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

In this paper we study the Hilbert-Schmidt norm of time-frequency localization operators $L_{\mathrm{\Omega }}\text{\hspace{0.17em}⁣}:L^2({\mathbb{R}}^d)\to L^2({\mathbb{R}}^d)$, with Gaussian window, associated with a subset $\mathrm{\Omega }\subset {\mathbb{R}}^{2d}$ of finite measure. We prove, in particular, that the Hilbert-Schmidt norm of $L_{\mathrm{\Omega }}$ is maximized, among all subsets $\mathrm{\Omega }$ of a given finite measure, when $\mathrm{\Omega }$ 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.