On \(\alpha\)-Bloch spaces and multipliers of Dirichlet spaces
Suppose \(B^\alpha, \alpha\in (0, \infty); Q_p, p\in (0, \infty)\) and \(D_\alpha, \alpha\in (-\infty, 2)\) are the spaces of all analytic functions \(f\) in the unit disc \(D:| z| <1\), respectively with \(\sup_D (1-| | ^2)^\alpha | f'(z)| <\infty; \sup_{w\in D}\int_D | f'(z)| ^2 (log| \frac{1-\bar w z}{w-z}| )^p dx dy<\infty\) and \(\int_D | f'(z)| ^2 (1-| z| )^{1-\alpha}dx dy<\infty\). Also suppose \(M(D_\alpha)\) is the set of all multipliers of \(D_\alpha\). This paper mainly establishes the following 1. An inclusion between \(B^\alpha\) and \(Q_p\). 2. Some conditions for an analytic function with non-negative Taylor coefficients to be in \(B^\alpha\). 3. Relations between \(B^\alpha\) and \(D_\alpha\). 4. Inclusions between \(B^\beta \) and \(M(D_\alpha)\) and the answer to a question of Brown and Shields on cyclic vectors of the Dirichlet space. 5. A criterion for \(f\in Q_p\) in term of the Möbius transform of \(f\). 6. A sharp version of a result of Baernstein about univalent Bloch functions. 7. The meromorphic case of 1.
scientific article; zbMATH DE number 1031556
scientific article