  In this paper we study the Hilbert-Schmidt norm of time-frequency
localization operators $L_{\Omega} \colon L^2(\mathbb{R}^d) \rightarrow
L^2(\mathbb{R}^d)$, 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.
