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

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 $n\ge 2$, 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 $\frac{2\log 2}{n}$. Moreover, the normalized stability constant of the fractional Sobolev inequality converges to the stability constant of the log-Sobolev inequality as $s\to 0^{+}$. After Hersch normalization, every sequence of $L^{p_{s_k}}$-normalized minimizers for stability of the fractional Sobolev inequality, with $s_k\to 0^{+}$, has a subsequence converging strongly in $L^2({\mathbb{S}}^n)$ to an $L^2$-normalized minimizer of stability of the log-Sobolev inequality.