Composition Operators from Bloch Type Spaces to Hardy and Besov Spaces*

Boundedness and compactness of composition operators from Bloch type spaces to Hardy spaces and analytic Besov spaces are characterized by function theoretic properties of their inducing maps. For the case of the Bloch space, the characterizations involve the hyperbolic versions of Hardy and Besov classes. The conditions for the hyperbolic Besov classes are then interpreted geometrically when the symbols are univalent, and strict inclusion between different hyperbolic Besov classes is shown by an example.