Uniform stability of concentration inequalities and applications

TL;DR

This paper establishes a sharp, uniform stability version of concentration inequalities related to wavelet transforms, Hardy spaces, and Poisson extensions, with implications for geometric and complex-analytic analysis.

Contribution

It provides a novel, sharp stability version of concentration inequalities for wavelet transforms and Hardy spaces, unifying geometric and complex analysis techniques.

Findings

Sharp stability version of concentration inequalities for wavelet transforms

Uniform results across Cauchy wavelet parameters

Improved understanding of extremal functions' geometry

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 . 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 a previous result by the first and fourth authors together with A. Guerra and P. Tilli, further improving the current understanding of the geometry of near extremals in all contexts under consideration.