Volterra-Composition Operators Acting on S^p Spaces and Weighted Zygmund Spaces

Let 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 are defined as (I₆^f) (z) = ₀^z f^ ( () ) g () d 0. 07in and 0. 07in (T₆^f) (z) = ₀^z f ( () ) g^ () d. For 1 p<, Sᵖ (D) is the space of all analytic functions on D whose first derivative f^ lies in the Hardy space Hᵖ (D), endowed with the norm \|f\|ₒ㵵=|f (0) |+\|f^\|₇㵵. Let: (0, 1] (0, ) be a positive continuous function on D such that for z we define (z) = (|z|). The weighted Zygmund space Z_ (D) is the space of all analytic functions f on D such that ₙ ₃ (z) |f^ (z) | is finite. In this paper, we characterize the boundedness and compactness of the Volterra-type composition operators that act between Sᵖ spaces and weighted Zygmund spaces.