A Lower Bound in Nehari’s Theorem on the Polydisc

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 for d > 1, i.e., if Hψ is a bounded Hankel form on H(D) with analytic symbol ψ, then there is a function φ in L∞(Td) such that ψ is the Riesz projection of φ. A method proposed in Helson’s last paper is used to show that the constant Cd in the estimate ‖φ‖∞ ≤ Cd‖Hψ‖ grows at least exponentially with d; it follows that there is no analogue of Nehari’s theorem on the infinite-dimensional polydisc. This note solves the following problem studied by H. Helson [2, 3]: Is there an analogue of Nehari’s theorem on the infinite-dimensional polydisc? By using a method proposed in [3], we show that the answer is negative. The proof is of interest also in the finite-dimensional situation because it gives a nontrivial lower bound for the constant appearing in the norm estimate in Nehari’s theorem; we choose to present this bound as our main result. We first introduce some notation and give a brief account of Nehari’s theorem. Let d be a positive integer, D the open unit disc, and T the unit circle. We letH(D) be the Hilbert space of functions analytic in D with square-summable Taylor coefficients. Alternatively, we may view H(D) as a subspace of L(T) and express the inner product of H(D) as 〈f, g〉 = ∫ Td fg, where we integrate with respect to normalized Lebesgue measure on T. Every function ψ in H(D) defines a Hankel form Hψ by the relation Hψ(fg) = 〈fg, ψ〉; this makes sense at least for holomorphic polynomials f and g. Nehari’s theorem—a classical result [6] when d = 1 and a remarkable and relatively recent achievement of S. Ferguson and M. Lacey [1] (d = 2) and M. Lacey and E. Terwilleger [5] (d > 2) in the general case—says that Hψ extends to a bounded form on H(D) × H(D) if and only if ψ = P+φ for some bounded function φ on T; here P+ is the Riesz projection on T or, in other words, the orthogonal projection of L(T) onto H(D). We define Cd as the smallest constant C that can be chosen in the estimate ‖φ‖∞ ≤ C‖Hψ‖, where it is assumed that φ has minimal L∞ norm. Nehari’s original theorem says that C1 = 1. Theorem. For even integers d ≥ 2, the constant Cd is at least (π/8). The theorem thus shows that the blow-up of the constants observed in [4, 5] is not an artifact resulting from the particular inductive argument used there. 2000 Mathematics Subject Classification. 47B35, 42B30, 32A35. The first author is supported by the project MTM2008-05561-C02-01. The second author is supported by the Research Council of Norway grant 160192/V30. This work was done as part of the research program Complex Analysis and Spectral Problems at Centre de Recerca Matemàtica (CRM), Bellaterra in the spring semester of 2011. The authors are grateful to CRM for its support and hospitality. 1 2 J. ORTEGA-CERDÀ AND KRISTIAN SEIP Since clearly Cd increases with d and, in particular, we would need that Cd ≤ C∞ should Nehari’s theorem extend to the infinite-dimensional polydisc, our theorem gives a negative solution to Helson’s problem. Nehari’s theorem can be rephrased as saying that functions in H(D) (the subspace of holomorphic functions in L(T)) admit weak factorizations, i.e., every f in H(D) can be written as f = ∑ j gjhj with fj , gj in H (D) and ∑ j ‖gj‖2‖hj‖2 ≤ A‖f‖1 for some constant A. Taking the infimum of the latter sum when gj , hj vary over all weak factorizations of f , we get an alternate norm (a projective tensor product norm) on H(D) for which we write ‖f‖1,w. We let Ad denote the smallest constant A allowed in the norm estimate ‖f‖1,w ≤ A‖f‖1. Our proof shows that we also have Ad ≥ (π/8) when d is an even integer. Proof of the theorem. We will follow Helson’s approach [3] and also use his multiplicative notation. Thus we define a Hankel form on T∞ as