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

In this article we present various uniqueness and existence results for Q-curvature type equations with a Paneitz operator on ${\text{s}}^n$ 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.