The quantitative isoperimetric inequality for the Hilbert-Schmidt norm of localization operators

Abstract

In this paper we study the Hilbert-Schmidt norm of time-frequency localization operators $L_{\Omega }:L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$, with Gaussian window, associated with a subset $\Omega \subset {\mathbb{R}}^{2d}$ of finite measure. We prove, in particular, that the Hilbert-Schmidt norm of $L_{\Omega }$ is maximized, among all subsets $\Omega$ of a given finite measure, when $\Omega$ is a ball and that there are no other extremizers. Actually, the main result is a quantitative version of this estimate, with sharp exponent. A similar problem is addressed for wavelet localization operators, where rearrangements are understood in the hyperbolic setting.

arXiv comment24 pages, 3 figures. Added some observations, in particular Remark 5.3, about the Conjecture 5.2