Beckner inequalities for Moebius measures on spheres^☆

¹ College of Mathematics and Statistics, Shenzhen University, 518060 Shenzhen, China.
² College of Mathematics and Statistics, Wuhan University, 430072 Hubei, China.

^* Corresponding author: zhlzhang.math@whu.edu.cn

Received: 26 April 2017
Accepted: 26 November 2018

Abstract

In this paper, we consider the Moebius measures μ_xⁿ indexed by dimension n and |x| < 1 on the unit sphere Sⁿ⁻¹ in ℝⁿ (n ≥ 3), and provide a precise two-sided estimate on the order of the Beckner inequality constant with exponent p ∈ [1, 2) in the three parameters. As special cases for p = 1 and p tending to 2, our results cover those in Barthe et al. [Forum Math. (submitted for publication)] for n ≥ 3 and explore an interesting phenomenon.