On Beckner's Inequality for Axially Symmetric Functions on \mathbbS⁶

2023/04/11 by Gui, Changfeng, Li, Tuoxin, Wei, Juncheng +1
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.04955

Abstract

We prove that axially symmetric solutions to the Q-curvature type problem αP₆ u + 120(1-\frace^6u∫_\mathbbS⁶ e^6u)=0 on \mathbbS⁶ must be constants, provided that (1)/(2)≤ α<1. In view of the existence of non-constant solutions obtained by Gui-Hu-Xie \citeGHW2022 for (1)/(7)<α<(1)/(2), this result is sharp. This result closes the gap of the related results in \citeGHW2022, which proved a similar uniqueness result for α≥ 0.6168. The improvement is based on two types of new estimates: one is a better estimate of the semi-norm \lfloor G\rfloor², the other one is a family of refined estimates on Gegenbauer coefficients, such as pointwise decaying and cancellations properties.

Related

• Improved Beckner's inequality for axially symmetric functions on \mathbbS⁴
• Improved Beckner's inequality for axially symmetric functions on \mathbbSⁿ
• On Sharp Beckner's Inequality for Axially Symmetric Functions on \mathbbS⁴
• A Fourier restriction estimate for surfaces of positive curvature in R⁶
• Integral formulas for a class of curvature PDE's and applications to isoperimetric inequalities and to symmetry problems