Sharp quantitative stability of Poincare-Sobolev inequality in the hyperbolic space and applications to fast diffusion flows

Consider the Poincare-Sobolev inequality on the hyperbolic space: for every n >= 3 and 1<p <= n+2/n-2, there exists a best constant Sn,p,lambda(B-n)>0 such that Sn,p,lambda(Bn)( integral Bn|u|p+1dvBn)2p+1 <=integral Bn(|del Bnu|2-lambda u2)dvBn, holds for all u is an element of C-c(infinity)(B-n), and lambda <=(n-1)(2)/4, where (n-1)(2)/4 is the bottom of the L2-spectrum of -Delta(Bn). It is known from the results of Mancini and Sandeep (Ann. Sc. Norm. Super. Pisa Cl. Sci. 7 (4): 635-671, 2008) that under appropriate assumptions on n, p and lambda there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant S-n,S-p,S-lambda(B-n). In this article, we investigate the quantitative gradient stability of the above inequality and the corresponding Euler-Lagrange equation locally around a bubble. Our result generalizes the sharp quantitative stability of Sobolev inequality in R-n by Bianchi and Egnell (J. Funct. Anal. 100 (1): 18-24. 1991) and Ciraolo, Figalli and Maggi (Int. Math. Res. Not. IMRN (21): 6780-6797, 2018) to the Poincare-Sobolev inequality on the hyperbolic space. Furthermore, combining our stability results and implementing a novel and refined smoothing estimates in spirit of Bonforte and Figalli (Comm. Pure Appl. Math. 74 (4): 744-789, 2021), we prove a quantitative extinction rate towards its basin of attraction of the solutions of the sub-critical fast diffusion flow for radial initial data. In another application, we derive sharp quantitative stability of the Hardy-Sobolev-Maz'ya inequalities for the class of functions which are symmetric in the component of singularity.