Closures of finitely generated ideals in Hardy spaces

Let H∞ be the algebra of bounded analytic functions in the unit disk D. Let I=I(f1, … , fN) be the ideal generated by f1, … , fN∈H∞ and J=J(f1, … , fN) the ideal of the functions f∈H∞ for which there exists a constant C=C(f) such that |f(script z sign)|≤C(|f1(script z sign)|+…+ |fN(script z sign)|), script z sign∈D. It is clear that I⊆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 (Formula Presented) Our second result says that there is no analogue of Bourgain's example in any Hardy space Hp, 1≤p<∞. More concretely, if g∈Hp and the nontangential maximal function of |g(script z sign)|/ΣNj=1|fj(script z sign)| belongs to Lp(T), then g is in the Hp-closure of the ideal I.