March 10, 2025
We prove that, among all radial subsets \(\Omega\subset \mathbb{C}\) of prescribed measure, the ball is the only maximizer of the sum of the first \(K\) eigenvalues (\(K\geq 1\)) of the corresponding Toeplitz operator \(T_\Omega\) on the Fock space \(\mathcal{F}^2(\mathbb{C})\). As a byproduct, we prove that balls maximize any Schatten \(p\)-norm of \(T_\Omega\) for \(p>1\) (and minimize the corresponding quasinorm for \(p<1\)), and that the second eigenvalue is maximized by a particular annulus. Moreover, we extend some of these results to general radial symbols in \(L^p(\mathbb{C})\), with \(p > 1\), characterizing those that maximize the sum of the first \(K\) eigenvalues.
In recent years, there has been a growing interest in optimization problems arising in the context of time-frequency analysis and complex analysis, for example concerning the maximal concentration of the STFT with Gaussian window [1]–[3] or the wavelet transform with respect to the Cauchy wavelet [4] and related localization operators [5]–[8], optimal estimates for functionals and contractive estimates on various reproducing kernel Hilbert spaces [9]–[15]. A case of special interest is given by the Fock space \(\mathcal{F}^2(\mathbb{C})\) of entire functions \(F:\mathbb{C}\to\mathbb{C}\) with finite norm \[\|F\|_{\mathcal{F}^2(\mathbb{C})}^2=\int_\mathbb{C}|F(z)|^2 e^{-\pi |z|^2} dA(z),\] where \(dA(z)\) stands for the Lebesgue measure.
In [3] the first and third author proved the following concentration estimate for normalized functions \(F\in \mathcal{F}^2(\mathbb{C})\): \[\int_{\Omega} |F(z)|^2 e^{-\pi |z|^2}\, dA(z) \leq 1 - e^{-|\Omega|},\] where \(\Omega \subset \mathbb{C}\) is any subset with finite measure. They also showed that equality is achieved if and only if \(\Omega \subset \mathbb{C}\) is equivalent, up to a set a measure zero, to a ball (that is, its symmetric difference with some ball has measure zero).
If we denote with \(P\) the projection operator from \(L^2(\mathbb{C}, e^{-\pi |z|^2} dA(z))\) onto \(\mathcal{F}^2(\mathbb{C})\), this result can be seen as an optimal estimate for the first eigenvalue of the Toeplitz operator \(T_{\Omega} = P \chi_{\Omega} \colon \mathcal{F}^2(\mathbb{C}) \rightarrow \mathcal{F}^2(\mathbb{C})\), where \(\chi_{\Omega}\) is the characteristic function of the set \(\Omega\). These operators, as well known, are bounded, self-adjoint, compact and positive on \(\mathcal{F}^2(\mathbb{C})\) (see, e.g., [16]), and therefore their eigenvalues can be ordered in a decreasing way \(\lambda_1(\Omega) \geq \lambda_2(\Omega) \geq \cdots > 0\). Hence, the aforementioned result can be stated as follows: for any set \(\Omega \subset \mathbb{C}\) of finite measure it holds \[\lambda_1(\Omega) \leq 1 - e^{-|\Omega|}\] and equality is achieved if and only if \(\Omega\) is (equivalent up to a set of measure zero to) a ball. This estimate was already proved in [5] for radial sets, that is for sets of the kind \(\{z \in \mathbb{C}\colon |z| \in I\}\) for some \(I \subset [0,\infty)\).
In this paper we are interested in extending this result by considering the sum of the first \(K\) eigenvalues, for some natural number \(K\), in the case of radial sets. We are going to prove that, if \(\Omega\subset\mathbb{C}\) is a radial set of finite measure, we have \[\sum_{k=1}^K \lambda_k(\Omega)\leq K - \sum_{k=0}^{K-1} (K-k)\dfrac{|\Omega|^k}{k!}e^{-|\Omega|},\] and equality occurs if and only if \(\Omega\) is equivalent, up to a set of measure zero, to a ball.
The proof is based on a simple yet unexpected observation about integrals over the super-level sets of monomials in the Fock space and it is unrelated to the proof given in [3] of the above mentioned inequality for the first eigenvalue. However, while the latter also deals with the non-radial case, our proof is specific to the radial setting. An extension to the non-radial case seems at the moment out of reach and requires different techniques.
The paper is organized as follows. In Section 2 we set notation and some preliminary facts to state our main result, that is stated in Theorem 1 as a concentration inequality for orthonormal systems in the Fock space, and whose proof is detailed in Section 4.
In Section 3 we derive several interesting corollaries of Theorem 1. First, we rephrase the main theorem in terms of partial sums of the first eigenvalues of the operator \(T_{\Omega}\) and generalize this result to weighted sums. Then, using the Hardy–Littlewood majorization theory we prove that, among all radial subsets of \(\mathbb{C}\) of finite positive measure, balls maximize functionals of the kind \(\mathrm{tr}\,\Phi(T_{\Omega})\), where \(\Phi \colon (0,1) \to \mathbb{R}\) is an arbitrary convex function. Moreover, if \(\Phi\) is not affine in \((0,1)\) we also prove that balls are the only extremizers. In particular, taking \(\Phi(t) = t^p\) we deduce that the balls are the only maximizers for the Schatten \(p\)-(quasi-)norm of \(T_{\Omega}\) for \(p>1\), and minimizers for \(p<1\). Also, exploiting the Bargmann transform, we rephrase the results in terms of the short-time Fourier transform with Gaussian window and time-frequency localization operators.
In Section 5 we address a maximization problem for the second eigenvalue and we show that this is maximized (only) by a particular annulus.
Finally, in Section 6 we consider Toeplitz operators with a generic radial weight and we prove that the sum of the first eigenvalues increases under symmetric rearrangement of the weight. Moreover, we are also able to characterize the weights that maximize the aforementioned sum, among all radial weights with a fixed \(L^p(\mathbb{C})\) norm, for some \(p>1\).
It is likely that a quantitative version of our main theorem can be proved, using refinements of these techniques. Moreover, analogous questions can be raised in higher dimension or in different settings, such as for functions in weighted Bergman spaces or spaces of polynomials. We plan to investigate these issues in future work.
In the end, we also extend some of the previous results to Toeplitz operators with a generic radial symbol. As a simple consequence, we characterize those radial weights that, among all weights with the same \(L^p(\mathbb{C})\), maximize the sum of the first \(K\) eigenvalues.
We denote by \(\mathcal{F}^2(\mathbb{C})\) the Fock space of entire functions on \(\mathbb{C}\) that are in the space \(L^2(\mathbb{C}, e^{-\pi |z|^2}dA(z))\) of square-integrable functions with respect to the measure \(e^{-\pi |z|^2}dA(z)\), where \(dA(z)\) denotes the Lebesgue measure on \(\mathbb{C}\). It is a closed subspace of \(L^2(\mathbb{C}, e^{-\pi |z|^2}dA(z))\) and the orthogonal projection \(P \colon L^2(\mathbb{C}, e^{-\pi |z|^2}dA(z)) \rightarrow \mathcal{F}^2(\mathbb{C})\) is given by \[Pf(z) = \int_{\mathbb{C}} K(z,w) f(w) e^{-\pi |w|^2} \, dA(w),\] where \(K(z,w) = e^{\pi z \overline{w}}\) is the reproducing kernel of the Fock space. Given \(\Omega \subset \mathbb{C}\) measurable, we denote with \(|\Omega|\) its Lebesgue measure and we consider the corresponding Toeplitz operator \(T_{\Omega} \mathrel{\vcenter{:}}= P \chi_{\Omega}\) in \(\mathcal{F}^2(\mathbb{C})\). If \(|\Omega|\) is finite, then \(T_{\Omega}\) is bounded, compact, self-adjoint and positive (see, e.g., [16]). We denote by \(\{\lambda_k(\Omega)\}_{k=1}^{\infty}\) the sequence of its eigenvalues ordered in a decreasing way (with multiplicity).
When \(\Omega\) is radial its eigenfunctions are given by the normalized monomials (see [17], [18]) \[e_k(z) = \sqrt{\dfrac{\pi^k}{k!}}z^k, \quad z \in \mathbb{C},\;k \in \mathbb{N},\] with the understanding that \(\mathbb{N}\) starts with 0, while reserving the notation \(\mathbb{N}_+\) for the set of positive natural numbers. Hence, the corresponding eigenvalues are given by \[\label{eq:expression32eigenvalues} \int_{\Omega} |e_k(z)|^2 e^{-\pi|z|^2} \, dA(z), \quad k \in \mathbb{N}.\tag{1}\]
In relation with these monomials, we introduce the following notation: \[\varphi_k(s) = \dfrac{s^k}{k!},\;\quad |e_k(z)|^2 = \dfrac{\pi^k}{k!}|z|^{2k} = \varphi_k(\pi |z|^2), \quad k \in \mathbb{N}.\] For \(K \in \mathbb{N}_+\), we are going to consider vectors of natural numbers \(\alpha = (\alpha_1, \ldots, \alpha_k) \in \mathbb{N}^K\) that are strictly increasing, meaning that they satisfy \(\alpha_1 < \alpha_2 < \dots < \alpha_K\). Among such vectors, of particular importance is \(\alpha^* = (0,\ldots,K-1)\), which will be denoted \(\alpha^*(K)\) when its length needs to be specified. Given a strictly increasing vector \(\alpha \in \mathbb{N}^K\), we define the following operations:
\(\alpha+1 = (\alpha_1+1, \ldots, \alpha_K+1)\);
\(\alpha-1 = (\alpha_1-1, \ldots, \alpha_K-1)\); clearly this operation is allowed only if \(\alpha_1 > 0\);
we write \(\alpha = (\alpha', \alpha'')\), where \(\alpha' = (0,\ldots,l)\) for the largest possible \(l\). Both \(\alpha'\) and \(\alpha''\) may be absent. Some examples with \(K=3\) are:
\(\alpha=(0,1,2) \rightarrow \alpha'=(0,1,2),\;\alpha''=()\);
\(\alpha=(0,1,3) \rightarrow \alpha'=(0,1),\;\alpha''=(3)\);
\(\alpha=(1,2,3) \rightarrow \alpha'=(),\;\alpha''=(1,2,3)\).
To every increasing vector \(\alpha \in \mathbb{N}^K\) we associate the following function: \[u_{\alpha}(z) = \left(\sum_{k=1}^{K} |e_{\alpha_k}(z)|^2\right) e^{-\pi |z|^2} = \left(\sum_{k=1}^{K} \varphi_{\alpha_k}(\pi |z|^2) \right) e^{-\pi |z|^2} = v_{\alpha}(\pi |z|^2),\] where \[v_{\alpha}(s) = \left(\sum_{k=1}^{K} \varphi_{\alpha_k}(s)\right) e^{-s}, \quad s \geq 0.\] We denote by \(\Omega_{\alpha}\) any super-level set \(\{u_{\alpha} > t\}\) with \(t \in (0,1)\). Since \(u_{\alpha}\) is radial, \(u_{\alpha} \rightarrow 0\) as \(|z| \rightarrow \infty\) and \(v_{\alpha}\) has finitely many critical points, \(\Omega_{\alpha}\) is the union of a (possibly empty) open ball and finitely many annular regions, that is \[\label{eq:expression32of32Omega} \Omega_{\alpha} = B_{R_0} \cup \bigcup_{m=1}^{M} B_{R_m} \setminus \overline{B_{r_m}} = \bigcup_{m=0}^{M} B_{R_m} \setminus \overline{B_{r_m}},\tag{2}\] where \(r_0 = 0\), whereas \(R_0 \geq 0\) and is strictly positive if and only if \(\alpha_1 = 0\) (since otherwise \(u_{\alpha}(0) = 0\)) and \(0 = r_0 \leq R_0 < r_1 < R_1 < \dots < r_M < R_M\). We also introduce \(a_m = \pi r_m^2\) and \(b_m = \pi R_m^2\), \(m=0,\ldots,M\).
Having set the notation, we can state our main theorem.
Theorem 1. Let \(\Omega \subset \mathbb{C}\) be a radial set with finite positive measure and let \(\{F_1, \ldots, F_K\}\) be an orthonormal set in the Fock space \(\mathcal{F}^2(\mathbb{C})\) for some \(K \in \mathbb{N}_+\). Then \[\label{eq:main32estimate} \sum_{k=1}^K \int_{\Omega} |F_k(z)|^2 e^{-\pi |z|^2} \, dA(z) \leq \int_{\Omega_{\alpha^*}} u_{\alpha^*}(z) \, dA(z) = K - \sum_{k=0}^{K-1} (K-k)\dfrac{|\Omega|^k}{k!}e^{-|\Omega|},\qquad{(1)}\] where \(\Omega_{\alpha^*}\) is the unique super-level set of \(u_{\alpha^*}\) with the same measure as \(\Omega\), that is an open ball centered at 0.
Equality, among all possible radial sets with the same positive finite measure, is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega_{\alpha^*}\) and \[\begin{pmatrix} F_1(z) \\ \vdots \\ F_K(z) \end{pmatrix} = U \begin{pmatrix} e_0(z) \\ \vdots \\ e_{K-1}(z) \end{pmatrix}\] for every \(z \in \mathbb{C}\) and for some unitary matrix \(U \in \mathbb{C}^{K \times K}\).
Remark 2. An analogous problem can be posed for sets that are radial with respect to a given point \(z_0 \in \mathbb{C}\) different from the origin, that is sets of the kind \(\{z \in \mathbb{C}\colon |z-z_0| \in I\}\) for some \(I \subset [0,\infty)\). Our theorem can be easily transferred to this situation exploiting the unitary map on \(\mathcal{F}^2(\mathbb{C})\) given by \[U_{-z_0}F(z) = e^{-\pi |z_0|^2/2}e^{-\pi z \overline{z_0}}F(z+z_0), \quad F \in \mathcal{F}^2(\mathbb{C}),\;z \in \mathbb{C},\] which has the property that \(|F(z+z_0)|^2 e^{-\pi |z+z_0|^2} = |U_{-z_0} F(z)|^2 e^{-\pi |z|^2}\).
Theorem 1 has a number of interesting corollaries that we are going to state and prove in this section.
Using the variational characterization of eigenvalues, the next corollary follows immediately. In fact, we will prove this result together with Theorem 1 in Section 4.
Corollary 3. Let \(\Omega \subset \mathbb{C}\) be a radial set with finite positive measure and fix \(K \in \mathbb{N}_+\). Then \[\label{eq:main32estimate32for32eigenvalues} \sum_{k=1}^{K} \lambda_k(\Omega) \leq \sum_{k=1}^{K} \lambda_k(\Omega^*) = K - \sum_{k=0}^{K-1} (K-k)\dfrac{|\Omega|^k}{k!}e^{-|\Omega|},\qquad{(2)}\] where \(\Omega^* \subset \mathbb{C}\) is the open ball of center 0 and the same measure as \(\Omega\). Equality is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\).
Remark 4. We point out that the above formula is in agreement with two known results. Indeed, when \(K=1\) we have \[\lambda_1(\Omega) \leq 1 - e^{-|\Omega|},\] which was already mentioned in the Introduction. On the other hand, if we take the limit \(K \rightarrow \infty\) we obtain \[\sum_{k=1}^{\infty} \lambda_k(\Omega) \leq |\Omega|,\] which agrees with the fact that the left-hand side is the trace of \(P \chi_{\Omega}\), which is equal to \(|\Omega|\) (see [19]).
Corollary 3 can be extended to weighted sums of the first \(K\) eigenvalues with decreasing weights.
Corollary 5. Let \(t_1 \geq t_2 \geq \cdots \geq t_K > 0\) for some \(K \in \mathbb{N}_+\). Let \(\Omega \subset \mathbb{C}\) be a radial set of finite positive measure. Then \[\label{eq:inequality32for32weighted32sum} \sum_{k=1}^K t_k\lambda_k(\Omega) \leq \sum_{k=1}^K t_k\lambda_k(\Omega^*),\qquad{(3)}\] where \(\Omega^*\) is the open ball of center 0 and the same measure as \(\Omega\). Equality is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\).
Proof. First of all, if \(t_k = t_K\) for \(k=1,\ldots,K\) the statement immediately follows from Corollary 3. Hence, we may assume that not all the values in the sequence \(\{t_k\}_{k=1}^K\) are equal.
The proof is by strong induction on \(K\). The base case \(K=1\) is obvious. Now assume \(K \geq 2\) and that the statement holds for every \(N < K\). Letting \(N\) be the number of non-zero elements in the sequence \(\{t_k - t_K\}_{k=1}^K\) we have \(1 \leq N \leq K-1\), therefore \[\begin{align} \sum_{k=1}^K t_k \lambda_k(\Omega) &= \sum_{k=1}^K (t_k - t_K) \lambda_k(\Omega) + \sum_{k=1}^K t_K \lambda_k(\Omega) \\ &= \sum_{k=1}^N (t_k - t_K) \lambda_k(\Omega) + \sum_{k=1}^K t_K \lambda_k(\Omega) \\ &\leq \sum_{k=1}^N (t_k - t_K) \lambda_k(\Omega^*) + \sum_{k=1}^K t_K \lambda_k(\Omega^*) = \sum_{k=1}^K t_k\lambda_k(\Omega^*), \end{align}\] where for the first sum we used the inductive hypothesis (legitimate since \(t_k - t_K > 0\) for \(k=1,\ldots,N\) by definition of \(N\)), whereas for the second we used Corollary 3. Since in ?? equality is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\) and \(t_K > 0\) we also have uniqueness. ◻
Remark 6. If there exists \(k_0 \in \{1,\ldots,K-1\}\) such that \(t_1 = \cdots = t_{k_0} < t_{k_0+1}\) then the above conclusion is false. To see this, consider the function \[f(s) = \sum_{k=1}^K t_k \dfrac{s^{k-1}}{(k-1)!}e^{-s}, \quad s \geq 0.\] Since \(f\) is strictly increasing in a right neighborhood of 0, we can choose \(s_0, \varepsilon > 0\) such that \(f(s_0) > f(0) = t_1\), \(s_0 + \varepsilon < 1\) and consider \(\Omega = \{z \in \mathbb{C}\colon s_0 < \pi |z|^2 < s_0 + \varepsilon\}\) (so that \(|\Omega| = \varepsilon\)). Since the family of functions \(\{\frac{s^k}{k!}e^{-s}\}_{k \in \mathbb{N}}\) is strictly decreasing for \(s \in (0,1)\), we have \[\begin{align} \sum_{k=1}^K t_k \lambda_k(\Omega) &= \sum_{k=1}^K t_k \int_{\Omega} |e_{k-1}(z)|^2 e^{-\pi |z|^2} \, dA(z) \\ &= \sum_{k=1}^K t_k \int_{s_0}^{s_0+\varepsilon} \dfrac{s^{k-1}}{(k-1)!} e^{-s} \, ds \\ &= \int_{s_0}^{s_0 + \varepsilon} f(s) \, ds = f(s_0) \varepsilon + o(\varepsilon), \end{align}\] as \(\varepsilon \searrow 0\). On the other hand, for \(\Omega^*\) we have \[\sum_{k=1}^K t_k \lambda_k(\Omega^*) = \sum_{k=1}^K t_k \int_{0}^{\varepsilon} \dfrac{s^{k-1}}{(k-1)!} e^{-s} \, ds = t_1 \varepsilon + o(\varepsilon),\] as \(\varepsilon \searrow 0\). Since \(f(s_0) > t_1\), for \(\varepsilon\) sufficiently small we see that ?? is false.
Using the Hardy–Littlewood majorization theory (cf.[20]) we can prove the following result.
Corollary 7. Let \(\Phi \colon (0,1) \to \mathbb{R}\) be a convex function. Then, among all radial sets \(\Omega \subset \mathbb{C}\) of finite positive measure, the ball maximizes \(\mathrm{tr}\,\Phi(T_{\Omega}) = \sum_{k=1}^{\infty} \Phi(\lambda_k(\Omega))\), that is \[\label{eq:inequality32traces} \mathrm{tr}\, \Phi(T_{\Omega}) \leq \mathrm{tr}\, \Phi(T_{\Omega^*}),\qquad{(4)}\] where \(\Omega^*\) is the ball with center \(0\) and the same measure as \(\Omega\). Moreover, if \(\mathrm{tr}\,\Phi(T_{\Omega^*})\) is finite and if \(\Phi\) is not affine in the interval \((0,\lambda_1(\Omega^*))\), then equality in ?? is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\).
Proof. Corollary 3 implies that \(\{\lambda_k(\Omega)\}_{k=1}^{\infty}\) is majorized by \(\{\lambda_k(\Omega^*)\}_{k=1}^{\infty}\), in the sense that
both sequence are ordered in a decreasing way;
\(\sum_{k=1}^K \lambda_k(\Omega) \leq \sum_{k=1}^K \lambda_k(\Omega^*)\) for every \(K \in \mathbb{N}\);
\(\sum_{k=1}^{\infty} \lambda_k(\Omega) = \sum_{k=1}^{\infty} \lambda_k(\Omega^*)\).
Then, by Karamata’s inequality (alias Hardy–Littlewood majorization theory), in the form of Proposition 19 below, we immediately conclude that the desired estimate holds. Moreover, if equality is achieved in ?? , by Proposition 19 it must hold \(\sum_{k=1}^K \lambda_k(\Omega) = \sum_{k=1}^K \lambda_k(\Omega^*)\) for some \(K \in \mathbb{N}_+\), but by Theorem 1 this is possible if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\). ◻
When \(\Phi(t) = t^p\), for some \(p > 0\), we have \(\mathrm{tr}\,\Phi(T_{\Omega}) = \| T_{\Omega} \|_{\mathcal{S}_p}^p\), that is the Schatten \(p\)-norm of the operator \(T_{\Omega}\). Since \(\Phi\) is clearly not affine in \((0,1)\) for every \(p>0\) and \(p \neq 1\), the following result is an immediate consequence of the previous corollary.
Corollary 8. Let \(\Omega \subset \mathbb{C}\) be a radial set with finite positive measure. Denote with \(\Omega^* \subset \mathbb{C}\) the open ball with center 0 and the same measure as \(\Omega\). Then, for \(p>1\), it holds \[\label{eq:inequality32Schatten32p32with32p621} \| T_{\Omega} \|_{\mathcal{S}_p} \leq \| T_{\Omega^*} \|_{\mathcal{S}_p},\qquad{(5)}\] while for \(0 < p < 1\) it holds \[\| T_{\Omega} \|_{\mathcal{S}_p} \geq \| T_{\Omega^*} \|_{\mathcal{S}_p}.\] If \(p>1\), equality is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\). The same is true if \(0<p<1\), provided both sides are finite.
Remark 9. In [6], the first and second author proved that for \(p=2\), that is for the Hilbert–Schmidt norm, inequality ?? holds also without the assumption that \(\Omega\) is radial.
Theorem 1 and Corollary 3 can be rephrased in terms of the short-time Fourier transform with Gaussian window \(\varphi(x) = 2^{1/4} e^{-\pi x^2}\), defined as \[\mathcal{V}f (x, \omega) = \int_{\mathbb{R}} f(t) \varphi(t-x) e^{-2\pi i \omega t} \, dt, \quad (x,\omega) \in \mathbb{R}^2,\;f \in L^2(\mathbb{R}),\] Indeed, it is well known (cf. [21]) that the Bargmann transform \[\mathcal{B}f(z)\mathrel{\vcenter{:}}= 2^{1/4} \int_{\mathbb{R}} f(t) e^{\pi i t z - \pi t^2 - \pi |z|^2/2}\, dt, \quad z = x+i\omega \in \mathbb{C}\] is a unitary operator from \(L^2(\mathbb{R})\) on \(\mathcal{F}^2(\mathbb{C})\) and is related to the short-time Fourier transform via the relation \[\mathcal{V}f(x,-\omega) = e^{\pi i x \omega} e^{- \pi |z|^2/2} \mathcal{B}f(z), \quad z = x + i \omega \in \mathbb{C}.\] Using this fact and knowing that Hermite functions \[h_k(t) = \frac{2^{1/4}}{\sqrt{k!}}\left(-\frac{1}{2\sqrt{\pi}}\right)^k e^{\pi t^2} \frac{\mathrm{d}^k}{\mathrm{d}t^k}(e^{-2\pi t^2}), \quad t \in \mathbb{R},\;k \in \mathbb{N}\] are mapped into the normalized monomials in \(\mathcal{F}^2(\mathbb{C})\), the next corollary follows immediately from Theorem 1.
Corollary 10. Let \(\Omega \subset \mathbb{R}^2\) be a radial set with finite positive measure and let \(\{f_1, \ldots, f_K\}\) be an orthonormal set in \(L^2(\mathbb{R})\) for some \(K \in \mathbb{N}_+\). Then \[\sum_{k=1}^K \int_{\Omega} |\mathcal{V}f_k(x,\omega)|^2 \, dx d\omega \leq \sum_{k=0}^{K-1} \int_{\Omega^*} |\mathcal{V}h_k (x,\omega)|^2 \, dx d\omega = K - \sum_{k=0}^{K-1} (K-k)\dfrac{|\Omega|^k}{k!}e^{-|\Omega|},\] where \(\Omega^*\) is the open ball with center \(0\) and the same measure as \(\Omega\). Equality is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\) and \[\begin{pmatrix} f_1(t) \\ \vdots \\ f_K(t) \end{pmatrix} = U \begin{pmatrix} h_0(t) \\ \vdots \\ h_{K-1}(t) \end{pmatrix}\] for a.e. every \(t \in \mathbb{R}\) and for some unitary matrix \(U \in \mathbb{C}^{K \times K}\).
In this setting, the role of Toeplitz operators is played by the so-called time-frequency localization operators or anti-Wick operators (see [17]), defined as \(L_{\Omega} \mathrel{\vcenter{:}}= \mathcal{V}^* \chi_{\Omega} \mathcal{V}\colon L^2(\mathbb{R}) \to L^2(\mathbb{R})\), where \(\Omega \subset \mathbb{R}^2\) is a set of finite measure. With the natural identification \(\mathbb{R}^2 \simeq \mathbb{C}\) and using the definition of Bargmann transform, it is easy to see that if \(\Omega\) is radial then \(L_{\Omega} = \mathcal{B}^* T_{\Omega} \mathcal{B}\) and therefore \(L_{\Omega}\) has the same eigenvalues of \(T_{\Omega}\). With this in mind, the next corollary follows immediately from Corollary 3.
Corollary 11. Let \(\Omega \subset \mathbb{R}^2\) be a radial set with finite positive measure and denote with \(\{\lambda_k(\Omega)\}_{k=1}^{\infty}\) the eigenvalues of \(L_{\Omega}\) ordered in a decreasing way. Then, for every \(K \in \mathbb{N}_+\) it holds \[\sum_{k=1}^{K} \lambda_k(\Omega) \leq \sum_{k=1}^{K} \lambda_k(\Omega^*) = K - \sum_{k=0}^{K-1} (K-k)\dfrac{|\Omega|^k}{k!}e^{-|\Omega|},\] where \(\Omega^* \subset \mathbb{R}^2\) is the open ball of center 0 and the same measure as \(\Omega\). Equality is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega^*\).
The proof of Theorem 1 relies on the following lemmas. We use the notation from Section 2.
Lemma 12. Fix \(K \in \mathbb{N}_+\) and a strictly increasing vector \(\alpha \in \mathbb{N}^K\). Then \[\int_{\Omega_{\alpha+1}} u_{\alpha+1}(z) \, dA(z) < \int_{\Omega_{\alpha}} u_{\alpha}(z) \, dA(z).\]
Proof. From the decomposition 2 we have \[\int_{\Omega_{\alpha+1}} u_{\alpha+1}(z) \, dA(z) = \sum_{m=1}^{M} \int_{B_{R_m} \setminus \overline{B_{r_m}}} v_{\alpha+1}(\pi |z|^2) \, dA(z).\] We point out that the summation starts with \(m=1\) because the first entry of \(\alpha+1\) is clearly greater than 0, so \(R_0 = 0\).
Passing to polar coordinates \((\rho, \theta)\) and with the change of variable \(s = \pi \rho^2\) we obtain that the previous integral is equal to \[\sum_{m=1}^{M} \int_{a_m}^{b_m} v_{\alpha+1}(s) \, ds.\] Since \(\varphi_{k+1}'(s) = \varphi_k(s)\), it is immediate to see that \[\label{eq:equation32for32derivative32of32v32alpha} v_{\alpha+1}'(s) = -v_{\alpha+1}(s) + v_{\alpha}(s).\tag{3}\] So, plugging this in the integral leads to \[\sum_{m=1}^{M} \left[ \int_{a_m}^{b_m} v_{\alpha}(s) \, ds - v_{\alpha+1}\big\vert_{a_m}^{b_m} \right],\] but the contribution of the border term is null because we are integrating over the border of the super-level set \(\Omega_{\alpha+1}\) of \(u_{\alpha+1}\), so \(v_{\alpha+1}(a_m) = v_{\alpha+1}(b_m)\) for every \(m=1,\ldots,M\). Therefore, we obtain that \[\int_{\Omega_{\alpha+1}} u_{\alpha+1}(z) \, dA(z) = \sum_{m=1}^{M} \int_{a_m}^{b_m} v_{\alpha}(s) \, ds = \int_{\Omega_{\alpha+1}} u_{\alpha}(z) \, dA(z) < \int_{\Omega_{\alpha}} u_{\alpha}(z) \, dA(z).\] ◻
Loosely speaking, previous lemma states that, when possible, shifting the vector of indices down by one yields an increase in the integral. In the next lemma, we are going to prove an analogous result but for the last part of \(\alpha\), that is the one we denoted by \(\alpha''\), in the notation of Section 2.
Lemma 13. Fix \(K \in \mathbb{N}_+\), a strictly increasing vector \(\alpha \in \mathbb{N}^K\) and consider the partition \(\alpha = (\alpha', \alpha'')\) introduced in Section 2, that is \[\alpha = (0, \ldots, l-1, \alpha_{l+1}, \ldots, \alpha_{K}),\;\alpha' = (0, \ldots, l-1) = \alpha^*(l),\;\alpha'' = (\alpha_{l+1}, \ldots, \alpha_K),\] where \(1 \leq l \leq K-1\) and, by definition, \(\alpha_{l+1} > l\). Then \[\int_{\Omega_{(\alpha', \alpha'')}} u_{(\alpha', \alpha'')}(z) \, dA(z) < \int_{\Omega_{(\alpha', \alpha''-1)}} u_{(\alpha', \alpha''-1)}(z) \, dA(z).\]
Proof. As before, we have \[\int_{\Omega_{(\alpha', \alpha'')}} u_{(\alpha',\alpha'')}(z) \, dA(z) = \sum_{m=0}^M \int_{a_m}^{b_m} v_{(\alpha',\alpha'')}(s) \, ds = \sum_{m=0}^M
\int_{a_m}^{b_m} \left[ v_{\alpha'}(s) + v_{\alpha''}(s) \right] \, ds .\] For \(m=0\) we have \(b_0 > 0\) (because \(\alpha'\) is
not absent) so, using 3 , we obtain \[\int_{0}^{b_0} v_{\alpha''}(s) \, ds = \int_{0}^{b_0} v_{\alpha''-1}(s) \, ds -
v_{\alpha''}(s)\big\vert_0^{b_0} \leq \int_0^{b_0} v_{\alpha''-1}(s) \, ds,\] since \(v_{\alpha''}(0) = 0\). For \(m \geq 1\), using again 3 and the fact that \(v_{\alpha^*(l)}'(s) = - v_{\alpha^*(l)}(s) + v_{\alpha^*(l-1)}(s)\) we have \[\begin{align}
&\int_{a_m}^{b_m} \left[v_{\alpha'}(s) + v_{\alpha''}(s)\right] \, ds = \int_{a_m}^{b_m} \left[v_{\alpha^*(l-1)}(s) + v_{\alpha''-1}(s)\right] \, ds - v_{\alpha}\big\vert_{a_m}^{b_m} \\ \leq& \int_{a_m}^{b_m} \left[
v_{\alpha'}(s) + v_{\alpha''-1}(s) \right]\, ds,
\end{align}\] where in the last inequality we used the fact that the border term is 0 since we are integrating over the super-level set of \(v_{\alpha}\) and that \(v_{\alpha^*(l-1)}(s) <
v_{\alpha^*(l)}(s) = v_{\alpha'}(s)\).
Summing up, we have \[\begin{align} &\int_{\Omega_{(\alpha', \alpha'')}} u_{(\alpha', \alpha'')}(z) \, dA(z) \\ &= \int_{0}^{b_0} v_{\alpha'}(s) \, ds + \int_{0}^{b_0} v_{\alpha''}(s)
\, ds + \sum_{m=1}^M \int_{a_m}^{b_m} \left[ v_{\alpha'}(s) + v_{\alpha''}(s) \right] \, ds\\ &< \int_{0}^{b_0} v_{\alpha'}(s) \, ds + \int_{0}^{b_0} v_{\alpha''-1}(s) \, ds + \sum_{m=1}^M \int_{a_m}^{b_m} \left[
v_{\alpha'}(s) + v_{\alpha''-1}(s) \right] \, ds \\ &= \int_{\Omega_{(\alpha', \alpha'')}} u_{(\alpha', \alpha''-1)}(z) \, dA(z) < \int_{\Omega_{(\alpha', \alpha''-1)}} u_{(\alpha',
\alpha''-1)}(z) \, dA(z).
\end{align}\] ◻
We can now combine previous lemmas to prove the main theorem.
Proof of Theorem 1 and Corollary 3. Consider the eigenfunctions corresponding to the first \(K\) eigenvalues of \(T_{\Omega}\). As we already know, these are the monomials \(e_{\alpha_1}, \ldots, e_{\alpha_K}\) for some \(0 \leq \alpha_1 < \cdots < \alpha_K\) and we set \(\alpha = (\alpha_1, \ldots, \alpha_K) \in \mathbb{N}^K\). For the sake of clarity, we remark that \(\alpha_k\) does not necessary correspond to the \(k\)-th eigenvalue of \(T_{\Omega}\).
It is clear that \[\begin{align} \label{eq:first32estimate32proof32theorem} &\sum_{k=1}^K \int_{\Omega} |F_k(z)|^2 e^{-\pi |z|^2} \, dA(z) \leq \sum_{k=1}^K \int_{\Omega} |e_{\alpha_k}(z)|^2 e^{-\pi |z|^2} \, dA(z) \\ =& \int_{\Omega} u_{\alpha}(z) \, dA(z) \leq \int_{\Omega_{\alpha}} u_{\alpha}(z) \, dA(z), \nonumber \end{align}\tag{4}\] where \(\Omega_{\alpha}\) is the unique super-level set of \(u_{\alpha}\) that has the same measure as \(\Omega\).
If \(\alpha_1 \geq 1\), we can write \(\alpha = \tilde{\alpha} + n\) for \(n = \alpha_1\), so that the first element of \(\tilde{\alpha}\) is 0. Iterating Lemma 12 we obtain that \[\int_{\Omega_{\tilde{\alpha}+n}} u_{\tilde{\alpha}+n}(z) \, dA(z) < \int_{\Omega_{\tilde{\alpha}+n-1}} u_{\tilde{\alpha}+n-1}(z) \, dA(z) < \dots < \int_{\Omega_{\tilde{\alpha}}} u_{\tilde{\alpha}}(z) \, dA(z),\] and therefore it suffices to prove the theorem for those \(\alpha\) such that \(\alpha_1 = 0\). In this case, we consider the decomposition \(\alpha = (\alpha', \alpha'')\) (cf. Section 2) and we may suppose that \(\alpha''\) is not empty. Assume that \(\alpha'\) has \(l\) entries, which means that \(\alpha'=(0,\ldots,l-1)\). Using Lemma 13 we obtain \[\int_{\Omega_{(\alpha', \alpha'')}} u_{(\alpha', \alpha'')}(z) \, dA(z) < \int_{\Omega_{(\alpha', \alpha''-1)}} u_{(\alpha', \alpha''-1)}(z) \, dA(z),\] and this procedure can be iterated until the first element of \(\alpha''\) reaches the value \(l\). At this point, we end up with a vector of indices for which the length of the remainder part \(\alpha''\) is decreased. This process of iterations ends exactly when \(\alpha = \alpha^*\), hence \[\sum_{k=1}^K \int_{\Omega} |F_k(z)|^2 e^{-\pi |z|^2} \, dA(z) \leq \int_{\Omega_{\alpha^*}} u_{\alpha^*}(z) \, dA(z).\] Since whenever we apply Lemma 12 or Lemma 13 we have a strict inequality, the only possibility to have equality in ?? is that the starting vector of indices of the eigenfunctions is exactly \(\alpha^* = (0,\ldots,K-1)\). This implies that \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega_{\alpha^*}\) and, since all the eigenvalues of the Toeplitz operator associated to the ball \(\Omega_{\alpha^*}\) are simple, equality in 4 occurs if and only if \[\mathrm{span}\{ F_1, \ldots, F_K \} = \mathrm{span}\{ e_0, \ldots, e_{K-1} \},\] which implies \[\begin{pmatrix} F_1(z) \\ \vdots \\ F_K(z) \end{pmatrix} = U \begin{pmatrix} e_0(z) \\ \vdots \\ e_{K-1}(z) \end{pmatrix}\] for every \(z \in \mathbb{C}\), where \(U = (\langle F_j, e_{k-1} \rangle )_{1 \leq j,k \leq K} \in \mathbb{C}^{K \times K}\) is a unitary matrix because \(\{ F_1, \ldots, F_K \}\) is an orthonormal set. ◻
Using Lemma 12 with \(K=1\) we can prove that, among all radial sets of fixed positive finite measure, the second eigenvalue is maximized by a particular annulus.
Proposition 14. Let \(\Omega \subset \mathbb{C}\) be a radial set of finite positive measure. Then \[\label{eq:inequality32second32eigenvalue} \lambda_2(\Omega) \leq \lambda_2(\Omega_1),\qquad{(6)}\] where \(\Omega_1\) is the super-level set of \(u_1(z) = \pi |z|^2 e^{-\pi|z|^2}\) with the same measure as \(\Omega\). Equality is achieved if and only if \(\Omega\) is equivalent, up to a set of measure zero, to \(\Omega_1\).
Proof. With the notation of Section 2, we have \[\lambda_2(\Omega) = \int_{\Omega} u_j(z)\, dA(z) \leq \int_{\Omega} u_k(z)\, dA(z)\] for some \(j,k \in \mathbb{N}\), \(j \neq k\). Let \(\ell=\max\{j,k\}.\) Then \(\ell\geq 1\) and therefore, by Lemma 12, we have \[\lambda_2(\Omega)\leq \int_{\Omega} u_\ell(z) \, dA(z) \leq \int_{\Omega_\ell} u_\ell(z) \, dA(z) \leq \int_{\Omega_1} u_1(z) \, dA(z).\] Since, by the computation in the proof of Lemma 12, \[\int_{\Omega_1} u_1(z) \, dA(z) = \int_{\Omega_1} u_0(z) \, dA(z),\] \(T_{\Omega_1}\) has at least two eigenvalues greater or equal than \(\int_{\Omega_1} u_1(z) \, dA(z)\). Therefore \[\int_{\Omega_1} u_1(z) \, dA(z) \leq \lambda_2(\Omega_1),\] and combining all previous inequalities we end up with \[\lambda_2(\Omega) \leq \lambda_2(\Omega_1).\] Equality is achieved if and only if \(\ell=1\) (since the inequality in Lemma 12 is strict) and therefore \(\Omega = \Omega_\ell = \Omega_1\). ◻
Remark 15. For the third eigenvalue, it is not true that for every radial set \(\Omega \subset \mathbb{C}\) with finite positive measure it holds \[\lambda_3(\Omega) \leq \lambda_3(\Omega_2),\] where \(\Omega_2\) is the unique super-level set of \(u_2(z) = \frac{\pi^2}{2}|z|^4 e^{-\pi |z|^2}\) with the same measure as \(\Omega\). To see this, let \(f_k(s) = \frac{s^k}{k!}e^{-s}\), \(s \geq 0\), and fix \(\varepsilon > 0\). The super-level set of \(\Omega_2\) of the \(u_2\) is given by \[\Omega_2 = \{z \in \mathbb{C}\colon a < \pi |z|^2 < b\},\] where \((a,b) \subset \mathbb{R}\) is a neighborhood of \(2\) (that is the maximum point of \(f_2\)) with length \(\varepsilon\). At \(s=2\) it holds \[f_1(2) = f_2(2) > f_3(2) > f_0(2) > f_k(2) \quad \forall k \geq 4,\] therefore, for \(\varepsilon\) sufficiently small, we have \[\lambda_3(\Omega_2) = \int_{\Omega_2} u_3(z) \, dA(z) = \int_a^b f_3(s) \, ds = f_3(2) \varepsilon + o(\varepsilon)\] as \(\varepsilon \searrow 0\). On the other hand, consider \[\Omega = \{z \in \mathbb{C}\colon \sqrt{2} < \pi |z|^2 < \sqrt{2}+\varepsilon\},\] which has measure \(\varepsilon\). For \(\varepsilon\) sufficiently small, for \(s \in (\sqrt{2},\sqrt{2}+\varepsilon)\) it holds \[f_1(s) > f_2(s) > f_0(s) > f_k(s) \quad \forall k \geq 3,\] therefore we have \[\lambda_3(\Omega) = \int_{\Omega} u_0(z) \, dA(z) = \int_{\sqrt{2}}^{\sqrt{2}+\varepsilon} f_0(s) \, ds = f_0(\sqrt{2}) \varepsilon + o(\varepsilon)\] as \(\varepsilon \searrow 0\). Since \(f_0(\sqrt{2}) > f_3(2)\), for \(\varepsilon\) sufficiently small it holds \(\lambda_3(\Omega) > \lambda_3(\Omega_2)\).
The definition of Toeplitz operators can be easily generalized from characteristic functions to generic functions \(F \colon \mathbb{C}\rightarrow \mathbb{C}\) by letting \(T_F \mathrel{\vcenter{:}}= PF\), where we recall that \(P\) is the orthogonal projection on \(\mathcal{F}^2(\mathbb{C})\). The function \(F\) is usually called the symbol of the operator. It is well known that if \(F \in L^p(\mathbb{C})\) for some \(p \in [1,+\infty)\) then \(T_F\) is bounded and compact. Moreover, if \(F\) is real-valued and nonnegative then \(T_F\) is also self-adjoint and nonnegative. Thus, for symbols \(F \in L^p(\mathbb{C})\), \(p \in [1,\infty)\), that are real-valued and nonnegative it is still true that \(T_F\) admits a sequence of decreasing eigenvalues \(\lambda_1(F) \geq \lambda_2(F) \geq \cdots \geq 0\) and so it still makes sense to ask, among a given class of symbols, which functions maximize the sum of the first eigenvalues. We start proving that, given a radial nonnegative symbol \(F \in L^p(\mathbb{C})\) for some \(p \in [1,\infty)\), this sum increases if we replace \(F\) with its decreasing rearrangement \(F^*\), that is \[F^*(z) = \int_0^{\infty} \chi_{\{F>t\}^*}(z) \, dt,\] where \(\{F>t\}^*\) is the open ball of center 0 and the same measure as \(\{F>t\}\). It is well known that \(F^*\) is radially decreasing and equimeasurable with \(F\), hence \(\|F\|_p = \|F^*\|_p\) (see, e.g., [20]).
Define \[G_K(s) = K - \sum_{k=0}^{K-1} (K-k)\dfrac{s^k}{k!}e^{-s},\] that is the function that appears on the right-hand side of ?? .
Proposition 16. Let \(F \in L^p(\mathbb{C})\), for some \(p \in [1,\infty)\), be a radial nonnegative symbol and let \(F^*\) be the decreasing rearrangement of \(F\). Then, for every \(K \in \mathbb{N}_+\) it holds \[\label{eq:sum32of32the32eigenvalues32increases32under32rearrangements} \sum_{k=1}^{K} \lambda_k(F) \leq \sum_{k=1}^{K} \lambda_k(F^*) = \int_0^{\infty} G_K(\mu(t)) \, dt,\qquad{(7)}\] where \(\mu(t) = |\{F>t\}|\) is the distribution function of \(F\). Equality is achieved if and only if \(F=F^*\) a.e. on \(\mathbb{C}\).
Proof. Since \(F\) is radial, its eigenfunctions are still the monomials (see [17]) and its eigenvalues are therefore given by \[\int_{\mathbb{C}} F(z) |e_k(z)|^2 e^{-\pi|z|^2} \, dA(z), \quad k \in \mathbb{N}.\]
Again, we denote by \(\alpha \in \mathbb{N}^K\) the strictly increasing vector of the indices of the eigenfunctions corresponding to the first \(K\) eigenvalues of \(T_F\). Then, using the layer cake representation and Theorem 1 we have \[\begin{align} \sum_{k=1}^{K} \lambda_k(F) &= \int_{\mathbb{C}} F(z) u_{\alpha}(z) \, dA(z) = \int_{\mathbb{C}} \left(\int_0^{\infty} \chi_{\{F>t\}}(z) \, dt \right) u_{\alpha} (z) \, dA(z) \\ &= \int_0^{\infty} \left( \int_{\{F>t\}} u_{\alpha}(z) \, dA(z) \right) \, dt \leq \int_0^{\infty} \left( \int_{\{F>t\}^*} u_{\alpha^*}(z) \, dA(z) \right) \, dt \\ &= \int_{\mathbb{C}} \left(\int_0^{\infty} \chi_{\{F>t\}^*}(z) \, dt \right) u_{\alpha^*} (z) \, dA(z) = \int_{\mathbb{C}} F^*(z) u_{\alpha^*} (z) \, dA(z) \\ &= \sum_{k=1}^{K} \lambda_k(F^*), \end{align}\] where the last equality is justified since \(F^*\) is radially decreasing, therefore its \(k\)-th eigenfunction is exactly \(e_k\) and its eigenvalues are simple. This latter fact is well known when \(F\) is the characteristic function of a ball [17], and using the layer cake representation it is immediate to see that \[\begin{align} &\int_{\mathbb{C}} F^*(z) |e_k(z)|^2 e^{-\pi|z|^2} \, dA(z) = \int_{\mathbb{C}} \left( \int_0^{\infty} \chi_{\{F^*>t\}}(z) \, dt \right) \, |e_k(z)|^2 e^{-\pi|z|^2} dA(z)\\ &= \int_0^{\infty} \left( \int_{\{F^*>t\}} |e_k(z)|^2 e^{-\pi|z|^2}\, dA(z) \right) \, dt\\ &> \int_0^{\infty} \left( \int_{\{F^*>t\}} |e_{k+1}(z)|^2 e^{-\pi|z|^2}\, dA(z) \right) \, dt\\ &= \int_{\mathbb{C}} F^*(z) |e_{k+1}(z)|^2 e^{-\pi|z|^2} \, dA(z), \end{align}\] where we used the fact that the super-level sets of \(F^*\) are balls.
From the previous computations, it is easy to see that equality in ?? is achieved if and only if \(\{F>t\} = \{F>t\}^* = \{F^*>t\}\) for a.e. \(t \in (0,\infty)\), which implies that \(F=F^*\) a.e. on \(\mathbb{C}\).
Moreover \[\int_{\{F>t\}^*} u_{\alpha^*}(z) \, dA(z) = G_K(|\{F>t\}^*|) = G_K(|\{F>t\}|)\] and therefore \[\sum_{k=1}^{K} \lambda_k(F^*) = \int_0^{\infty} G_K(\mu(t)) \, dt.\] ◻
Arguing as in [7], [8] we can find those weights that maximize the sum of the first \(K\) eigenvalues of \(T_F\), among a given class. For simplicity, we restrict ourselves to the case of an \(L^p\) constraint with \(p>1\), but the same technique can be used to deal with more than one constraint.
Proposition 17. Let \(p > 1\), \(A>0\) and \(K \in \mathbb{N}_+\). Then, for all radial nonnegative symbols \(F \in L^p(\mathbb{C})\) such that \(\|F\|_p \leq A\), it holds \[\label{eq:optimal32estimate32sum32eigenvalues32operators} \sum_{k=1}^{K} \lambda_k(F) \leq \sum_{k=1}^{K} \lambda_k(F_p),\qquad{(8)}\] where \(F_p \in L^p(\mathbb{C})\) is given by \[F_p(z) = \alpha \left( \sum_{k=0}^{K-1} \frac{(\pi |z|^2)^k}{k!}\right)^{\frac{1}{p-1}}e^{-\frac{\pi |z|^2}{p-1}}, \quad z \in \mathbb{C},\] with \(\alpha=A/C\), where \(C>0\) is given by \(C^p = \int_0^{\infty} (G_K'(s))^{\frac{p}{p-1}}\,ds\). Equality in ?? is achieved if and only if \(F(z)=F_p(z)\) for a.e. \(z \in \mathbb{C}\).
Proof. We know from Proposition 16 that for any radial nonnegative \(F \in L^p(\mathbb{C})\) it holds \[\sum_{k=1}^{K} \lambda_k(F) \leq \int_0^{\infty} G_K(\mu(t)) \, dt,\] where \(\mu(t) = |\{F > t\}|\). Hence, we have to maximize the right-hand side. The proper variational problem to consider is \[\label{eq:variational32problem} \sup_{\mu \in \mathcal{C}} \int_0^{\infty} G_K(\mu(t)) \, dt,\tag{5}\] where
\(\mathcal{C} = \{ \mu \colon (0,\infty) \to [0,\infty)\) decreasing s.t \(p\int_0^{\infty} t^{p-1} \mu(t) \, dt \leq A^p \}\)
is the class of all possible distribution functions coming from symbols satisfying \(\|F\|_p \leq A\). For \(K=1\) this variational problem has already been solved in [7]. Then, a careful inspection of the proof contained in the aforementioned paper reveals that the same argument can be used here with minor modifications. In this way, one proves that 5 admits a unique maximizer \(\mu_p\) given implicitly by \[\label{eq:implicit32expression32of32mu95p} G_K'(\mu_p(t)) = (t/\alpha)^{p-1}, \quad t \in (0,\alpha),\tag{6}\] while \(\mu_p(t)=0\) for \(t \geq \alpha\). The constant \(\alpha > 0\) is the only positive number for which \(\mu_p\) achieves equality in the constraint, that is \[p\int_0^{\alpha} t^{p-1} \mu_p(t) \, dt = A^p.\] A direct computation easily shows that this is equivalent to \[\alpha^p \int_0^{\infty} (G_K'(s))^{\frac{p}{p-1}} \, ds = \alpha^p C^p = A^p,\] which gives the expression of \(\alpha\). We can now reconstruct the optimal weight \(F_p\). Indeed, we have \[\sum_{k=1}^{K} \lambda_k(F) \leq \int_0^{\infty} G_K(\mu(t)) \, dt \leq \int_0^{\infty} G_K(\mu_p(t)) \, dt,\] with equality if and only if \(F(z) = F^*(z)\) and \(\mu(t) = \mu_p(t)\), which means that the optimal weight function \(F_p\) must be radially decreasing, that is \(F_p(z) = \rho(|z|)\) for some decreasing \(\rho \colon [0,\infty) \to [0,\infty)\), and its distribution function must be \(\mu_p\).
To find the expression of \(\rho\), fix \(t \in (0,\alpha)\). Then, the super-level set \(\{F_p > t\}\) is a ball of radius \(r\) and measure \(\mu_p(t)\), which means \[\pi r^2 = \mu_p(t)\] and, moreover, it is clear that \(\rho(r) = t\). However, from 6 we have \[t = \alpha (G_K'(\mu_p(t)))^{\frac{1}{p-1}} = \lambda (G_K'(\pi r^2))^{\frac{1}{p-1}}\] and, in the end \[F_p(z) = \rho(|z|) = \alpha (G_K'(\pi |z|^2))^{\frac{1}{p-1}} = \alpha \left( \sum_{k=0}^{K-1} \frac{(\pi |z|^2)^k}{k!} \right)^{\frac{1}{p-1}} e^{-\frac{\pi |z|^2}{p-1}}, \quad z \in \mathbb{C}.\] ◻
Remark 18. When \(K=1\) we recover [7], at least in the radial case. Indeed, we have \(G_1(s) = 1 - e^{-s}\) and from a direct computation it is easy to see that \(C^p = (p-1)/p\), therefore \(\alpha = A (\frac{p-1}{p})^{1/p}\) and the optimal weight function is \[F_p(z) = A \left( \frac{p-1}{p} \right)^{1/p} e^{-\frac{\pi |z|^2}{p-1}}, \quad z \in \mathbb{C}.\]
We state and prove a version of Karamata’s inequality (cf.[20]) for infinite sequences and convex functions \(\Phi:(0,1)\to\mathbb{R}\), where \(\Phi'\) is allowed to be unbounded below. We also study the cases of equality when \(\Phi\) is not assumed strictly convex but is merely not affine.
Proposition 19. Let \(\{x_k\}_{k=1}^{\infty}, \{y_k\}_{k=1}^{\infty} \subset (0,1)\) be two decreasing sequences such that
\(\sum_{k=1}^K x_k \geq \sum_{k=1}^K y_k\) for every \(K \geq 1\);
\(\sum_{k=1}^{\infty} x_k = \sum_{k=1}^{\infty} y_k < \infty\).
Then, for every convex function \(\Phi \colon (0,1) \to \mathbb{R}\) it holds \[\label{eq:Karamata32inequality} \sum_{k=1}^{\infty} \Phi(x_k) \geq \sum_{k=1}^{\infty} \Phi(y_k).\qquad{(9)}\] Assume that, in addition, \(\Phi\) is not affine in \((0,x_1)\). Then, if the series \(\sum_{k=1}^{\infty} \Phi(x_k)\) converges and equality occurs in ?? , we have \(\sum_{k=1}^K x_k = \sum_{k=1}^K y_k\) for some \(K \geq 1\).
Proof. First of all we notice that, since \(\Phi\) is convex, it has constant sign on \((0,\varepsilon)\) for some \(\varepsilon>0\), so \(\{\Phi(x_k)\}_{k=1}^{\infty}\) and \(\{\Phi(y_k)\}_{k=1}^{\infty}\) have eventually constant sign and therefore both the series \(\sum_{k=1}^{\infty} \Phi(x_k)\) and \(\sum_{k=1}^{\infty} \Phi(y_k)\) have a limit. Moreover, given \(\lim_{x \to 0^+} \Phi(x) = L \in \mathbb{R}\cup \{+\infty\}\), if \(L \neq 0\) both series diverge to \(+\infty\) or both diverge to \(-\infty\). Hence, we may suppose \(L=0\), in which case the series may diverge to \(-\infty\) but not to \(+\infty\). Then, setting \(\Phi(0):=0\) we have that \(\Phi\) is still convex on \([0,1)\) and we denote by \(\Phi'_+(0)\) its right-derivative at \(0\). We now prove the inequality ?? .
Step 1. We start supposing that \(\Phi'_+(0)\) is finite (that is, \(\Phi'_+(0)>-\infty\)), which implies that both series converge.
Since equal terms of the sequences do not contribute to the difference \(\sum_{k=1}^{\infty} \Phi(x_k) - \sum_{k=1}^{\infty} \Phi(y_k)\), we may suppose that \(x_k \neq y_k\) for every \(k \geq 1\) and that the sequences obtained after removing equal terms have still infinite terms (since otherwise we could simply apply the classical Karamata’s inequality, see, e.g., [20]).
We introduce the following notation:
\(c_k = \frac{\Phi(x_k) - \Phi(y_k)}{x_k - y_k}\), so \(c_k \geq c_{k+1}\) (since \(\Phi\) is convex);
\(A_K = \sum_{k=1}^K x_k\), \(B_K = \sum_{k=1}^K y_k\), with the convention \(A_0 = B_0 = 0\), therefore \(x_k = A_k - A_{k-1}\) and \(y_k = B_k - B_{k-1}\).
So, we have \[\begin{align} \sum_{k=1}^K \left( \Phi(x_k) - \Phi(y_k)\right) &= \sum_{k=1}^K c_k(x_k - y_k)\\ &= \sum_{k=1}^K c_k(A_k - A_{k-1} - B_k + B_{k-1})\\ &= \sum_{k=1}^K c_k(A_k - B_k) - \sum_{k=1}^K c_k(A_{k-1}-B_{k-1})\\ &= c_K(A_K - B_K) + \sum_{k=1}^{K-1} c_k(A_k - B_k) - \sum_{k=2}^K c_k(A_{k-1}-B_{k-1})\\ &= c_K(A_K - B_K) + \sum_{k=1}^{K-1}(c_k - c_{k+1})(A_k - B_k). \end{align}\] Since \(\Phi'_+(0)\) is finite, the sequence \(\{c_K\}_{K=1}^{\infty}\) is bounded, therefore taking the limit \(K \to \infty\) leads to \[\label{eq32karamata95intermedia} \sum_{k=1}^{\infty} \left( \Phi(x_k) - \Phi(y_k)\right) = \sum_{k=1}^{\infty}(c_k-c_{k+1})(A_k-B_k).\tag{7}\] The right-hand side is nonnegative because, for every \(k \geq 1\), \(c_k \geq c_{k+1}\) by the convexity of \(\Phi\) and \(A_k \geq B_k\) by the hypothesis.
Step 2. We now assume \(\Phi'_+(0) = -\infty\) and we consider \[\Phi_n(t) = \begin{cases} n\Phi(1/n)t, &t \in [0,1/n)\\ \Phi(t), &t \in [1/n,1). \end{cases}\] The functions \(\Phi_n\) are such that \((\Phi_n)'_+(0) > -\infty\) so, it holds \[\sum_{k=1}^{\infty} \Phi_n(x_k) \geq \sum_{k=1}^{\infty} \Phi_n(y_k)\] and since \(\Phi_n \searrow \Phi\), from monotone convergence theorem we have \[\sum_{k=1}^{\infty} \Phi(x_k) \geq \sum_{k=1}^{\infty} \Phi(y_k).\]
Step 3. Finally, we have to prove the “uniqueness". Assume that \(\Phi\) is not affine in \((0,x_1)\) and that the series \(\sum_{k=1}^{\infty} \Phi(x_k)\) and \(\sum_{k=1}^{\infty} \Phi(y_k)\) converge. Let us prove that, if \(A_k>B_k\) for every \(k\geq 1\), then the inequality in ?? is strict.
Observe that, if \(A_k>B_k\) for every \(k\geq 1\), then the sequences obtained from \(x_k\) and \(y_k\) by removing the equal terms contain, indeed, infinite terms. We still denote by \(x_k\) and \(y_k\) these sequences. With the above notation, we have \(c_k=c_k\) for some \(k\geq 1\) if and only if \(\Phi\) is affine on the interval \([\min\{x_{k+1},y_{k+1}\},\max\{x_{j},y_{j}\}]\). Hence, since \(\Phi\) is not affine on \((0,x_1)\), there must be a \(k_0 \geq 1\) such that \(c_{k_0} > c_{k_0+1}\). Therefore, if \(A_k > B_k\) for every \(k\geq 1\) and \(\Phi'_+(0)>-\infty\) we see from 7 that \[\sum_{k=1}^{\infty} \left( \Phi(x_k) - \Phi(y_k)\right) = \sum_{k=1}^{\infty}(c_k-c_{k+1})(A_k-B_k) \geq (c_{k_0} - c_{k_0+1})(A_{k_0} - B_{k_0}) > 0.\] In the case \(\Phi'_+(0) = -\infty\), pick \(\delta < \mathrm{min}\{x_{k_0+1}, y_{k_0+1}\}\) and consider the decomposition \(\Phi(t) = \Phi_1(t) + \Phi_2(t)\), where \[\Phi_1(t) = \begin{cases} \Phi(t), & t \in [0,\delta)\\ \Phi(\delta) + \Phi'_-(\delta)(t-\delta), &t \in [\delta,1) \end{cases},\] where \(\Phi'_-\) denotes the left derivative of \(\Phi\). Then, \(\Phi_2\) is not affine in \((0,x_1)\) and its right derivative at 0 is finite so, since all the series are convergent, we have \[\begin{align} \sum_{k=1}^{\infty} \Phi(x_k) &= \sum_{k=1}^{\infty} \Phi_1(x_k) + \sum_{k=1}^{\infty} \Phi_2(x_k)\\ &\geq \sum_{k=1}^{\infty} \Phi_1(y_k) + \sum_{k=1}^{\infty} \Phi_2(x_k)\\ &> \sum_{k=1}^{\infty} \Phi_1(y_k) + \sum_{k=1}^{\infty} \Phi_2(y_k) = \sum_{k=1}^{\infty}\Phi(y_k). \end{align}\] ◻
Remark 20. We point out that the hypothesis “\(\Phi\) not affine in \((0,x_1)\)” is necessary. This can be easily seen by taking \(\Phi(t) = (t-x_1)_+\), where \((\cdot)_+\) denotes the positive part. Clearly \(\Phi\) is not affine on \((0,1)\) but \(\sum_{k=1}^{\infty} \Phi(x_k) = \sum_{k=1}^{\infty} \Phi(y_k) = 0\), independently of the sequences \(\{x_k\}_{k=1}^{\infty}\) and \(\{y_k\}_{k=1}^{\infty}\). In general, the behaviour of the series does not depend on the values \(\Phi(t)\) for \(t > x_1\), so the proposition extends to every convex function on \((0,x_1]\).
Moreover, we point out that equality in ?? may occur even if \(x_k \neq y_k\) for every \(k\). As example, take \(\Phi(t) = (t - 1/2)_+\), \(1 > x_1 > y_1 > y_2 > x_2 > 1/2\) such that \(x_1 + x_2 = y_1 + y_2\), while \(x_k\) and \(y_k\) for \(k \geq 3\) are decreasing sequences with values in \((0,1/2)\), such that \(x_k \neq y_k\) for every \(k\geq 3\), \(\sum_{k=3}^K x_k \geq \sum_{k=3}^K y_k\) for every \(K \geq 3\) and \(\sum_{k=1}^{\infty} x_k = \sum_{k=1}^{\infty} y_k < \infty\).