A lower bound in Nehari's theorem on the polydisc

TL;DR

This paper demonstrates that the constant in Nehari's theorem on the polydisc grows exponentially with dimension, indicating the theorem does not extend to infinite dimensions, using a method inspired by Helson.

Contribution

It establishes a lower bound on the growth of the constant in Nehari's theorem with respect to dimension, showing the theorem's limitations in higher and infinite dimensions.

Findings

The constant C_d grows at least exponentially with d.

Nehari's theorem does not extend to the infinite-dimensional polydisc.

A method from Helson's work is used to derive the lower bound.

Abstract

By theorems of Ferguson and Lacey (d=2) and Lacey and Terwilleger (d>2), Nehari's theorem is known to hold on the polydisc D^d for d>1, i.e., if H_\psi is a bounded Hankel form on H^2(D^d) with analytic symbol \psi, then there is a function \phi in L^\infty(\T^d) such that \psi is the Riesz projection of \phi. A method proposed in Helson's last paper is used to show that the constant C_d in the estimate \|\phi\|_\infty\le C_d \|H_\psi\| grows at least exponentially with d; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.