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 (i.e., if Hψ is a bounded Hankel form on H2(Dd) with analytic symbol ψ, then there is a function φ in L∞(Td ) such that ψ is the Riesz projection of g4) is known to hold on the polydisc Dd for d > 1. 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? 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 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 ( 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 H ψ extends to a bounded form on 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 to be the smallest constant C that can be chosen in the estimate ϕ ∞ ≤ C H ψ , where it is assumed that ϕ has minimal L ∞ norm. Nehari's original theorem says that C 1 = 1.

Theorem. For even integers d
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 C d clearly increases with d and, in particular, we would need C d ≤ C ∞ for Nehari's theorem to 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.
We follow Helson's approach [3] and use his multiplicative notation. Thus, we define a Hankel form on T ∞ as H ψ ( fg) = ∞ j,k =1 ρ jk a j b k . 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 consider only the finite-dimensional case, which means that the coefficients are nonzero only for indices j of the form p ν 1 1 · · · p ν d d . The prime numbers play no role in the proof except for serving as a convenient tool for bookkeeping.
We now assume that d is an even integer and introduce the set q j and q j = p 2 j −1 or q j = p 2 j .
We define a Hankel form H ψ on D d by setting 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 c j = 2 − ( j )/2 , where ( j ) is the number of prime factors in j . We then get 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 . We choose for which a n = ρ n and thus, on the one hand, H ψ ( f ) = 2 d/2 . On the other hand, an explicit computation shows that f 1 = (4/π) d/2 , so that H ψ , viewed as a linear functional on H 1 (D d ), has norm at least (π/2) d/2 . This concludes the proof, since it follows that (π/2) d/2 ≤ ϕ ∞ and we know from above that H ψ ≤ 2 d/4 .
It is worth noting that our application of the Schur test shows that, in fact, H ψ = 2 d/4 since f 2 = 2 d/4 . The fact that |H ψ ( f )| = H ψ f 2 implies that f 1,w = f 2 . In other words, the trivial factorization f · 1 is an optimal weak factorization of the function f defined in (1).