Vol. 10, No. 4, 2017

 Recent Issues
 The Journal Cover About the Cover Editorial Board Editors’ Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print)
Improving Beckner's bound via Hermite functions

Paata Ivanisvili and Alexander Volberg

Vol. 10 (2017), No. 4, 929–942
Abstract

We obtain an improvement of the Beckner inequality $\parallel f{\parallel }_{2}^{2}-\parallel f{\parallel }_{p}^{2}\le \left(2-p\right)\parallel \nabla f{\parallel }_{2}^{2}$ valid for $p\in \left[1,2\right]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p\in \left(1,2\right)$, and moreover, we find the natural extension of the inequality for any real $p$.

Keywords
Poincaré inequality, log-Sobolev inequality, Sobolev inequality, Beckner inequality, Gaussian measure, log-concave measures, semigroups, Hermite polynomials, Hermite differential equation, confluent hypergeometric functions, Turán's inequality, error term in Jensen's inequality, phi-entropy, phi-Sobolev, F-Sobolev, phi-divergence, information theory, backwards heat, Monge–Amperè with drift, exterior differential systems
Mathematical Subject Classification 2010
Primary: 42B37, 52A40, 35K55, 42C05, 60G15
Secondary: 33C15, 46G12