  In this paper, we are concerned with the optimal asymptotic lower bound for
the stability of fractional Sobolev inequality: \begin{equation}\label{Sob sta
ine} \left\|(-\Delta)^{s/2} U \right\|_2^2 - \mathcal S_{s,n} \|
U\|_{\frac{2n}{n-2s}}^2\geq C_{n,s} d^{2}(U, \mathcal{M}_s), \end{equation}
where $\mathcal{M}_s$ is the set of maximizers of the fractional Sobolev
inequality of order $s$, $s\in (0,\min\{1,n/6\})$ and $C_{n,s}$ denotes the
optimal lower bound of stability. We prove that the optimal lower bound
$C_{n,s}$ is equal to $O(\frac{1}{n})$ when $n\rightarrow +\infty$ for any
$s\in (0,1)$, which extends the work by Dolbeault-Esteban-Figalli-Frank-Loss
[18] when $s=1$ and quantify the asymptotic behavior for lower bound of
stability of fractional Sobolev inequality established by the author's previous
work in [15] in the case of $s\in (0,\min\{1,n/6\})$. Moreover, $C_{n,s}$ is
equal to $O(s)$ when $s\rightarrow 0$ for any dimension $n$. (See Theorem 1.1
for these asymptotic estimates.) As an application, we derive the global
stability for the log-Sobolev inequality 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 1.3). This sharpens the earlier work by the authors [14] on the
local stability for the log-Sobolev inequality on the sphere. We also obtain
the asymptotically optimal lower bound for the Hardy-Littlewood-Sobolev
inequality when $s\to 0$ and $n\to \infty$ (See Theorem 1.4 and the subsequent
Remark 1.5).
