View in EDS

Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality.

Saved in:
Bibliographic Details
Title: Stability of hypercontractivity, the logarithmic Sobolev inequality, and Talagrand's cost inequality.
Authors: Bez, Neal1 (AUTHOR) nealbez@mail.saitama-u.ac.jp, Nakamura, Shohei1,2 (AUTHOR) srmkn@math.sci.osaka-u.ac.jp, Tsuji, Hiroshi2 (AUTHOR) u302167i@ecs.osaka-u.ac.jp
Source: Journal of Functional Analysis. Nov2023, Vol. 285 Issue 10, pN.PAG-N.PAG. 1p.
Subjects: Fokker-Planck equation, Hamilton-Jacobi equations, Transportation costs, Cost
Geographic Terms: Nelson (N.Z.)
Abstract: We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker–Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton–Jacobi equation, the Poincaré inequality, and for Beckner's inequality. [ABSTRACT FROM AUTHOR]
Copyright of Journal of Functional Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
Database: Engineering Source
Description
Abstract:We provide deficit estimates for Nelson's hypercontractivity inequality, the logarithmic Sobolev inequality, and Talagrand's transportation cost inequality under the restriction that the inputs are semi-log-subharmonic, semi-log-convex, or semi-log-concave. In particular, our result on the logarithmic Sobolev inequality complements a recently obtained result by Eldan, Lehec and Shenfeld concerning a deficit estimate for inputs with small covariance. Similarly, our result on Talagrand's transportation cost inequality complements and, for a large class of semi-log-concave inputs, improves a deficit estimate recently proved by Mikulincer. Our deficit estimates for hypercontractivity will be obtained by using a flow monotonicity scheme built on the Fokker–Planck equation, and our deficit estimates for the logarithmic Sobolev inequality will be derived as a corollary. For Talagrand's inequality, we use an optimal transportation argument. An appealing feature of our framework is robustness and this allows us to derive deficit estimates for the hypercontracivity inequality associated with the Hamilton–Jacobi equation, the Poincaré inequality, and for Beckner's inequality. [ABSTRACT FROM AUTHOR]
ISSN:00221236
DOI:10.1016/j.jfa.2023.110121