Generalized composition operators from Bloch type spaces to \(Q_K\) type spaces
Let \(\mathbb{D}\) denote the open unit disk in the complex plane, \(H(\mathbb{D})\) the space of all analytic functions on \(\mathbb{D}\) and \(\phi\) an analytic self-map of \(\mathbb{D}\). In [Comment. Math. Helv. 52, 591--602 (1977; Zbl 0369.30012)], \textit{Ch. Pommerenke} introduced the so-called Volterra type operator, which for \(f,g \in H(\mathbb{D})\) is defined by \[ J_gf(z):= \int_0^z f(\xi) g'(\xi) \; d \xi, \; z \in \mathbb{D}. \] The importance of this operator comes from its close relation to the multiplication operator \[ M_gf(z):= g(z)f(z) \text{ for every } z \in \mathbb{D}, \] given by the equation \[ J_gf + Igf= M_g f - f(0)g(0), \] where the operator \(I_g\) is defined by \[ I_gf(z):= \int_0^z f'(\xi) g(\xi) \; d \xi, \; z \in \mathbb{D}, \] see [\textit{R. Yoneda}, Complex Variables, Theory Appl. 49, No.~14, 1045--1061 (2004; Zbl 1084.47026)]. In the present paper, the authors study Volterra composition operators \[ I_{g,\phi}(z)= \int_0^z f'(\phi(\xi)) g(\xi) \; d \xi \] acting between Bloch type spaces \[ B^{\alpha}:= \{ f \in H(\mathbb{D}); \; \sup_{z \in \mathbb{D}} |f'(z)| (1-|z|^2)^{\alpha} < \infty\}, \] endowed with the norm \(\|f\|_{\alpha}:= |f(0)| + \sup_{z \in \mathbb{D}} |f'(z)| (1-|z|^2)^{\alpha}\), and the spaces \[ Q_K(p,q):= \left \{ f \in H(\mathbb{D}); \; \|f\|_{K,p,q}^p= \sup_{a \in \mathbb{D}} \int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^q K(g(z,a)) \; dA(z) < \infty \right \}, \] where \(p>0\), \(q > -2\), \(K:[0, \infty) \to [0, \infty)\) is a right-continuous and nondecreasing function and \(g(z,a)= \log\left (\frac{1}{|\varphi_a(z)|} \right )\) is the Green's function with pole at the point \(a\). The authors show that the operator \(I_{g,\phi}: B^{\alpha} \to Q_K(p,q)\) is bounded if and only if \[ \sup_{a \in \mathbb{D}} \int_{\mathbb{D}} \frac{|g(z)|^p (1-|z|^2)^q}{(1-|\phi(z)|^2)^{p \alpha}} K(g(z,a)) \; dA(z) < \infty. \] Moreover, it is proved that the above operator is compact if and only if the following conditions hold: {\parindent=6mm \begin{itemize}\item[(a)] \(\sup_{a \in \mathbb{D}} \int_{\mathbb{D}} |g(z)|^p (1-|z|^2)^q K(g(z,a)) \; dA(z) < \infty\), \item[(b)] \( \lim_{r \to 1} \sup_{a \in \mathbb{D}} \int_{|\phi(z)| >r} \frac{|g(z)|^p (1-|z|^2)^q}{(1-|\phi(z)|^2)^{p \alpha}} K(g(z,a)) \; dA(z) =0.\) \end{itemize}}
scientific article; zbMATH DE number 5704734