\(\mathcal F_\varphi\) and \(\mathcal F_g\) classes
For the parameter values \(0<p<\infty\), \(-2<q<\infty\), and \(0<s<\infty\), \textit{R. Zhao} [Ann. Acad. Sci. Fenn. Math. Diss. 105, 1--56 (1996; Zbl 0851.30017)] introduced the so called \(F(p,q,s)\)-spaces as follows: Let \(f\) be an analytic function defined in the unit disk \(\mathbb D\). Let \(g(z,a)\) be the Green function of \(\mathbb D\) with logarithmic singularity at \(a\). Then \(f\in F(p,q,s)\) if \[ \sup_{a\in\mathbb D}\iint_{\mathbb D}|f'(z)|^p\big(1-|z|^2\big)^qg^s(z,a)\,dxdy<\infty. \] In this definition, the Green function \(g(z,a)\) could be replaced by the expression \(1-|\varphi_a(z)|^2\), where \(\varphi_a(z)=\frac{a-z}{1-\bar az}\) is a Möbius transformation of \(\mathbb D\). In later papers, the derivative \(|f'|\) in case of meromorphic functions \(g\) was replaced by the spherical derivative \(g^{\#}=|g'|/(1+|g|^2)\), and the corresponding classes \(F^{\#}(p,q,s)\) were introduced by \textit{H. Wulan} [Ann. Acad. Sci. Fenn. Math. Diss. 116, 57p. (1998; Zbl 0912.30021)], and studied by \textit{J. Rättyä} [Ann. Acad. Sci. Fenn. Math. Diss. 124, 1--73 (2001; Zbl 0984.30019)]. For bounded analytic functions \(|f|<1\), \textit{X. Li} [Ann. Acad. Sci. Fenn. Math. Diss. 145, 1--65 (2005; Zbl 1089.30031)] considered the hyperbolic classes \(Q^*\) with the hyperbolic derivative \(f^*=|f'|/(1-|f|^2)\). In the paper under review, the authors generalize these classes by bringing into play operators \(\mathcal F\) whose main property is only continuity. Compared to the above-mentioned, these operators correspond to \(\mathcal F(f)(z)=|f'(z)|^p(11-|z|^2)^q\) in the analytic case, \(\mathcal F(f)(z)=(f^{\#}(z))^p(1-|z|^2)^q\) in the meromorphic case, and \(\mathcal F(f)(z)=(f^*(z))^2\) for bounded analytic functions. In this way, the authors are able to unify and to simplify several important results, also about the classical \(D_p\)-spaces (weighted Dirichlet space), \(Q_p\)-spaces, \(\alpha\)-Bloch spaces \(\mathcal B^{\alpha}\), etc.
scientific article; zbMATH DE number 5695446