Arkiv för Matematik

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  • Volume 39, Number 1 (2001), 137-149.

Closures of finitely generated ideals in Hardy spaces

Artur Nicolau and Jordi Pau

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LetH be the algebra of bounded analytic functions in the unit disk D. Let I=I(f1,..., fN) be the ideal generated by f1,..., fNH and J=J(f1,..., fN) the ideal of the functions f∈H for which there exists a constant C=C(f) such that |f(z)|≤C(|f1(z)|+...; +|fN(z)|), zD. 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 Hp, 1≤p<∞. More concretely, if g∈Hp and the nontangential maximal function of $|g(z)|/\sum\nolimits_{j = 1}^N {|f_j (z)|} $ belongs to Lp (T), then g is in the Hp-closure of the ideal I.


Both authors are supported in part by DGICYT grant PB98-0872 and CIRIT grant 1998SRG00052.

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Ark. Mat., Volume 39, Number 1 (2001), 137-149.

Received: 23 July 1999
First available in Project Euclid: 31 January 2017

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2001 © Institut Mittag-Leffler


Nicolau, Artur; Pau, Jordi. Closures of finitely generated ideals in Hardy spaces. Ark. Mat. 39 (2001), no. 1, 137--149. doi:10.1007/BF02388795.

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