Attainment of the sharp stability constant of Beckner conformal log-Sobolev inequality

Lu Chen; Bohan Wang

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arXiv · 2610.18327

Attainment of the sharp stability constant of Beckner conformal log-Sobolev inequality

Lu Chen·Bohan Wang

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Abstract

The existence of minimizer for stability of the fractional Sobolev inequality and the stability of Beckner conformal Log-Sobolev inequality (the endpoint case of fractional Sobolev inequality) have been established by König and Chen-Lu-Tang. The existence of minimizer for stability of the Log-Sobolev inequality is unsolved. In this paper, we solve this problem. For every $n\geq2$, we prove that the sharp $L^2$-stability constant of Beckner conformal log-Sobolev inequality on $\mathbb S^n$ is attained and is strictly less than $\frac{2\log2}{n}$. Moreover, the normalized stability constant of the fractional Sobolev inequality converges to the stability constant of the log-Sobolev inequality as $s\to0^+$. After Hersch normalization, every sequence of $L^{p_{s_k}}$-normalized minimizers for stability of the fractional Sobolev inequality, with $s_k\to0^+$, has a subsequence converging strongly in $L^2(\mathbb S^n)$ to an $L^2$-normalized minimizer of stability of the log-Sobolev inequality.

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BibTeX RIS

Lu Chen, Bohan Wang. 2026-10-13. Attainment of the sharp stability constant of Beckner conformal log-Sobolev inequality. https://arxiv.org/abs/2610.18327

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