Improved Beckner's inequality for axially symmetric functions on $\mathbb{S}^n$

TL;DR

This paper improves Beckner's inequality constants for axially symmetric functions on spheres and establishes related uniqueness, existence, and limit theorems for Q-curvature equations involving the Paneitz operator.

Contribution

It provides new bounds for Beckner's inequality in specific dimensions and addresses uniqueness and existence of solutions for Q-curvature equations with symmetry constraints.

Findings

Improved Beckner's inequality constants for n=6,8

Uniqueness results for Q-curvature equations in these dimensions

Proof of the first Szeg"o limit theorem for axially symmetric functions

Abstract

In this article we present various uniqueness and existence results for Q-curvature type equations with a Paneitz operator on $\s^n$ in axially symmetric function spaces. In particular, we show uniqueness results for $n=6, 8$ and improve the best constant of Beckner's inequality in these dimensions for axially symmetric functions under the constraint that their centers of mass are at the origin. As a consequence, the associated first Szeg\"o limit theorem is also proven for axially symmetric functions.