Closures of finitely generated ideals in Hardy spaces

LetH ^∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f ₁,...,f _N) be the ideal generated byf ₁,...,f _N∈H ^∞ andJ=J(f ₁,...,f _N) the ideal of the functionsf∈H ^∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f ₁ (z)|+...;+|f _N (z)|),z∈D. It is clear that , but an example due to J. Bourgain shows thatJ is not, in general, in the norm closure ofI. Our first result asserts thatJ is included in the norm closure ofI ifI contains a Carleson-Newman Blaschke product, or equivalently, if there existss>0 such that Our second result says that there is no analogue of Bourgain's example in any Hardy spaceH ^p, 1≤p<∞. More concretely, ifg∈H ^p and the nontangential maximal function of belongs toL ^p (T), theng is in theH ^p-closure of the idealI. Both authors are supported in part by DGICYT grant PB98-0872 and CIRIT grant 1998SRG00052.