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: ‖(−Δ)^s/2U‖₂²−S_s,n‖U‖_{[Formula presented]}²≥C_n,sd²(U,M_s), where M_s is the set of maximizers of the fractional Sobolev inequality of order s, s∈(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 [Formula presented] when n→+∞ for any fixed s∈(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∈(0,1). Moreover, C_n,s behaves asymptotically at the order of s when s→0 for any given dimension n. (See Theorem 1.1 for these asymptotic estimates.) As an important application of this asymptotic estimate as s→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→0 for fixed dimension n and n→∞ for fixed s∈(0,1) (see Theorem 1.4 and the subsequent Remark 1.5).