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July/August 2021 Deficit estimates for the Logarithmic Sobolev inequality
Emanuel Indrei, Daesung Kim
Differential Integral Equations 34(7/8): 437-466 (July/August 2021). DOI: 10.57262/die034-0708-437

Abstract

Carlen showed that the Beckner-Hirschman entropic uncertainty principle, which is an improvement of Heisenberg's uncertainty principle, is equivalent to a quantitative logarithmic Sobolev inequality involving the entropy of the Fourier-Wiener transform. In this work, we identify sharp spaces and prove quantitative and non-quantitative stability results for the logarithmic Sobolev inequality involving Wasserstein and $L^p$ metrics. Inter-alia, we prove a new inequality relating the entropy of the Fourier-Wiener transform with the Fisher information and Wasserstein metric via the Monge-Ampère equation. Moreover, we investigate the dependence on the dimension of the stability constant and show that in certain configurations it can be taken independently and identify spaces in which the dimension-dependence cannot be removed. The techniques are based on optimal transport theory and Fourier analysis. Lastly, we discuss a probabilistic approach to stability estimates based on Cramér's convolution theorem.

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Emanuel Indrei. Daesung Kim. "Deficit estimates for the Logarithmic Sobolev inequality." Differential Integral Equations 34 (7/8) 437 - 466, July/August 2021. https://doi.org/10.57262/die034-0708-437

Information

Published: July/August 2021
First available in Project Euclid: 23 June 2021

Digital Object Identifier: 10.57262/die034-0708-437

Subjects:
Primary: 35J96 , ‎43A32 , 49Q22 , 81R30 , 82B31 , 82Bxx

Rights: Copyright © 2021 Khayyam Publishing, Inc.

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Vol.34 • No. 7/8 • July/August 2021
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