Pérez-González, Fernando; Rättyä, Jouni Forelli–Rudin estimates, Carleson measures and \(F(p,q,s)\)-functions. (English) Zbl 1097.30040 J. Math. Anal. Appl. 315, No. 2, 394-414 (2006). For \(0< p<\infty\), \(-2< q<\infty\), and \(0\leq s<\infty\), the space \(F(p, q, s)\) is defined as the set of all analytic functions \(f\) on the unit disc \(\mathbb D =\{z\in \mathbb C: | z| <1\}\) such that \(\sup_{a\in\mathbb D}\int_{\mathbb D}| f'(z)| ^p(1-| z| ^2)^qg^s(z, a)\,dA(z)<\infty\), where \(g\) is the Green’s function of \(\mathbb D\) and \(dA(z)\) denotes the Lebesgue area measure on \(\mathbb D\). If \(1< s<\infty\), then \(F(p, q, s)\) coincides with \({\mathcal B}^{(q+2)/p}\), where the \(\alpha\)-Bloch space \({\mathcal B}^{\alpha}\) consists of those analytic functions \(f\) on \(\mathbb D\) for which \(\sup_{z\in\mathbb D}(1-| z| ^2)^{\alpha}| f'(z)| <\infty\).A positive measure \(\mu\) on \(\mathbb D\) is said to be a bounded \(s\)-Carleson measure, \(0<s<\infty\), if \(\sup_{| I| \leq 1}\mu(S(I))/| I| ^{s}<\infty\), where \(| I| \) denotes the the arc length of a subarc \(I\) of \(\partial \mathbb D\) and \(S(I)=\{z\in\mathbb D: z/| z| \in I, 1-| I| \leq | z| \}\). A classical theorem of L. Carleson states that the injection map from the Hardy space \(H^p\) into \(L^p(d\,\mu)\) is bounded if and only if the positive measure \(\mu\) on \(\mathbb D\) is a bounded \(1\)-Carleson measure. The authors obtain Forelli-Rudin type estimates for arbitrary positive measures on \(\mathbb D\). These estimates are applied to characterize bounded \(s\)-Carleson measures in terms of \(\alpha\)-Bloch- and \(F(p, q, s)\)-functions. Reviewer: Saulius Norvidas (Vilnius) Cited in 25 Documents MSC: 30D55 \(H^p\)-classes (MSC2000) 46E15 Banach spaces of continuous, differentiable or analytic functions Keywords:Carleson measures; Hardy spaces; Bloch spaces; \(F(p, q, s)\) spaces PDFBibTeX XMLCite \textit{F. Pérez-González} and \textit{J. Rättyä}, J. Math. Anal. Appl. 315, No. 2, 394--414 (2006; Zbl 1097.30040) Full Text: DOI References: [1] Arazy, J.; Fisher, S. D.; Peetre, J., Möbius invariant function spaces, J. Reine Angew. Math., 363, 110-145 (1985) · Zbl 0566.30042 [2] Ahern, P.; Jevtić, M., Inner multipliers of the Besov space, \(0 < p \leqslant 1\), Rocky Mountain J. Math., 20, 753-764 (1990) · Zbl 0725.30024 [3] Aulaskari, R.; Lappan, P., Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, (Complex Analysis and Its Applications. 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