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

Abstract

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 $N\geq 3$.

Disclosure

“DF Profes- sorial Fellowship of Mathematics, 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 [1] W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser–Tr”

PDF page 23

Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-23T05:09:55.495765Z

Structural counts