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

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 ${\mathbb{R}}^n$, $n\ge 2$, of boundary with a $C^2$ neighborhood of nonzero curvature. Furthermore we prove that for a polytope $P$ in ${\mathbb{R}}^n$, the inequality $${\int }_{2P}\frac{|\hat{f}(x)|}{m(P\cap (x-P))}dx\le C(P)\Vert f{\Vert }_{L^1},$$ 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.

21 pages