On sum of weighted differentiation composition operators from Bergman spaces with admissible weights to Zygmund type spaces

Let \({\mathbb D}\) be the open unit disk in the complex plane. We characterize the boundedness and compactness of the sum of weighted differentiation composition operators

$$\begin{aligned} (T_{\overrightarrow{\psi }, \varphi } f)(z)=\sum _{j=0}^{n}(D^j_{\psi _j, \varphi }f)(z)=\sum _{j=0}^n\psi _{j}(z) f^{(j)} (\varphi (z)),\quad z\in {\mathbb D}, \end{aligned}$$

where \(n\in {\mathbb N}_0\), \(\psi _j\), \(j\in \overline{0,n}\), are holomorphic functions on \({\mathbb D}\), and \(\varphi \), a holomorphic self-maps of \({\mathbb D}\), acting from Bergman spaces with admissible weights to Zygmund type spaces.