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A lower bound in Nehari's theorem on the polydisc
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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 $\varphi$ in $L^\infty(\T^d)$ such that $\psi$ is the Riesz projection of $\varphi$. A method proposed in Helson's last paper is used to show that the constant $C_d$ in the estimate $\|\varphi\|_\infty\le C_d \|H_\psi\|$ grows at least exponentially with $d$; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.
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ORTEGA CERDÀ, Joaquim and SEIP, Kristian. A lower bound in Nehari's theorem on the polydisc. Journal d'Analyse Mathematique. 2012. Vol. 118, num. 1, pags. 339-342. ISSN 0021-7670. [consulted: 16 of October of 2026]. Available at: https://hdl.handle.net/2445/34463