Uniform stability of concentration inequalities and applications

Abstract

We prove a sharp quantitative version of recent Faber–Krahn inequalities for the continuous Wavelet transforms associated to a certain family of Cauchy wavelet windows [Ramos and Tilli, Soc. 55 (2023), no. 4, 2018–2034]. Our results are uniform on the parameters of the family of Cauchy wavelets, and asymptotically sharp in both directions. As a corollary of our results, we are able to recover not only the original result for the short-time Fourier transform as a limiting procedure, but also a new concentration result for functions in Hardy spaces. This is a completely novel result about optimal concentration of Poisson extensions, and our proof automatically comes with a sharp stability version of that inequality. Our techniques highlight the intertwining of geometric and complex-analytic arguments involved in the context of concentration inequalities. In particular, in the process of deriving uniform results, we obtain a refinement over the proof of the result in [Gómez et al., Invent. Math. 236 (2024), no. 2, 779–836], further improving the current understanding of the geometry of near extremals in all contexts under consideration.