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

Abstract

Let $\varphi$ be an analytic selfmap of the open unit disk $\mathbb{D}$ and $g$ be an analytic function on $\mathbb{D}$. The Volterra-type composition operators induced by the maps $g$ and $\varphi$ are defined as  $$\left(I_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f^{\prime}(\varphi(\zeta)) g(\zeta) d\zeta  \hspace{0.07in} \text{and} \hspace{0.07in}  \left(T_{g}^{\varphi}f\right)(z)= \int_{0}^{z} f(\varphi(\zeta)) g^{\prime}(\zeta) d\zeta .$$
For $1\leq p<\infty$, $S^p(\mathbb{D})$ is the space of all analytic functions on $\mathbb{D}$ whose first derivative $f^{\prime}$ lies in the Hardy space $H^p(\mathbb{D})$, endowed with the norm $\displaystyle \|f\|_{S^p}=|f(0)|+\|f^{\prime}\|_{H^p}$. Let $\mu: (0,1] \rightarrow (0, \infty)$ be a positive continuous function on $\mathbb{D}$ such that for $z\in\mathbb{D}$ we define $\mu(z) = \mu(|z|)$. The weighted Zygmund space  $\mathcal{Z}_{\mu}(\mathbb{D})$ is the space of all analytic functions $f$ on $\mathbb{D}$ such that $\sup_{z\in \mathbb{D}} \mu(z) |f^{\prime\prime}(z)|$ 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.