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

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 $$(Igφf)(z)= ∫₀ᶻ f^{}(φ(ζ)) g(ζ) dζ {0.07in} and {0.07in} (Tgφf)(z)= ∫₀ᶻ f(φ(ζ)) g^{}(ζ) dζ .$$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 $z$ we define $μ(z) = μ(|z|)$. The weighted Zygmund space $Zμ(D)$ is the space of all analytic functions $f$ on $D$ such that $z∈ D μ(z) |f^{}(z)|$ 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.