Vol. 10, No. 4, 2017

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Improving Beckner's bound via Hermite functions

Paata Ivanisvili and Alexander Volberg

Vol. 10 (2017), No. 4, 929–942

We obtain an improvement of the Beckner inequality f22 fp2 (2 p)f22 valid for p [1,2] and the Gaussian measure. Our improvement is essential for the intermediate case p (1,2), and moreover, we find the natural extension of the inequality for any real p.

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
Received: 28 June 2016
Revised: 30 January 2017
Accepted: 18 March 2017
Published: 9 May 2017
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