Uniform stability of concentration inequalities and applications
Jaime Gómez, David Kalaj, Petar Melentijević, João P. G. Ramos
Source abstract
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.
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