Abstract
There is a bounded Hankel operator on the Paley–Wiener space of a disc in \({\mathbb {R}}^2\) which does not arise from a bounded symbol.
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1 Introduction
Let \({\mathbb {D}}\) be the unit disc in \({\mathbb {R}}^2\). The Paley–Wiener space \({{\,\mathrm{PW}\,}}({\mathbb {D}})\) is the subspace of \(L^2({\mathbb {R}}^2)\) comprised of functions f whose Fourier transforms \({\widehat{f}}\) are supported in \(\overline{{\mathbb {D}}}\). For a tempered distribution \(\varphi \), we consider the Hankel operator \({\mathbf {H}}_\varphi \) defined by the equation
on the dense subset of \({{\,\mathrm{PW}\,}}({\mathbb {D}})\) comprised of functions f such that \({\widehat{f}}\) is smooth and compactly supported in \({\mathbb {D}}\).
We are interested in the characterization of the symbols \(\varphi \) such that \({\mathbf {H}}_\varphi \) extends by continuity to a bounded operator on \({{\,\mathrm{PW}\,}}({\mathbb {D}})\). If \(\varphi \) is in \(L^\infty ({\mathbb {R}}^2)\), then clearly
Since \(\xi +\eta \) is in \(2{\mathbb {D}}\) whenever \(\xi \) and \(\eta \) are in \({\mathbb {D}}\), \({\mathbf {H}}_\varphi = {\mathbf {H}}_\psi \) for any \(\psi \) such that the restrictions of \({\widehat{\psi }}\) and \({\widehat{\varphi }}\) to \(2{\mathbb {D}}\) coincide (as distributions in \(2{\mathbb {D}}\)). We thus find that
We say that the Hankel operator \({\mathbf {H}}_\varphi \) has a bounded symbol if the quantity on the right hand side of (3) is finite. We have just demonstrated that if \({\mathbf {H}}_\varphi \) has a bounded symbol, then \({\mathbf {H}}_\varphi \) is bounded. We wish to explore the converse.
Question
Does every bounded Hankel operator on \({{\,\mathrm{PW}\,}}({\mathbb {D}})\) have a bounded symbol?
In the classical one-dimensional setting, where the role of \({\mathbb {D}}\) is played by the half-line \({\mathbb {R}}_+ = [0,\infty )\), Nehari [6] gave a positive answer to this question. We therefore refer to affirmative answers to analogous questions as Nehari theorems. Our question for \({{\,\mathrm{PW}\,}}({\mathbb {D}})\) was first raised implicitly by Rochberg [9, Sec. 7], after he had proved that Nehari’s theorem holds for the Paley–Wiener space \({{\,\mathrm{PW}\,}}(I)\) of a finite interval \(I \subseteq {\mathbb {R}}\).
It was conditionallyFootnote 1 shown in [1] that the Nehari theorem holds for the Paley–Wiener space \({{\,\mathrm{PW}\,}}({\mathbb {P}})\) of any convex polygon \({\mathbb {P}}\). However, in view of C. Fefferman’s negative resolution [3] of the disc conjecture for the Fourier multiplier of a disc, it would not be surprising to see differing results for \({{\,\mathrm{PW}\,}}({\mathbb {P}})\) and \({{\,\mathrm{PW}\,}}({\mathbb {D}})\).
The main purpose of the present note is to establish the following.
Theorem 1
There is a bounded Hankel operator on \({{\,\mathrm{PW}\,}}({\mathbb {D}})\) which does not have a bounded symbol.
Minor modifications of our proof show that if \({\mathbb {P}}_n\) is an n-sided regular polygon, then the optimal constant in the inequality
satisfies \(C_n \ge c_\varepsilon n^{1/2 - \varepsilon }\) for any fixed \(\varepsilon > 0\). Here, \(c_\varepsilon >0\) denotes a constant which depends only on \(\varepsilon \). Conversely, the conditional argument of [1] yields that \(C_n \le cn\) for some absolute constant \(c > 0\). Analogous estimates for Fourier multipliers associated with polygons were considered in [2].
Finally, let us remark that Ortega-Cerdà and Seip [7] have shown that Nehari’s theorem also fails for (small) Hankel operators on the infinite-dimensional torus. However, Helson [4] proved that if the Hankel operator is in the Hilbert–Schmidt class \(S_2\), then it is induced by a bounded symbol. We are led to the following.
Question
Does every Hankel operator on \({{\,\mathrm{PW}\,}}({\mathbb {D}})\) in \(S_2\) have a bounded symbol?
In this context, we mention that Peng [8] has characterized when \({\mathbf {H}}_\varphi \) is in the Schatten class \(S_p\), for \(1 \le p \le 2\), in terms of the membership of \(\varphi \) in certain Besov spaces adapted to \(2{\mathbb {D}}\). In particular, \({\mathbf {H}}_\varphi \) is in \(S_2\) if and only if
2 Proof of Theorem 1
If the Nehari theorem were to hold for \({{\,\mathrm{PW}\,}}({\mathbb {D}})\), there would by the closed graph theorem exist an absolute constant \(C < \infty \) such that
for every bounded Hankel operator on \({{\,\mathrm{PW}\,}}({\mathbb {D}})\). To prove Theorem 1, we will construct a sequence of symbols which demonstrates that no such \(C<\infty \) can exist.
We begin with an upper bound for \(\Vert {\mathbf {H}}_\varphi \Vert \). Guided by the following lemma, our plan is to construct \(\varphi \) such that \({\mathbf {H}}_\varphi \) admits an orthogonal decomposition. For a symbol \(\varphi \), define
Lemma 2
Suppose that \(\varphi = \varphi _1 + \varphi _2\) and that \(D_{\varphi _1} \cap D_{\varphi _2} = \emptyset \). Then,
Proof
Let f be any function in \({{\,\mathrm{PW}\,}}({\mathbb {D}})\) such that \({\widehat{f}}\) is smooth and compactly supported in \({\mathbb {D}}\) . Since \({\mathbf {H}}_\varphi f = {\mathbf {H}}_{\varphi _1} f + {\mathbf {H}}_{\varphi _2} f\) by linearity of the integral (1), it is sufficient to demonstrate that \({\mathbf {H}}_{\varphi _1} f \perp {\mathbf {H}}_{\varphi _2} f\). It follows directly from the definition of the Hankel operator (1) that
By the assumption that \(D_{\varphi _1} \cap D_{\varphi _2} = \emptyset \), we therefore conclude that
\(\square \)
In particular, if \(D_{\varphi _1} \cap D_{\varphi _2}= \emptyset \), then
Let us next explain the construction of \(\varphi \). Consider a radial smooth bump function \({\widehat{b}}\) which is bounded by 1, equal to 1 on \(\frac{1}{2}{\mathbb {D}}\) and compactly supported in \({\mathbb {D}}\). For a real number \(0<r < 1/2\), set \({\widehat{b}}_r(\xi ) = {\widehat{b}}(\xi /r)\). Note that
For \(j=1,2,\ldots ,n\), we let \({\widehat{\varphi }}_j\) be the function obtained by translating \({\widehat{b}}_r\) by \(2-r\) units in the direction \(\theta _j = 2 \pi (j-1)/n\), as measured with respect to the positive \(\xi _1\)-axis in the \(\xi _1\xi _2\)-plane. We set
Since \(0< r < 1/2\), it is clear that \({{\,\mathrm{supp}\,}}{{\widehat{\varphi }}} \subseteq 2{\mathbb {D}} \setminus {\mathbb {D}}\). Let \(r_0 = 1-\frac{1}{\sqrt{2}}=0.29\ldots \).
Plots of D(w) and the corresponding disc sector from the proof of Lemma 3, for \(w=1.1\), \(w=1.5\), and \(w = 1.8\)
Lemma 3
If \(n \ge 2\) and \(r=\min (r_0,(2/n)^2)\), then
for every \(1 \le j \ne k \le n\).
Proof
Throughout this proof, we identify \({\mathbb {R}}^2\) with \({\mathbb {C}}\). We consider first a simpler situation. For a point w in \(2 {\mathbb {D}} \setminus {\mathbb {D}}\), let
In other words, D(w) is the intersection of the discs defined by \(|\xi |<1\) and \(|w-\xi |<1\). To find the intersection of the corresponding circles, we set \(\xi = e^{i\theta }\) and let \(\theta ^{\pm }\) denote the solutions of the equation
Let \(P_0\) denote the origin, \(P_\pm \) the points \(e^{i\theta ^\pm }\), and \(P_w\) the point w. The law of cosines implies that the angle \(\angle P_0 P_\pm P_w\) is greater than or equal to \(\pi /2\) if and only if \(|w|\ge \sqrt{2}\). If this holds, then the intersection of the two discs is contained in the disc sector defined by the origin and the two points \(P_\pm \). See Fig. 1.
Suppose therefore that \(|w|\ge \sqrt{2}\) and set \(I(w) = (\theta ^-,\theta ^+)\). If \(\xi \) is in D(w), we have just seen that \(\arg {\xi }\) is in I(w). It follows that if \(w_1\) and \(w_2\) are points in \(2{\mathbb {D}} \setminus \sqrt{2}{\mathbb {D}}\), then
Our goal is now to estimate
Since \({{\,\mathrm{supp}\,}}{{\widehat{\varphi }}_j}\) is contained in a disc with center \((2-r) e^{i\theta _j}\) and radius r, straightforward geometric arguments show that if w is in \({{\,\mathrm{supp}\,}}{{\widehat{\varphi }}_j}\), then
To ensure that \(|w|\ge \sqrt{2}\) we require that \(r \le r_0 = 1-\frac{1}{\sqrt{2}}\). Moreover, if \(\theta ^\pm \) correspond to the point w as above, then
Here, we used that \(2-r \ge 1\) and that \(\arctan {r}\le r\) for \(0\le r \le 1\). This shows that
Since \(|\theta _j-\theta _k| \ge 2 \pi /n\) for every \(1 \le j \ne k \le n\) and since \(\pi >3\), it follows that if we choose \(r = \min (r_0,(\frac{2}{n})^2)\), then we guarantee that \(I_{\varphi _j} \cap I_{\varphi _k}= \emptyset \) for every \(1 \le j \ne k \le n\). The proof is completed by appealing to (7). \(\square \)
Let \(\varphi \) be as in (6), with \(n\ge 2\) and \(r=\min (r_0,(2/n)^2)\). It then follows from Lemmas 2, 3, (2), and (5) that
A lower bound for the left hand side in (4) will be established through duality.
Lemma 4
Suppose that \({\widehat{f}}\) is smooth and compactly supported in \(2 {\mathbb {D}}\). Then,
Proof
Obviously,
and when \({\widehat{f}}\) is supported in \(2{\mathbb {D}}\) and \({\widehat{\psi }}|_{2{\mathbb {D}}} = {\widehat{\varphi }}|_{2{\mathbb {D}}}\), we have that
\(\square \)
We now need to choose a test function f adapted to the symbol \(\varphi \) of (6). It turns out that \(f=f_1+f_2+\cdots +f_n\), where \(f_j = \varphi _j\) for \(j=1,2,\ldots ,n\), will do. By our choice of \(n\ge 2\) and \(r=\min (r_0,(2/n)^2)\), it is clear that \({{\,\mathrm{supp}\,}}{{\widehat{f}}_j} \cap {{\,\mathrm{supp}\,}}{{\widehat{f}}_k}=\emptyset \) for every \(1 \le j \ne k \le n\), since the converse statement would contradict Lemma 3.
Exploiting this, we find that
To get an upper bound for \(\Vert f\Vert _1\), we split the integral at some \(R>0\),
For the first integral, we use the Cauchy–Schwarz inequality,
where we again exploited that \({{\,\mathrm{supp}\,}}{{\widehat{f}}_j} \cap {{\,\mathrm{supp}\,}}{{\widehat{f}}_k} = \emptyset \) for \(1 \le j \ne k \le n\). For the second integral, we note that b is rapidly decaying, since \({\widehat{b}}\) is smooth and compactly supported. In particular, for every \(\kappa \ge 1\), there is a constant \(A_\kappa \) such that
holds for every \(\varrho >0\). We constructed \(\widehat{f_j}\) by translating \({\widehat{b}}_r\) by \(2-r\) units in direction \(\theta _j\), so there is a unimodular function \(g_j\) such that
Thus \(|f(x)| \le n r^2 b(rx)\) and (10), with \(\varrho = Rr\), yields
Combining our estimates for \(I_1\) and \(I_2\) and choosing \(R = n^{1/(2\kappa )}/r\), we find that
Inserting the estimates (9) and (11) into Lemma 4, we obtain
Final part of the proof of Theorem 1
To finish the proof of Theorem 1, we combine (8) and (12) to conclude that the constant C in (4) must satisfy
for any fixed \(\kappa \ge 1\) and every integer \(n\ge 2\). Choosing some \(\kappa >1\) and letting \(n\rightarrow \infty \), we obtain a contradiction. \(\square \)
Notes
The arguments in [1] rely on Nehari’s theorem for \({\mathbb {R}}_+ \times {\mathbb {R}}_+\) as a black box. It was long believed that the Nehari theorem had been proven in this setting, but a significant flaw was recently observed in the available reasoning. We refer to [5, Sect. 10] for a detailed discussion.
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Brevig, O.F., Perfekt, KM. The Nehari Problem for the Paley–Wiener Space of a Disc. J Geom Anal 33, 16 (2023). https://doi.org/10.1007/s12220-022-01063-2
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DOI: https://doi.org/10.1007/s12220-022-01063-2