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In this paper we study the Hilbert-Schmidt norm of time-frequencylocalization operators $L_{\Omega} \colon L^2(\mathbb{R}^d) \rightarrowL^2(\mathbb{R}^d)$, with Gaussian window, associated with a subset$\Omega\subset\mathbb{R}^{2d}$ of finite measure. We prove, in particular, thatthe 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 otherextremizers. Actually, the main result is a quantitative version of thisestimate, with sharp exponent. A similar problem is addressed for waveletlocalization operators, where rearrangements are understood in the hyperbolicsetting.

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