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

Abstract

Let $\phi$ 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 $\phi$ are defined as  $$\left(I_g^{\phi }f\right)(z)={\int }_0^z{f}^{\prime }(\phi (\zeta ))g(\zeta )d\zeta \text{and}\left(T_g^{\phi }f\right)(z)={\int }_0^{z}f(\phi (\zeta ))g^{\prime }(\zeta )d\zeta .$$
For $1\le p<\mathrm{\infty }$, $S^p(\mathbb{D})$ 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 ${\displaystyle \Vert f{\Vert }_{S^p}=|f(0)|+\Vert f^{\prime }{\Vert }_{H^p}}$. Let $\mu :(0,1]\to (0,\mathrm{\infty })$ 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 $\underset{z\in \mathbb{D}}{sup}\mu (z)|f^{\prime \prime }(z)|$ 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.