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.