(untitled) (European Journal of Pure and Applied Mathematics, 2024)
Let φ 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 φ are defined as (I_g^φf)(z)= _0^z f^′(φ(ζ)) g(ζ) dζ 0.07in and 0.07in (T_g^φf)(z)= _0^z f(φ(ζ)) g^′(ζ) dζ . For 1≤ p<∞, 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 \|f\|_S^p=|f(0)|+\|f^′\|_H^p. Let μ: (0,1] → (0, ∞) be a positive continuous function on D such that for zD we define μ(z) = μ(|z|). The weighted Zygmund space Z_μ(D) is the space of all analytic functions f on D such that _zD μ(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.