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_Ω : L^2(R^d ) → L^2(R^d ), with Gaussian window, associated with a subset Ω ⊂ R^2d of finite measure. We prove, in particular, that the Hilbert–Schmidt norm of L_Ω is maximized, among all subsets Ω of a given finite measure, when Ω 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.