Time-frequency localization operators (with Gaussian window) L F : L 2 ( R d ) → L 2 ( R d ) L_F:L^2(\mathbb {R}^d)\to L^2(\mathbb {R}^d) , where F F is a weight in R 2 d \mathbb {R}^{2d} , were introduced in signal processing by I. Daubechies [IEEE Trans. Inform. Theory 34 (1988), pp. 605–612], 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 → L 2 \|L_F\|_{L^2\to L^2} , assuming F ∈ L p ( R 2 d ) F\in L^p(\mathbb {R}^{2d}) , 1 > p > ∞ 1>p>\infty or F ∈ L p ( R 2 d ) ∩ L ∞ ( R 2 d ) F\in L^p(\mathbb {R}^{2d})\cap L^\infty (\mathbb {R}^{2d}) , 1 ≤ p > ∞ 1\leq p>\infty . It turns out that two regimes arise, depending on whether the quantity ‖ F ‖ L p / ‖ F ‖ L ∞ \|F\|_{L^p}/\|F\|_{L^\infty } is less or greater than a certain critical value. In the first regime the extremal weights F F , for which equality occurs in the estimates, are certain Gaussians, whereas in the second regime they are proved to be Gaussians truncated above, degenerating into a multiple of a characteristic function of a ball for p = 1 p=1 . This phase transition through Gaussians truncated above 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.