The Faber-Krahn inequality for the Short-time Fourier transform

In this paper we solve an open problem concerning the characterization of those measurable sets $\Omega\subset \mathbb{R}^{2d}$ that, among all sets having a prescribed Lebesgue measure, can trap the largest possible energy fraction in time-frequency space, where the energy density of a generic function $f\in L^2(\mathbb{R}^d)$ is defined in terms of its Short-time Fourier transform (STFT) $\mathcal{V} f(x,\omega)$, with Gaussian window. More precisely, given a measurable set $\Omega\subset\mathbb{R}^{2d}$ having measure $s>0$, we prove that the quantity \[ \Phi_\Omega=\max\Big\{\int_\Omega|\mathcal{V} f(x,\omega)|^2\,dxd\omega: f\in L^2(\mathbb{R}^d),\ \|f\|_{L^2}=1\Big\}, \] is largest possible if and only if $\Omega$ is equivalent, up to a negligible set, to a ball of measure $s$, and in this case we characterize all functions $f$ that achieve equality. This result leads to a sharp uncertainty principle for the"essential support"of the STFT (when $d=1$, this can be summarized by the optimal bound $\Phi_\Omega\leq 1-e^{-|\Omega|}$, with equality if and only if $\Omega$ is a ball). Our approach, using techniques from measure theory after suitably rephrasing the problem in the Fock space, also leads to a local version of Lieb's uncertainty inequality for the STFT in $L^p$ when $p\in [2,\infty)$, as well as to $L^p$-concentration estimates when $p\in [1,\infty)$, thus proving a related conjecture. In all cases we identify the corresponding extremals.


Introduction
The notion of energy concentration for a function f ∈ L 2 (R) in the time-frequency plane is an issue of great theoretical and practical interest and can be formalised in terms of time-frequency distributions such as the so-called Short-time Fourier transform (STFT), defined as where ϕ is the "Gaussian window" normalized in such way that ϕ L 2 = 1. It is well known that Vf is a complex-valued, real analytic, bounded function and V : L 2 (R) → L 2 (R 2 ) is an isometry (see [27,33,41,50]).
It is customary to interpret |Vf (x, ω)| 2 as the time-frequency energy density of f (see [33,41]). Consequently, the fraction of energy captured by a measurable subset Ω ⊆ R 2 of a function f ∈ L 2 (R) \ {0} will be given by the Rayleigh quotient (see [2,3,18,42]) (1.2) Φ Ω (f ) : The bounded, nonnegative and self-adjoint operator V * 1 Ω V on L 2 (R) is known in the literature under several names, e.g. localization, concentration, Anti-Wick or Toeplitz operator, as well as time-frequency or timevarying filter. Since its first appearance in the works by Berezin [8] and Daubechies [18], the applications of such operators have been manifold and the related literature is enormous: we refer to the books [9,51] and the survey [16], and the references therein, for an account of the main results. Now, when Ω has finite measure, V * 1 Ω V is a compact (in fact, trace class) operator. Its norm V * 1 Ω V L(L 2 ) , given by the quantity represents the maximum fraction of energy that can in principle be trapped by Ω for any signal f ∈ L 2 (R), and explicit upper bounds for Φ Ω are of considerable interest. Indeed, the analysis of the spectrum of V * 1 Ω V was initiated in the seminal paper [18] for radially symmetric Ω, in which case the operator is diagonal in the basis of Hermite functions -and conversely [1] if an Hermite function is an eigenfunction and Ω is simply connected then Ω is a ball centered at 0-and the asymptotics of the eigenvalues (Weyl's law), in connection with the measure of Ω, has been studied by many authors; again the literature is very large and we address the interested reader to the contributions [2,3,20,42,43] and the references therein.
The study of the time-frequency concentration of functions, in relation to uncertainty principles and under certain additional constraints (e.g. on subsets of prescribed measure in phase space, or under limited bandwidth etc.) has a long history which, as recognized by Landau and Pollak [39], dates back at least to Fuchs [29], and its relevance both to theory and applications has been well known since the seminal works by Landau-Pollack-Slepian, see e.g. [28,38,48], and other relevant contributions such as those of Cowling and Price [17], Donoho and Stark [23], and Daubechies [18].
However, in spite of the abundance of deep and unexpected results related to this circle of ideas (see e.g. the visionary work by Fefferman [25]) the question of characterizing the subsets Ω ⊂ R 2 of prescribed measure, which allow for the maximum concentration, is still open. In this paper we provide a complete solution to this problem proving that the optimal sets are balls in phase space, and, in dimension one, our result can be stated as follows (see Theorem 4.1 for the same result in arbitrary dimension). Theorem 1.1 (Faber-Krahn inequality for the STFT). Among all measurable subsets Ω ⊂ R 2 having a prescribed (finite, non zero) measure, the quantity achieves its maximum if and only if Ω is equivalent, up to a set of measure zero, to a ball.
Moreover, when Ω is a ball of center (x 0 , ω 0 ), the only functions f that achieve the maximum in (1.3) are the functions of the kind that is, the scalar multiples of the Gaussian window ϕ defined in (1.1), translated and modulated according to (x 0 , ω 0 ).
This "Faber-Krahn inequality" (see Remark 1.3 at the end of this section) proves, in the L 2 -case, a conjecture by Abreu and Speckbacher [4] (the full conjecture is proved in Theorem 5.3), and confirms the distinguished role played by the Gaussian (1.4), as the first eigenfunction of the operator V * 1 Ω V when Ω has radial symmetry (see [18]; see also [23] for a related conjecture on band-limited functions, and [17, page 162] for further insight).
When Ω is a ball of radius r, one can see that Φ Ω = 1−e −πr 2 (this follows from the results in [18], and will also follow from our proof of Theorem 1.1). Hence we deduce a more explicit form of our result, which leads to a sharp form of the uncertainty principle for the STFT. Theorem 1.2 (Sharp uncertainty principle for the STFT). For every subset Ω ⊂ R 2 whose Lebesgue measure |Ω| is finite we have and, if |Ω| > 0, equality occurs if and only if Ω is a ball. As a consequence, if for some ∈ (0, 1), some function f ∈ L 2 (R) \ {0} and some Ω ⊂ R 2 we have Φ Ω (f ) ≥ 1 − , then necessarily Theorem 1.2 solves the long-standing problem of the optimal lower bound for the measure of the "essential support" of the STFT with Gaussian window. The best result so far in this direction was obtained by Gröchenig (see [33,Theorem 3.3.3]) as a consequence of Lieb's uncertainly inequality [40] for the STFT, and consists of the following (rougher, but valid for any window) lower bound (see Section 5 for a discussion in dimension d). Notice that the sup in (1.7) is a bounded function of ∈ (0, 1), as opposite to the optimal bound in (1.6) (see Fig. 1 in the Appendix for a graphical comparison).
We point out that, although in this introduction the discussion of our results is confined (for ease of notation and exposition) to the one dimensional case, our results are valid in arbitrary space dimension, as discussed in Section 4 (Theorem 4.1 and Corollary 4.2).
While addressing the reader to [12,28,34] for a review of the numerous uncertainty principles available for the STFT (see also [10,19,21,26]), we observe that inequality (1.5) is nontrivial even when Ω has radial symmetry: in this particular case it was proved in [32], exploiting the already mentioned diagonal representation in the Hermite basis.
Some concentration-type estimates were recently provided in [4] as an application of the Donoho-Logan large sieve principle [22] and the Selberg-Bombieri inequality [11]. However, though this machinery certainly has a broad applicability, as observed in [4] it does not seem to give sharp bounds for the problem above. For interesting applications to signal recovery we refer to [5,44,45,49] and the references therein.
Our proof of Theorem 1.1 (and of its multidimensional analogue Theorem 4.1) is based on techniques from measure theory, after the problem has been rephrased as an equivalent statement (where the STFT is no longer involved explicitly) in the Fock space. In order to present our strategy in a clear way and to better highlight the main ideas, we devote Section 3 to a detailed proof of our main results in dimension one, while the results in arbitrary dimension are stated and proved in Section 4, focusing on all those things that need to be changed and adjusted.
In Section 5 we discuss some extensions of the above results in different directions, such as a local version of Lieb's uncertainty inequality for the STFT in L p when p ∈ [2, ∞) (Theorem 5.2), and L p -concentration estimates for the STFT when p ∈ [1, ∞) (Theorem 5.3, which proves [4, Conjecture 1]), identifying in all cases the extremals f and Ω, as above. We also study the effect of changing the window ϕ by a dilation or, more generally, by a metaplectic operator.
We believe that the techniques used in this paper could also shed new light on the Donoho-Stark uncertainty principle [23] and the corresponding conjecture [23,Conjecture 1], and that also the stability of (1.5) (via a quantitative version when the inequality is strict) can be investigated. We will address these issues in a subsequent work, together with applications to signal recovery. Remark 1.3. The maximization of Φ Ω among all sets Ω of prescribed measure can be regarded as a shape optimization problem (see [14]) and, in this respect, Theorem 1.1 shares many analogies with the celebrated Faber-Krahn inequality (beyond the fact that both problems have the ball as a solution). The latter states that, among all (quasi) open sets Ω of given measure, the ball minimizes the first Dirichlet eigenvalue λ Ω := min On the other hand, if T Ω : H 1 0 (Ω) → H 1 0 (Ω) is the linear operator that associates with every (real-valued) u ∈ H 1 0 (Ω) the weak solution T Ω u ∈ H 1 0 (Ω) of the problem −∆(T Ω u) = u in Ω, integrating by parts we have ∇u · ∇(T Ω u) dz = T Ω u, u H 1 0 , so that Faber-Krahn can be rephrased by claiming that is maximized (among all open sets of given measure) by the ball. Hence the statement of Theorem 1.1 can be regarded as a Faber-Krahn inequality for the operator V * 1 Ω V.

Rephrasing the problem in the Fock space
It turns out that the optimization problems discussed in the introduction can be conveniently rephrased in terms of functions in the Fock space on C. We address the reader to [33,Section 3.4] and [52] for more details on the relevant results that we are going to review, in a self-contained form, in this section.
The Bargmann transform of a function f ∈ L 2 (R) is defined as It turns out that Bf (z) is an entire holomorphic function and B is a unitary operator from L 2 (R) to the Fock space F 2 (C) of all holomorphic functions F : C → C such that In fact, B maps the orthonormal basis of Hermite functions in R into the orthonormal basis of F 2 (C) given by the monomials In particular, for the first Hermite function ϕ(x) = 2 1/4 e −πx 2 , that is, the window in (1.1), we have Bϕ(z) = e 0 (z) = 1.
The connection with the STFT is based on the following crucial formula (see e.g. [33,Formula (3.30)]): which allows one to rephrase the functionals in (1.2) as is a unitary operator, we can safely transfer the optimization problem in Theorem 1.1 directly on F 2 (C), observing that We will adopt this point of view in Theorem 3.1 below.
In the meantime, two remarks are in order. First, we claim that the maximum in (2.4) is invariant under translations of the set Ω. To see this, consider for any z 0 ∈ C, the operator U z 0 defined as The map z → U z turns out to be a projective unitary representation of C on F 2 (C), satisfying which proves our claim. Invariance under rotations in the plane is also immediate. Secondly, we observe that the Bargmann transform intertwines the action of the representation U z with the so-called "time-frequency shifts": are the translation and modulation operators. This allows us to write down easily the Bargmann transform of the maximizers appearing in Theorem For future reference, we explicitly set The following result shows the distinguished role played by the functions F z 0 in connection with extremal problems. A proof can be found in [52,Theorem 2.7]. For the sake of completeness we present a short and elementary proof which generalises in higher dimension.
and |F (z)| 2 e −π|z| 2 vanishes at infinity. Moreover the equality in (2.8) occurs at some point z 0 ∈ C if and only if F = cF z 0 for some c ∈ C.
Proof. By homogeneity we can suppose F F 2 = 1, hence F = k≥0 c k e k (cf. (2.2)), with k≥0 |c k | 2 = 1. By the Cauchy-Schwarz inequality we obtain Equality in this estimate occurs at some point z 0 ∈ C if and only if c k = ce −π|z 0 | 2 /2 e k (z 0 ), for some c ∈ C, |c| = 1, which gives Finally, the fact that |F (z)| 2 e −π|z| 2 vanishes at infinity is clearly true if F (z) = z k , k ≥ 0, and therefore holds for every F ∈ F 2 (C) by density, because of (2.8).

Proof of the main results in dimension 1
In this section we prove Theorems 1.1 and 1.2. In fact, by the discussion in Section 2, cf. (2.4), these will follow (without further reference) from the following result, which will be proved at the end of this section, after a few preliminary results have been established.
Theorem 3.1. For every F ∈ F 2 (C)\{0} and every measurable set Ω ⊂ R 2 of finite measure, we have Moreover, recalling (2.7), equality occurs (for some F and for some Ω such that 0 < |Ω| < ∞) if and only if F = cF z 0 (for some z 0 ∈ C and some nonzero c ∈ C) and Ω is equivalent, up to a set of measure zero, to a ball centered at z 0 .
Throughout the rest of this section, in view of proving (3.1), given an arbitrary function F ∈ F 2 (C) \ {0} we shall investigate several properties of the function in connection with its super-level sets its distribution function (note that u is bounded due to (2.8)), and the decreasing rearrangement of u, i.e. the function (for more details on rearrangements, we refer to [7]). Since F (z) in (3.2) is entire holomorphic, u (which letting z = x + iω can be regarded as a real-valued function u(x, ω) on R 2 ) has several nice properties which will simplify our analysis. In particular, u is real analytic and hence, since u is not a constant, every level set of u has zero measure (see e.g. [37]), i.e. Since u(z) > 0 a.e. (in fact everywhere, except at most at isolated points), µ(0) = lim t→0 + µ(t) = +∞, while the finiteness of µ(t) when t ∈ (0, max u] is entailed by the fact that u ∈ L 1 (R 2 ), according to (3.2) and (2.1) (in particular µ(max u) = 0). Moreover, by (3.6) µ(t) is continuous (and not just right-continuous) at every point t ∈ (0, max u]. Since µ is also strictly decreasing, we see that u * , according to (3.5), is just the elementarly defined inverse function of µ (restricted to (0, max u]), i.e.
In the following we will strongly rely on the following result.
Lemma 3.2. The function µ is absolutely continuous on the compact subintervals of (0, max u], and Similarly, the function u * is absolutely continuous on the compact subintervals of [0, +∞), and These properties of µ and u * are essentially well known to the specialists in rearrangement theory, and follow e.g. from the general results of [6,13], which are valid within the framework of W 1,p functions (see also [15] for the framework of BV functions, in particular Lemmas 3.1 and 3.2). We point out, however, that of these properties only the absolute continuity of u * is valid in general, while the others strongly depend on (3.7) which, in the terminology of [6], implies that u is coarea regular in a very strong sense, since it rules out the possibility of a singular part in the (negative) Radon measure µ (t) and, at the same time, it guarantees that the density of the absolutely continuous part is given (only) by the right-hand side of (3.10). As clearly explained in the excellent Introduction to [6], there are several subtleties related to the structure of the distributional derivative of µ(t) (which ultimately make the validity of (3.11) highly nontrivial), and in fact the seminal paper [13] was motivated by a subtle error in a previous work, whose fixing since [13] has stimulated a lot of original and deep research (see e.g. [15,31] and references therein).
However, since unfortunately we were not able to find a ready-to-use reference for (3.11) (and, moreover, our u is very smooth but strictly speaking it does not belong to W 1,1 (R 2 ), which would require to fix a lot of details when referring to the general results from [6,13,15]), here we present an elementary and self-contained proof of this lemma, specializing to our case a general argument from [13] based on the coarea formula.
Proof of Lemma 3.2. The fact that u is locally Lipschitz guarantees the validity of the coarea formula (see e.g. [13,24]), that is, for every Borel where H 1 denotes the one-dimensional Hausdorff measure (and with the usual convention that 0 · ∞ = 0 in the first integral). In particular, when h(z) = χ At (z)|∇u(z)| −1 (where |∇u(z)| −1 is meant as +∞ if z is a critical point of u), by virtue of (3.7) the function h(z)|∇u(z)| coincides with χ At (z) a.e., and recalling (3.4) one obtains therefore we see that µ(t) is absolutely continuous on the compact subintervals of (0, max u], and (3.10) follows. Now let D ⊆ (0, max u) denote the set where µ (t) exists, coincides with the integral in (3.10) and is strictly positive, and let D 0 = (0, max u] \ D. By (3.10) and the absolute continuity of µ, and since the integral in (3.10) is strictly positive for every t ∈ (0, max u) (note that H 1 ({u = t}) > 0 for every t ∈ (0, max u), otherwise we would have that |{u > t}| = 0 by the isoperimetric inequality), we infer that |D 0 | = 0, so that letting D = µ(D) and D 0 = µ(D 0 ), one has | D 0 | = 0 by the absolute continuity of µ, and D = [0, +∞) \ D 0 since µ is invertible. On the other hand, by (3.9) and elementary calculus, we see that (u * ) (s) exists for every s ∈ D and which implies (3.11) since | D 0 | = 0. Finally, since u * is differentiable everywhere on D, it is well known that u * maps every negligible set N ⊂ D into a negligible set. Since D ∪ D 0 = [0, +∞), and moreover u * ( D 0 ) = D 0 where |D 0 | = 0, we see that u * maps negligible sets into negligible sets, hence it is absolutely continuous on every compact interval [0, a].
The following estimate for the integral in (3.11), which can be of some interest in itself, will be the main ingredient in the proof of Theorem 3.1.
Proof. Letting for simplicity t = u * (s) and recalling that, for a.e. t ∈ (0, max u) (or, equivalently, for a.e. s > 0, since u * and its inverse µ are absolutely continuous on compact sets) the super-level set A t in (3.3) has a smooth boundary as in (3.8), we can combine the Cauchy-Schwarz inequality with the isoperimetric inequality in the plane to obtain, after division by t, The reason for dividing by t is that, in this form, the right-hand side turns out to be (quite surprisingly, at least to us) independent of t. Indeed, since along ∂A t = {u = t} we have |∇u| = −∇u · ν where ν is the outer normal to ∂A t , along {u = t} we can interpret the quotient |∇u|/t as −(∇ log u)·ν, and hence But by (3.2), since log |F (z)| is a harmonic function, we obtain so that the last integral equals 4π|A t |. Plugging this into (3.17), one obtains that the quotient on the right equals 1, and (3.13) follows. Finally, (3.14) follows on combining (3.11) with (3.13).
The following lemma establishes a link between the integrals of u on its super-level sets (which will play a major role in our main argument) and the function u * .
Moreover, it is easy to see that the same inequality is true also when h < 0 (provided s + h > 0), now using the reverse set inclusion A u * (s+h) ⊂ A u * (s) according to the fact that u * is decreasing. Since u * is continuous, (3.20) follows letting h → 0 when s > 0, and letting h → 0 + when s = 0. Finally, by Lemma 3.2, I = u * is absolutely continuous on [0, a] for every a ≥ 0, I = (u * ) , and (3.21) follows from (3.14).
We are now in a position to prove Theorem 3.1.
Proof of Theorem 3.1. By homogeneity we can assume F F 2 = 1 so that, defining u as in (3.2), (3.1) is equivalent to for every s ≥ 0 and every Ω ⊂ R 2 such that |Ω| = s. It is clear that, for any fixed measure s ≥ 0, the integral on the left is maximized when Ω is the (unique by (3.6)) super-level set A t = {u > t} such that |A t | = s (i.e. µ(t) = s), and by (3.9) we see that the proper cut level is given by t = u * (s). In other words, if |Ω| = s then with strict inequality unless Ω coincides -up to a negligible set-with A u * (s) (to see this, it suffices to let E := Ω∩A u * (s) and observe that, if |Ω\E| > 0, then the integral of u on Ω \ E, where u ≤ u * (s), is strictly smaller than the integral of u on A u * (s) \ E, where u > u * (s)). Thus, to prove (3.1) it suffices to prove (3.22) when Ω = A u * (s) , that is, recalling (3.19), prove that for every F such that F F 2 = 1. Now assume that equality occurs in (3.1), for some F (we may still assume F F 2 = 1) and for some set Ω of measure s 0 > 0: then, when s = s 0 , equality occurs everywhere in 1 = max u = max |F (z)| 2 e −π|z| 2 ≤ F 2 F 2 = 1 and, since equality is attained, by Proposition 2.1 we infer that F = cF z 0 for some z 0 , c ∈ C. We have already proved that Ω = A u * (s 0 ) (up to a negligible set) and, since by (2.7) has radial symmetry about z 0 and is radially decreasing, Ω is (equivalent to) a ball centered at z 0 . This proves the "only if part" of the final claim being proved.
The "if part" follows by a direct computation. For, assume that F = cF z 0 and Ω is equivalent to a ball of radius r > 0 centered at z 0 . Then using (3.30) we can compute, using polar coordinates and equality occurs in (3.1) because cF z 0 2 F 2 = |c| 2 . Remark 3.5. The "only if part" in the final claim of Theorem 3.1, once one has established that I(s) = 1 − e −s for every s ≥ 0, instead of using (3.29), can also be proved observing that there must be equality, for a.e. t ∈ (0, max u), both in (3.15) and in (3.16) (otherwise there would be a strict inequality in (3.14), hence also in (3.24), on a set of positive measure). But then, for at least one value (in fact, for infinitely many values) of t we would have that A t is a ball B(z 0 , r) (by the equality in the isoperimetric estimate (3.16)) and that |∇u| is constant along ∂A t = {u = t} (by the equality in (3.15)).
By applying the "translation" U z 0 (cf. (2.5) and (2.6)) we can suppose that the super-level set A t = B(z 0 , r) is centred at the origin, i.e. that z 0 = 0, and in that case we have to prove that F is constant (so that, translating back to z 0 , one obtains that the original F had the form cF z 0 ). Since now both u and e −|z| 2 are constant along ∂A t = ∂B(0, r), also |F | is constant there (and does not vanish inside B(0, r), since u ≥ t > 0 there). Hence log |F | is constant along ∂B(0, r), and is harmonic inside B(0, r) since F is holomorphic: therefore log |F | is constant in B(0, r), which implies that F is constant over C.
Note that the constancy of |∇u| along ∂A t has not been used. However, also this property alone (even ignoring that A t is a ball) is enough to conclude. Letting w = log u, one can use that both w and |∇w| are constant along ∂A t , and moreover ∆w = −4π as shown in (3.18): hence every connected component of A t must be a ball, by a celebrated result of Serrin [46]. Then the previous argument can be applied to just one connected component of A t , which is a ball, to conclude that F is constant.

The multidimensional case
In this Section we provide the generalisation of Theorems 1.1 and 1.2 (in fact, of Theorem 3.1) in arbitrary dimension.
We recall that the STFT of a function f ∈ L 2 (R d ), with a given window Consider now the Gaussian function and the corresponding STFT in (4.1) with window g = ϕ; let us write shortly V = V ϕ . Let ω 2d be the measure of the unit ball in R 2d . Recall also the definition of the (lower) incomplete γ function as where k ≥ 1 is an integer and s ≥ 0, so that

Theorem 4.1 (Faber-Krahn inequality for the STFT in dimension d).
For every measurable subset Ω ⊂ R 2d of finite measure and for every f ∈ where c Ω := π(|Ω|/ω 2d ) 1/d is the symplectic capacity of the ball in R 2d having the same volume as Ω. Moreover, equality occurs (for some f and for some Ω such that 0 < |Ω| < ∞) if and only if Ω is equivalent, up to a set of measure zero, to a ball centered at some (x 0 , ω 0 ) ∈ R 2d , and where ϕ is the Gaussian in (4.2).
We recall that the symplectic capacity of a ball of radius r in phase space is πr 2 in every dimension and represents the natural measure of the size of the ball from the point of view of the symplectic geometry [19,35,36].
Proof of Theorem 4.1. We give only a sketch of the proof, because it follows the same pattern as in dimension 1.
The definition of the Fock space F 2 (C) extends essentially verbatim to C d , with the monomials (π |α| /α!) 1/2 z α , z ∈ C d , α ∈ N d (multi-index notation) as orthonormal basis. The same holds for the definition of the functions F z 0 in (2.7), now with z, z 0 ∈ C d , and Proposition 2.1 extends in the obvious way too. Again one can rewrite the optimization problem in the Fock space F 2 (C d ), the formula (2.3) continuing to hold, with x, ω ∈ R d . Hence we have to prove that and Ω ⊂ C d of finite measure, and that equality occurs if and only if F = cF z 0 and Ω is equivalent, up to a set of measure zero, to a ball centered at z 0 . To this end, for F ∈ F 2 (C d )\{0}, F F 2 = 1, we set u(z) = |F (z)| 2 e −π|z| 2 , z ∈ C d , exactly as in (3.2) when d = 1, and define A t , µ(t) and u * (s) as in Section 3, replacing R 2 with R 2d where necessary, now denoting by |E| the 2d-dimensional Lebesgue measure of a set E ⊂ R 2d , in place of the 2-dimensional measure. Note that, now regarding u as a function of 2d real variables in R 2d , properties (3.6), (3.7) etc. are still valid, as well as formulas (3.10), (3.11) etc., provided one replaces every occurrence of H 1 with the (2d − 1)-dimensional Hausdorff measure H 2d−1 . Following the same pattern as in Proposition 3.3, now using the isoperimetric inequality in R 2d (see e.g. [30] for an updated account) 2d |{u > t}| (2d−1)/d and the fact that log u = −4πd on {u > 0}, we see that now u * satisfies the inequality in place of (3.13), and hence (3.14) is to be replaced with in place of (3.24). This, with the substitution (recall (4.4)), turns into Again G extends to a continuous function on [0, 1], with G(0) = 1, G(1) = 0. At this point one observes that, regarding σ as a function of s, Since the function e π(s/ω 2d ) 1/d I (s) is non decreasing, we see that G is non increasing on (0, 1], hence G is convex on [0, 1] and one concludes as in the proof of Theorem 3.1. Finally, the "if part" follows from a direct computation, similar to that at the end of the proof of Theorem 3.1, now integrating on a ball in dimension 2d, and using (4.3) to evaluate the resulting integral.
As a consequence of Theorem 4.1 we deduce a sharp form of the uncertainty principle for the STFT, which generalises Theorem 1.2 to arbitrary dimension. To replace the function log(1/ ) in (1.6) (arising as the inverse function of e −s in the right-hand side of (1.5)), we now denote by ψ d ( ), 0 < ≤ 1, the inverse function of 4.4)).
Corollary 4.2. If for some ∈ (0, 1), some f ∈ L 2 (R d ) \ {0}, and some with equality if and only if Ω is a ball and f has the form (4.6), where (x 0 , ω 0 ) is the center of the ball.
So far, the state-of-the-art in this connection has been represented by the lower bound (which reduces to (1.7) when d = 1, see [33,Theorem 3.3.3]). See Figure 1 in the Appendix for a graphical comparison with (4.8) in dimension d = 2. Figure 2 in the Appendix illustrates Theorem 4.1 and Corollary 4.2.
Remark. Notice that ψ 1 ( ) = log(1/ ), and ψ d ( ) is increasing with d. Moreover, it is easy to check that On the contrary, the right-hand side of (4.9) is bounded by e d ; see Figure  1 in the Appendix.

Some generalizations
In this Section we discuss some generalizations in several directions.
5.1. Local Lieb's uncertainty inequality for the STFT. An interesting variation on the theme is given by the optimization problem where Ω ⊂ R 2 is measurable subset of finite measure and 2 ≤ p < ∞. Again, we look for the subsets Ω, of prescribed measure, which maximize the above supremum.
Observe, first of all, that by the Cauchy-Schwarz inequality, Vf L ∞ ≤ f L 2 , so that the supremum in (5.1) is finite and, in fact, it is attained. Proof. The desired conclusion follows easily by the direct method of the calculus of variations. We first rewrite the problem in the complex domain via (2.3), as we did in Section 2, now ending up with the Rayleigh quotient It is easy to see that this expression attains a maximum at some F ∈ F 2 (C) \ {0}. In fact, let F n ∈ F 2 (C), F n F 2 = 1, be a maximizing sequence, and let u n (z) = |F n (z)| p e −pπ|z| 2 /2 . Since u n (z) = (|F n (z)| 2 e −π|z| 2 ) p/2 ≤ F n p F 2 = 1 by Proposition 2.1, we see that the sequence F n is equibounded on the compact subsets of C. Hence there is a subsequence, that we continue to call F n , uniformly converging on the compact subsets to a holomorphic function F . By the Fatou theorem, F ∈ F 2 (C) and F F 2 ≤ 1. Now, since Ω has finite measure, for every > 0 there exists a compact subset K ⊂ C such that |Ω \ K| < , so that Ω\K u n < and Ω\K |F (z)| p e −pπ|z| 2 /2 dz < . Together with the already mentioned convergence on the compact subsets, this implies that Ω u n (z) dz → Ω |F (z)| p e −pπ|z| 2 /2 dz. As a consequence, F = 0 and, since The reverse inequality is obvious, because F n is a maximizing sequence.
Theorem 5.2 (Local Lieb's uncertainty inequality for the STFT). Let 2 ≤ p < ∞. For every measurable subset Ω ⊂ R 2 of finite measure, and every f ∈ L 2 (R) \ {0}, Moreover, equality occurs (for some f and for some Ω such that 0 < |Ω| < ∞) if and only if Ω is equivalent, up to a set of measure zero, to a ball centered at some (x 0 , ω 0 ) ∈ R 2 , and where ϕ is the Gaussian in (1.1).
Observe that when p = 2 this result reduces to Theorem 1.1. Moreover, by monotone convergence, from (5.2) we obtain which is Lieb's inequality for the STFT with a Gaussian window (see [40] and [33,Theorem 3.3.2]). Actually, (5.3) will be an ingredient of the proof of Theorem 5.2.
Proof of Theorem 5.2. Transfering the problem in the Fock space F 2 (C), it is sufficient to prove that and that the extremals are given by the functions F = cF z 0 in (2.7), together with the balls Ω of center z 0 . We give only a sketch of the proof, since the argument is similar to the proof of Theorem 3.1. Assuming F F 2 = 1 and setting u(z) = |F (z)| p e −pπ|z| 2 /2 , arguing as in the proof of Proposition 3.3 we obtain that for a.e. s > 0, which implies (u * ) (s) + p 2 u * (s) ≥ 0 for a.e. s ≥ 0. With the notation of Lemma 3.4 we obtain I (s) + p 2 I (s) ≥ 0 for a.e. s ≥ 0, i.e. e sp/2 I (s) is non decreasing on [0, +∞). Arguing as in the proof of Theorem 3.1 we reduce ourselves to study the inequality I(s) ≤ 2 p (1−e −ps/2 ) or equivalently, changing variable s = − 2 p log σ, σ ∈ (0, 1], We can prove this inequality and discuss the case of strict inequality as in the proof of Theorem 3.1, the only difference being that now G(0) := lim σ→0 + G(σ) = R 2 u(z) dz ≤ 2/p by (5.3) (hence, at σ = 0 strict inequality may occur in (5.4), but this is enough) and, when in (5.4) the equality occurs for some (and hence for every) σ ∈ [0, 1], in place of (3.29) we will have 1 = max u = max |F (z)| p e −pπ|z| 2 /2 = (max |F (z)| 2 e −π|z| 2 ) p/2 ≤ F p F 2 = 1. The "if part" follows by a direct computation. 5.2. L p -concentration estimates for the STFT. We consider now the problem of the concentration in the time-frequency plane in the L p sense, which has interesting applications to signal recovery (see e.g. [5]). More precisely, Theorem 5.3 below proves a conjecture of Abreu and Speckbacher [4, Conjecture 1] (note that when p = 2 one obtains Theorem 1.1).
Let S (R) be the space of temperate distributions and, for p ≥ 1, consider the subspace (called modulation space in the literature [33]) In fact, the definition of STFT (with Gaussian or more generally Schwartz window) and the Bargmann transform B extend (in an obvious way) to injective operators on S (R). It is clear that V : M p (R) → L p (R 2 ) is an isometry, and it can be proved that B is a surjective isometry from M p (R) onto the space F p (C) of holomorphic functions F (z) satisfying see e.g. [47] for a comprehensive discussion. Moreover the formula (2.3) continues to hold for f ∈ S (R).
Moreover, equality occurs (for some f and for some Ω such that 0 < |Ω| < ∞) if and only if Ω is equivalent, up to a set of measure zero, to a ball centered at some (x 0 , ω 0 ) ∈ R 2 , and where ϕ is the Gaussian in (1.1).
We omit the proof, that is very similar to that of Theorem 3.1. We just observe that (2.8) extends to any p ∈ [1, ∞) as and again the equality occurs at some point z 0 ∈ C if and only if F = cF z 0 , for some c ∈ C (in particular, F z 0 p F p = 2/p); see e.g. [52, Theorem 2.7]. As a consequence we obtain at once the following sharp uncertainty principle for the STFT. We point out that, in the case p = 1, the following rougher -but valid for any window in M 1 (R) \ {0}-lower bound |Ω| ≥ 4(1 − ) 2 was obtained in [34,Proposition 2.5.2]. Arguing as in Section 4 it would not be difficult to suitably generalise Theorem 5.3 and Corollary 5.4 in arbitrary dimension. 5.3. Changing window. Theorem 4.1 can be suitably reformulated when the Gaussian window ϕ in (4.2) is dilated or, more generally, replaced by µ(A)ϕ, where µ(A) is a metaplectic operator associated with a symplectic matrix A ∈ Sp(d, R) (recall that, in dimension 1, Sp(1, R) = SL(2, R) is the special linear group of 2 × 2 real matrices with determinant 1).
We address to [34,Section 9.4] for a detailed introduction to the metaplectic representation. Roughly speaking one associates, with any matrix A ∈ Sp(d, R), a unitary operator µ(A) on L 2 (R d ) defined up to a phase factor, providing a projective unitary representation of Sp(d, R) on L 2 (R d ).