#### 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
##### 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