Product type operators acting between Dirichlet and Zygmund type spaces

Abstract

Let $\mathbb{D}= \{\zeta\in\mathbb{C}:|\zeta|<1\}$ be the open unit disk in the complex plane $\mathbb{C}$. By $H(\mathbb{D})$, we denote the space of all holomorphic functions on $\mathbb{D}$. For an analytic self map $\xi$ on $\mathbb{D}$ and $\phi_{1}, \phi_{2}\in H(\mathbb{D})$, we have a product type operator $T_{\phi_{1},\phi_{2},\xi}$ defined by
\begin{align*}
T_{\phi_{1},\phi_{2},\xi}f(\zeta)= \phi_{1}(\zeta)f(\xi(\zeta))+ \phi_{2}(\zeta)f'(\xi(\zeta)), \quad f\in H(\mathbb{D}), \;\zeta\in\mathbb{D},
\end{align*}
This operator is basically a combination of three other operators namely composition operator $C_{\xi}$, multiplication operator $M_{\phi}$ and differentiation operator $D$. We study the boundedness and compactness of this operator from Dirichlet-type spaces to Zygmund-type spaces.