors |
| Neal Bez | |
| Department of Mathematics Graduate School of Science and Engineering Saitama University Saitama Japan |
Graduate School of Mathematics Nagoya University Nagoya Japan |
| Shohei Nakamura | |
| Department of Mathematics Graduate School of Science Osaka University Osaka Japan |
School of Mathematics University of Birmingham Birmingham United Kingdom |
| © 2025 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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