On Sharp Beckner's Inequality for Axially Symmetric Functions on $\mathbb{S}^4$

TL;DR

This paper establishes a precise form of Beckner's inequality specifically for axially symmetric functions on the 4-sphere, utilizing properties of Gegenbauer polynomials.

Contribution

It introduces a sharp version of Beckner's inequality for axially symmetric functions on $ ext{S}^4$, based on new pointwise properties of Gegenbauer polynomials.

Findings

Proved a sharp Beckner's inequality for axially symmetric functions on $ ext{S}^4$

Identified key properties of Gegenbauer polynomials used in the proof

Enhanced understanding of inequalities on symmetric spheres

Abstract

We prove a sharp Beckner's inequality for axially symmetric functions on $S^4$. The key ingredients of our proof is some pointwise quantitative properties of Gegenbauer polynomials.