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Positive Gaussian Kernels also Have Gaussian Minimizers
About this Title
Franck Barthe and Paweł Wolff
Publication: Memoirs of the American Mathematical Society
Publication Year:
2022; Volume 276, Number 1359
ISBNs: 978-1-4704-5143-1 (print); 978-1-4704-7025-8 (online)
DOI: https://doi.org/10.1090/memo/1359
Published electronically: February 22, 2022
Keywords: Brascamp-Lieb inequality,
Gaussian kernel,
optimal transport,
positivity improving property,
reversed Gaussian hypercontractivity
Table of Contents
Chapters
- 1. Introduction
- 2. Well-posedness of the Minimization Problem and the Minimum Value
- 3. Proof of the Main Theorem
- 4. Geometric Brascamp-Lieb Inequality
- 5. Dual Form of Inverse Brascamp-Lieb Inequalities
- 6. Interpolation
- 7. Positivity in the Rank One Case
- 8. Positivity Condition in the General Case
Abstract
We study lower bounds on multilinear operators with Gaussian kernels acting on Lebesgue spaces, with exponents below one. We put forward natural conditions when the optimal constant can be computed by inspecting centered Gaussian functions only, and we give necessary and sufficient conditions for this constant to be positive. Our work provides a counterpart to Lieb’s results on maximizers of multilinear operators with real Gaussian kernels, also known as the multidimensional Brascamp-Lieb inequality. It unifies and extends several inverse inequalities.- S. Alesker, S. Dar, and V. Milman, A remarkable measure preserving diffeomorphism between two convex bodies in $\textbf {R}^n$, Geom. Dedicata 74 (1999), no. 2, 201–212. MR 1674116, DOI 10.1023/A:1005087216335
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