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 are concerned with the optimal asymptotic lower bound for the stability of fractional Sobolev inequality: (-Δ)^s/2 U _2^2 - 𝒮_s,n U_2n/n-2s^2≥ 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∈ (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(1/n) when n→ +∞ for any s∈ (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∈ (0,min{1,n/6}). Moreover, C_n,s is equal to O(s) when s→ 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→ 0 and n→∞ (See Theorem 1.4 and the subsequent Remark 1.5).