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^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.

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 let H 2 (D d ) be the Hilbert space of functions analytic in D d with square-summable Taylor coefficients. Alternatively, we may view H 2 (D d ) as a subspace of L 2 (T d ) and express the inner product of H 2 (D d ) as f, g = T d f g, where we integrate with respect to normalized Lebesgue measure on T d . Every function ψ in H 2 (D d ) defines a Hankel form H ψ by the relation H ψ (f g) = f g, 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 H ψ extends to a bounded form on H 2 (D d ) × H 2 (D d ) if and only if ψ = P + ϕ for some bounded function ϕ on T d ; here P + is the Riesz projection on T d or, in other words, the orthogonal projection of L 2 (T d ) onto H 2 (D d ). We define C d as the smallest constant C that can be chosen in the estimate where it is assumed that ϕ has minimal L ∞ norm. Nehari's original theorem says that C 1 = 1.

Theorem. For even integers
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. Since clearly C d increases with d and, in particular, we would need that C d ≤ 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 1 (D d ) (the subspace of holomorphic functions in L 1 (T d )) admit weak factorizations, i.e., every f in H 1 (D d ) can be written as f = j g j h j with f j , g j in H 2 (D d ) and j g j 2 h j 2 ≤ A f 1 for some constant A. Taking the infimum of the latter sum when g j , h j vary over all weak factorizations of f , we get an alternate norm (a projective tensor product norm) on H 1 (D d ) for which we write f 1,w . We let A d denote the smallest constant A allowed in the norm estimate f 1,w ≤ A f 1 . Our proof shows that we also have A d ≥ (π 2 /8) d/2 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 here (a j ), (b j ), and (ρ j ) are the sequences of coefficients of the power series of the functions f , g, and ψ, respectively. More precisely, we let p 1 , p 2 , p 3 , ... denote the prime numbers; if j = p ν 1 1 · · · p ν k k , then a j (respectively b j and ρ j ) is the coefficient of f (respectively of g and ψ) with respect to the monomial z ν 1 1 · · · z ν k k . We will only consider the finite-dimensional case, which means that the coefficients will be nonzero only for indices j of the form p ν 1 1 · · · p ν d d . The prime numbers will play no role in the proof except serving as a convenient tool for bookkeeping.
We now assume that d is an even integer and introduce the set We define a Hankel form H ψ on D d by setting ρ n = 1 if n is in I and ρ n = 0 otherwise. We follow [3, pp. 81-82] and use the Schur test to estimate the norm of H ψ . It suffices to choose a suitable finite sequence of positive numbers c j with j ranging over those positive integers that divide some number in I; for such j we choose where Ω(j) is the number of prime factors in j. We then get k ρ jk c k = 2 d/2−Ω(j) · 2 −(d/2−Ω(j))/2 = 2 d/4 c j , so that H ψ ≤ 2 d/4 by the Schur test.
If f is a function in H 1 (D d ) with associated Taylor coefficients a n , then H ψ (f ) = n a n ρ n .