Optimal asymptotic lower bound for stability of fractional Sobolev inequality and the global stability of log-Sobolev inequality on the sphere

In this paper, we establish the optimal asymptotic lower bound for the stability of fractional Sobolev inequality:(0.1)${\Vert {(-\Delta )}^{s/2}U\Vert }_2^2-S_{s,n}{\Vert U\Vert }_{\frac{2n}{n-2s}}^2\ge C_{n,s}{d}^2(U,M_s),$ where $M_s$ is the set of maximizers of the fractional Sobolev inequality of order s, $s\in (0,1)$ and $C_{n,s}$ denotes the optimal lower bound of stability. We prove that the optimal lower bound $C_{n,s}$ behaves asymptotically at the order of $\frac{1}{n}$ when $n\to +\infty$ for any fixed $s\in (0,1)$. This extends the work by Dolbeault-Esteban-Figalli-Frank-Loss [22] on the stability of the first order Sobolev inequality (i.e., $s=1$) and quantify the asymptotic behavior for lower bound of stability of fractional Sobolev inequality established by the current author's previous work in [15] in the case of $s\in (0,1)$. Moreover, $C_{n,s}$ behaves asymptotically at the order of s when $s\to 0$ for any given dimension n. (See Theorem 1.1 for these asymptotic estimates.) As an important application of this asymptotic estimate as $s\to 0$, we derive the global stability for the log-Sobolev inequality on the sphere established by Beckner in [3], [5] with the optimal asymptotic lower bound on the sphere through the stability of fractional Sobolev inequalities with optimal asymptotic lower bound and the end-point differentiation method (see Theorem 1.6). This sharpens the earlier work by the authors in [14] where only the local stability for the log-Sobolev inequality on the sphere was proved. We also obtain the asymptotically optimal lower bound for the Hardy-Littlewood-Sobolev inequality when $s\to 0$ for fixed dimension n and $n\to \infty$ for fixed $s\in (0,1)$ (see Theorem 1.4 and the subsequent Remark 1.5).