Improving Beckner's bound via Hermite functions

Abstract

We obtain an improvement of the Beckner's inequality $\Vert f{\Vert }_2^2-\Vert f{\Vert }_p^2\le (2-p)\Vert \nabla f{\Vert }_2^2$ 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 $p$.

We have extended the result to any real power p

References in corpus (2)

• Moment inequalities for functions of independent random variables
• Bellman partial differential equation and the hill property for classical isoperimetric problems

Cited by in corpus (1)

• Martingale transform and Square function: some weak and restricted weak sharp weighted estimates