On the generalized Hardy spaces (Q963190)

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 \(H_{F,p}(U)\).