On Beckner's Inequality for Axially Symmetric Functions on ${\mathbb{S}}^6$

We prove that axially symmetric solutions to the $Q$-curvature type problem $$\alpha P_{6}u+120(1-\frac{e^{6u}}{{\int }_{{\mathbb{S}}^6}{e}^{6u}})=0on{\mathbb{S}}^6$$ must be constants, provided that $\frac{1}{2}\le \alpha <1$. In view of the existence of non-constant solutions obtained by Gui-Hu-Xie \cite{GHW2022} for $\frac{1}{7}<\alpha <\frac{1}{2}$, this result is sharp. This result closes the gap of the related results in \cite{GHW2022}, which proved a similar uniqueness result for $\alpha \ge 0.6168$. The improvement is based on two types of new estimates: one is a better estimate of the semi-norm $\lfloor G{\rfloor }^2$, the other one is a family of refined estimates on Gegenbauer coefficients, such as pointwise decaying and cancellations properties.