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

Abstract

Consider the Poincaré-Sobolev inequality on the hyperbolic space: for every n≥3 and 1[removed]0 such that (Formula presented.) holds for all u∈Cc∞(Bn), and λ≤(n-1)24, where (n-1)24 is the bottom of the L2-spectrum of -ΔBn. 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 λ there exists an optimizer, unique up to the hyperbolic isometries, attaining the best constant Sn,p,λ(Bn). 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 Rn 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 Poincaré-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.