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

TL;DR

This work investigates Nehari’s theorem for Hankel operators on Paley–Wiener spaces $PW(\Omega)$ under Schatten class constraints. It proves that Nehari’s theorem fails for all $p>4$ when $\Omega\subset\mathbb{R}^n$ is convex with $C^{2}$ boundary neighborhood of nonzero curvature, highlighting geometric obstructions to bounded symbols. For polytopes $P$, it establishes a Hardy-type inequality $\int_{2P}\frac{|\widehat{f}(x)|}{\omega_{P}(x)}dx\le C(P)\|f\|_{L^{1}}$ for $f\in PW^{1}(2P)$, which implies that any Hilbert–Schmidt Hankel operator on $PW(2P)$ is generated by a bounded symbol and yields Nehari for $p=2$ in the polytope case. The paper then introduces a generalized Hardy inequality with $\rac{1}{\omega_{\Omega}^{d}}$ weights, identifying precise thresholds depending on geometry: $d\le 1$ for polytopes, $d<\frac{2}{n+1}$ (and fail for $d>1$) for smooth-boundary convex sets, and $d\le\frac{2}{n+1}$ in general, with sharp failures beyond these ranges. These results connect Schatten-class Hankel operator theory with geometric analysis of the ω-function and yield sharp bounds that inform when bounded-symbol Nehari-type characterizations can hold.

Abstract

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.

Figures (2)

• Figure 1: Some useful geometric observations.
• Figure 2: If $\Omega$ is the ellipse $\frac{x^2}{\gamma^2}+\frac{y^2}{\beta^2}<1$ then the vector $(0,\beta)$ belongs to the class $\mathscr{R}(\alpha,\beta,\beta)$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4