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On the generalized Hardy spaces. (English) Zbl 1191.30021

Let \(H(U)\) be the space of all analytic functions on the unit disk \(U\) of the complex plane. Given \(1\leq p<\infty\) and a one-to-one linear operator \(F:H(U)\to H(U)\), the author introduces the space \(H_{F, p}(U)\) consisting of all functions \(f\in H(U)\) such that
\[ \|f\|^p_{H_{F,p}}:=\sup_{0<r<1}\int_0^{2\pi}|F(f)(re^{i\theta})|^p\, {{d\theta}\over{2\pi}}<\infty. \]
Clearly, if \(F\) is the identity operator, then \(H_{F, p}(U)\) is the classical Hardy space. The author obtains some sufficient conditions that imply some basic properties of such a space, namely, the property of being a Banach space and the continuity of point evaluations. The author also introduces the notion of Carleson measures for \(H_{F,p}(U)\) and uses them to obtain results concerning boundedness and compactness of composition operators on \(H_{F,p}(U)\).

MSC:

30H10 Hardy spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
47B33 Linear composition operators
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References:

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