AI 中文总结
该研究针对$\mathbb{S}^N$上质心在原点的轴对称函数，通过盖根鲍尔系数的加权估计与稳定临界点刚性定理，证明了Sharp Beckner不等式成立，正面解答了轴对称情形下的广义Chang-Yang猜想。
AI 中文摘要
我们证明，对任意整数$N\geq 3$与$α\geq \frac{1}{2}$，Beckner不等式\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*}对所有质心位于原点的轴对称$u\in H^{\frac{N}{2}}(\mathbb{S}^N)$均成立。证明主要基于盖根鲍尔系数的加权$\ell ^2$估计与稳定临界点的刚性定理。由此，我们对任意整数$N\geq 3$，在轴对称情形下正面解答了广义Chang-Yang猜想。
英文摘要
We prove that for every integer $N\geq 3$ and $α\geq \frac{1}{2}$, 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 $\ell ^2$ 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 $N\geq 3$.
