Closures of finitely generated ideals in Hardy spaces
AbstractLetH∞ be the algebra of bounded analytic functions in the unit diskD. LetI=I(f1,...,fN) be the ideal generated byf1,...,fN∈H∞ andJ=J(f1,...,fN) the ideal of the functionsf∈H∞ for which there exists a constantC=C(f) such that |f(z)|≤C(|f1(z)|+...;+|fN(z)|),z∈D. It is clear that
$$I \subseteq J$$
, 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
$$\mathop {\inf }\limits_{z \in D} \sum\limits_{k = 0}^s {(1 - |z|)^k } \sum\limits_{j = 1}^N {|f_j^{(k)} (z)| > 0.} $$

Our second result says that there is no analogue of Bourgain's example in any Hardy spaceHp, 1≤p<∞. More concretely, ifg∈Hp and the nontangential maximal function of
$$|g(z)|/\sum\nolimits_{j = 1}^N {|f_j (z)|} $$
 belongs toLp (T), theng is in theHp-closure of the idealI.