Improving Beckner’s bound via Hermite functions

Ivanisvili, Paata; Volberg, Alexander

doi:10.2140/apde.2017.10.929

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Improving Beckner's bound via Hermite functions

Paata Ivanisvili and Alexander Volberg

Vol. 10 (2017), No. 4, 929–942

DOI: 10.2140/apde.2017.10.929

Abstract

We obtain an improvement of the Beckner inequality $∥ f ∥_{2}^{2} - ∥ f ∥_{p}^{2} \leq (2 - p) ∥ \nabla f ∥_{2}^{2}$ valid for $p \in [1, 2]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p \in (1, 2)$ , 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

Milestones

Received: 28 June 2016

Revised: 30 January 2017

Accepted: 18 March 2017

Published: 9 May 2017

Authors

Paata Ivanisvili
Department of Mathematics Kent State University Kent, OH 44240 United States

Alexander Volberg
Department of Mathematics Michigan State University East Lansing, MI 48824 United States

Vol. 10, No. 4, 2017