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

Abstract

We prove that for every integer $N\geq 3$ and $α\geq \frac{1}{2}$, Beckner's inequality \[ \fracα{2}\int_{\mathbb{S}^N}u(P_{N}u) dw+(N-1)!\int_{\mathbb{S}^N}u dw-\frac{(N-1)!}{N}\log\int_{\mathbb{S}^N}e^{Nu} dw\geq 0 \] 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 $N\geq 3$.

Disclosure

“ics, Macao SAR FDCT 0003/2023/RIA1 and Macao SAR FDCT 0024/2023/RIB1. The research of J. Wei is partially supported by General Research Grant of HKSAR GRF 14309824. The authors acknowledge the use of AI tools. All math- ematical arguments and proofs in the final manuscript were checked and written by the authors. R EFERENCES Beckner199”

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2026-08-23T05:14:18.559656Z

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