Nehari's Theorem and Hardy's inequality for Paley--Wiener spaces

Recently it was proven that for a convex subset of $\mathbb{R}^{n}$ that has infinitely many extreme vectors, the Nehari theorem fails, that is, there exists a bounded Hankel operator $\Ha_ϕ$ on the Paley–Wiener space $\PW(Ω)$ that does not admit a bounded symbol. In this paper we examine whether Nehari’s theorem can hold under the stronger assumption that the Hankel operator $\Ha_ϕ$ is in the Schatten class $S^{p}(\PW(Ω))$. We prove that this fails for $p>4$ for any convex subset of $\mathbb{R}^{n}$, $n\geq2$, of boundary with a $C^{2}$ neighborhood of nonzero curvature. Furthermore we prove that for a polytope $P$ in $\mathbb{R}^{n}$, the inequality $$\int_{2P}\dfrac{|\widehat{f}(x)|}{m(P\cap (x-P))}dx\leq C(P)|f|_{L^{1}},$$ holds for all $f\in \PW^{1}(2P)$, and consequently any Hilbert–Schmidt Hankel operator on a Paley–Wiener space of a polytope is generated by a bounded function.

💡 Research Summary

The paper investigates whether Nehari’s theorem—stating that every bounded Hankel operator on a Hardy space admits a bounded symbol—holds for Hankel operators acting on Paley–Wiener spaces PW(Ω) when the operators belong to a Schatten class S^p. The authors consider two complementary settings: convex domains with curved boundaries and polytopal domains with flat faces.

The first main result (Theorem 1.1) shows that if Ω⊂ℝ^n (n≥2) is a convex set whose boundary contains a C^2 neighbourhood with non‑zero Gaussian curvature, then Nehari’s theorem fails for all p>4. In other words, there exist bounded Hankel operators H_φ∈S^p(PW(Ω)) that cannot be represented by any bounded symbol φ∈L^∞(ℝ^n). The proof proceeds by introducing the overlap function ω_Ω(x)=|Ω∩(x−Ω)|, which controls the Hilbert–Schmidt norm of a Hankel operator via ∥H_φ∥{S^2}=∥\widehat φ·√ω_Ω∥{L^2}. For curved boundaries the authors compute ω for a unit ball and obtain the estimate ω_B(0,1)(x)≈(2−|x|)^{n+1} on the doubled ball. Using this estimate they construct, for any integer N, a family of Schwartz functions {ϕ_i} with disjoint interaction regions D_{ϕ_i}=Ω∩(supp \widehat ϕ_i−Ω). The sum ψ_N=∑_{i=1}^N ϕ_i yields orthogonal Hankel operators, and a careful analysis of the Schatten‑p norm versus the L^1 norm of ψ_N shows that the ratio required by a hypothetical Nehari bound grows without bound when p>4. Lemma 2.1 formalises the implication “Nehari holds ⇒ the infimum of L^∞ symbols is controlled by the Schatten norm”, and Lemma 2.2 translates the geometric configuration into a quantitative inequality. The contradiction proves the failure of Nehari’s theorem for p>4.

The second main result (Theorem 1.2) concerns polytopes P⊂ℝ^n. The authors prove a Hardy‑type inequality: for every f∈PW^1(2P) (i.e., f∈L^1 with Fourier support in 2P) one has
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