Sharp Beckner's Inequalities for Axially Symmetric Functions on ${\mathbb{S}}^N$

We prove that for every integer $N\ge 3$ and $\alpha \ge \frac{1}{2}$, Beckner’s inequality \begin{equation*} \fracα{2}\int_{$\mathbb{S}$^N}u(P_{N}u) dw+(N-1)!\int_{$\mathbb{S}$^N}u dw-\frac{(N-1)!}{N}\ln\int_{$\mathbb{S}$^N}e^{Nu} dw\geq 0 \end{equation*} holds for any axially symmetric $u\in H^{\frac{N}{2}}({\mathbb{S}}^N)$ whose center of mass is at the origin. The proof is mainly based on a weighted ${\mathrm{\ell }}^2$ estimate on Gegenbauer coefficients and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively in the axially symmetric case for every integer $N\ge 3$.