Volterra-Composition Operators Acting on Sp Spaces and Weighted Zygmund Spaces

Let phi be an analytic selfmap of the open unit disk D and g be an analytic function on D. The Volterra -type composition operators induced by the maps g and phi are defined as (I-g(phi) f ) ( z ) = integral(z)(0)f' ( phi ( zeta )) g (zeta) d zeta and (T-g(phi) f ) ( z ) = integral(z )(0)f ( phi (zeta)) g' (zeta)d zeta. For 1 <= p < infinity, S-p (D) is the space of all analytic functions on D whose first derivative f' lies in the Hardy space (H)(p) (D), endowed with the norm parallel to f parallel to(Sp) = |f(0)| + parallel to f 'parallel to(Hp). Let mu : (0 , 1] -> (0 , infinity) be a positive continuous function on D such that for z is an element of D we define mu (z) = mu (|z|). The weighted Zygmund space Z(mu)(D) is the space of all analytic functions f on D such that sup(z is an element of D) mu (z) | f" (z) | is finite. In this paper, we characterize the boundedness and compactness of the Volterra -type composition operators that act between S-p spaces and weighted Zygmund spaces.