Multivariable Nehari theorem for Hardy spaces

Establish whether every bounded Hankel operator on the multivariable Hardy spaces associated with the positive orthant is generated by a bounded function.

Background

For a distributional symbol, the paper defines Hankel operators on Paley–Wiener spaces by convolution in the Fourier domain. In one dimension, Nehari's theorem states that every bounded Hankel operator on the Hardy space associated with the positive half-line has a bounded-function symbol.

The analogous characterization for multivariable Hardy spaces remains unresolved. The paper proves a weaker Hilbert–Schmidt analogue for every convex frequency domain that contains no affine lines, but it does not resolve the full bounded-operator problem for multivariable Hardy spaces.

References

The analogous result for the multivariable Hardy spaces is one of the most famous open problems in complex analysis.

— Helson Inequality, Hankel Operators, and Weak Factorization on Paley-Wiener spaces of Convex Domains  (2609.10730 - Bampouras, 9 Sep 2026) in Section 1, Introduction; Section 3, “Hankel Operators and Nehari's Theorem”

The higher dimensional cases are open.

— Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes II  (2610.06728 - Bampouras, 5 Oct 2026) in Section 1, Introduction

For which $p>2$ is it true that every Schatten class Hankel operator $\Ha_\phi$ in $Sp(H2(\mathbb{T}\infty))$ admits a bounded symbol?

— Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes II  (2610.06728 - Bampouras, 5 Oct 2026) in Section 1, Introduction