The norm of time-frequency and wavelet localization operators

Time-frequency localization operators (with Gaussian window) $L_F:L^2(\mathbb{R}^d)\to L^2(\mathbb{R}^d)$, where $F$ is a weight in $\mathbb{R}^{2d}$, were introduced in signal processing by I. Daubechies in 1988, inaugurating a new, geometric, phase-space perspective. Sharp upper bounds for the norm (and the singular values) of such operators turn out to be a challenging issue with deep applications in signal recovery, quantum physics and the study of uncertainty principles. In this note we provide optimal upper bounds for the operator norm $\|L_F\|_{L^2\to L^2}$, assuming $F\in L^p(\mathbb{R}^{2d})$, $1<p<\infty$ or $F\in L^p(\mathbb{R}^{2d})\cap L^\infty(\mathbb{R}^{2d})$, $1\leq p<\infty$. It turns out that two regimes arise, depending on whether the quantity $\|F\|_{L^p}/\|F\|_{L^\infty}$ is less or greater than a certain critical value. In the first regime the extremal weights $F$, for which equality occurs in the estimates, are certain Gaussians, whereas in the second regime they are proved to be truncated Gaussians, degenerating in a multiple of a characteristic function of a ball for $p=1$. This phase transition through truncated Gaussians appears to be a new phenomenon in time-frequency concentration problems. For the analogous problem for wavelet localization operators -- where the Cauchy wavelet plays the role of the above Gaussian window -- a complete solution is also provided.