A lower bound in Nehari’s theorem on the polydisc

Abstract

By theorems of Ferguson and Lacey (d = 2) and Lacey and Terwilleger (d > 2), Nehari’s theorem (i.e., if H _ψ is a bounded Hankel form on H ²(D ^d) with analytic symbol ψ, then there is a function φ in L ^∞(T ^d ) such that ψ is the Riesz projection of g4) is known to hold on the polydisc D ^d for d > 1. A method proposed in Helson’s last paper is used to show that the constant C _d in the estimate ‖φ‖_∞ ≤ C _d ‖H _ψ ‖ grows at least exponentially with d; it follows that there is no analogue of Nehari’s theorem on the infinite-dimensional polydisc.