Closures of finitely generated ideals in Hardy spaces

Abstract

LetH^∞ be the algebra of bounded analytic functions in the unit disk D. Let I=I(f₁,..., f_N) be the ideal generated by f₁,..., f_N∈H^∞ and J=J(f₁,..., f_N) the ideal of the functions f∈H^∞ for which there exists a constant C=C(f) such that |f(z)|≤C(|f₁(z)|+...; +|f_N(z)|), z∈D. It is clear that $I \subseteq J$ , but an example due to J. Bourgain shows that J is not, in general, in the norm closure of I. Our first result asserts that J is included in the norm closure of I if I contains a Carleson-Newman Blaschke product, or equivalently, if there exists s>0 such that $\mathop {\inf }\limits_{z \in D} \sum\limits_{k = 0}^s {(1 - |z|)^k } \sum\limits_{j = 1}^N {|f_j^{(k)} (z)| > 0.} $

Our second result says that there is no analogue of Bourgain's example in any Hardy space H^p, 1≤p<∞. More concretely, if g∈H^p and the nontangential maximal function of $|g(z)|/\sum\nolimits_{j = 1}^N {|f_j (z)|} $ belongs to L^p (T), then g is in the H^p-closure of the ideal I.