A positive answer to the generalized Chang-Yang conjecture on ${\mathbb{S}}^N$

We prove that for every integer $N\ge 3$ and $\alpha \ge \frac{1}{2}$, Beckner’s inequality $$\frac{\alpha }{2}{\int }_{{\mathbb{S}}^N}u(P_{N}u)dw+(N-1)!{\int }_{{\mathbb{S}}^N}udw-\frac{(N-1)!}{N}\log {\int }_{{\mathbb{S}}^N}{e}^{Nu}dw\ge 0$$ holds for every $u\in H^{\frac{N}{2}}({\mathbb{S}}^N)$ whose center of mass is at the origin. The proof is mainly based on an integral representation formula and a rigidity theorem for stable critical points. Hence, we answer the generalized Chang-Yang conjecture positively for every integer $N\ge 3$.