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

Abstract

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 $n$. (See Theorem 1.1.) As an application of this asymptotic estimate as $s\to 0$ and through the end-point differentiation method, we also derive the global stability for the log-Sobolev inequality on the sphere established by Beckner in [3,4] with the optimal asymptotic lower bound on the sphere. (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 when $n\to \infty$ for fixed $s\in (0, 1)$ (See Theorem 1.4 and the subsequent Remark 1.5).