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

Understand

We establish 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 $s$, $s\in (0, 1)$ 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 $n\rightarrow +\infty$ for any fixed $s\in (0,1)$.
• This extends the work by Dolbeault-Esteban-Figalli-Frank-Loss [19] on the stability of the first order Sobolev inequality 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 $s\rightarrow 0$ for any given dimension $n$.