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arXiv · 2304.04955

On Beckner's Inequality for Axially Symmetric Functions on $\mathbb{S}^6$

Changfeng Gui·Tuoxin Li·Juncheng Wei·Zikai Ye
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Abstract

We prove that axially symmetric solutions to the $Q$-curvature type problem $$ \alpha P_6 u + 120(1-\frac{e^{6u}}{\int_{\mathbb{S}^6} e^{6u}})=0 \ \ \ \ \ \mbox{on} \ \mathbb{S}^6 $$ must be constants, provided that $ \frac{1}{2}\leq \alpha <1$. In view of the existence of non-constant solutions obtained by Gui-Hu-Xie \cite{GHW2022} for $\frac{1}{7}<\alpha<\frac{1}{2}$, this result is sharp. This result closes the gap of the related results in \cite{GHW2022}, which proved a similar uniqueness result for $\alpha \geq 0.6168$. The improvement is based on two types of new estimates: one is a better estimate of the semi-norm $\lfloor G\rfloor^2$, the other one is a family of refined estimates on Gegenbauer coefficients, such as pointwise decaying and cancellations properties.

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Changfeng Gui, Tuoxin Li, Juncheng Wei, Zikai Ye. 2023-04-11. On Beckner's Inequality for Axially Symmetric Functions on $\mathbb{S}^6$. https://arxiv.org/abs/2304.04955

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