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- W1993649037 abstract "LetH∞ 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(z)|≤C(|f1(z)|+...; +|fN(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} sumlimits_{k = 0}^s {(1 - |z|)^k } sumlimits_{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)|/sumnolimits_{j = 1}^N {|f_j (z)|} $ belongs to Lp (T), then g is in the Hp-closure of the ideal I." @default.
- W1993649037 created "2016-06-24" @default.
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- W1993649037 date "2001-03-01" @default.
- W1993649037 modified "2023-10-18" @default.
- W1993649037 title "Closures of finitely generated ideals in Hardy spaces" @default.
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- W1993649037 doi "https://doi.org/10.1007/bf02388795" @default.
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