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 ${\text{Ha}}_{\phi }$ on the Paley–Wiener space $\text{PW}(\mathrm{\Omega })$ 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 ${\text{Ha}}_{\phi }$ is in the Schatten class $S^p(\text{PW}(\mathrm{\Omega }))$. We prove that this fails for $p>4$ for any convex subset of ${\mathbb{R}}^n$, $n\ge 2$, of boundary with a $C^2$ neighborhood of nonzero curvature. Furthermore we prove that for a polytope $P$ in ${\mathbb{R}}^n$, the inequality $${\int }_{2P}{\displaystyle \frac{\mathrm{\mid }\widehat{f}(x)\mathrm{\mid }}{m(P\cap (x-P))}}dx\le C(P)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1},$$ holds for all $f\in {\text{PW}}^1(2P)$, and consequently any Hilbert–Schmidt Hankel operator on a Paley–Wiener space of a polytope is generated by a bounded function.