Set \({{\textbf {g}}}=(g_0,g_1,\cdots ,g_{n-1})\) with \(g_i\in H(\mathbb {C})\) for \(i=0,1,\cdots ,n-1\) and let \(T_{{\textbf{g}}}^{(n)}\) be the generalized Volterra-type operators on \(H(\mathbb {C}),\) which is represented as \(\begin{aligned}T_{{\textbf{g}}}^{(n)}f=I^n(fg_0+f'g_1+\cdots +f^{(n-1)}g_{n-1}),\end{aligned}\) where I denotes the integration operator \(\begin{aligned}(If)(z)=\int _{0}^{z}f(w)dw,\end{aligned}\) and \(I^n\) is the nth iteration of I. This operator is a generalization of the operator that was introduced by Chalmoukis in Ref. [1]. In this paper, we study the boundedness and compactness of the operators \(T_{{\textbf{g}}}^{(n)}\) acting on Fock spaces to another. As a consequence of these characterizations, we obtain conditions for certain linear differential equations to have solutions in Fock spaces. Then, we study the same properties, boundedness and compactness, of the following linear combination of weighted composition–differentiation operators: let \(\textbf{u}=(u_0,\cdots ,u_n)\) with \(u_k\in H(\mathbb {C})\) for \(0\le k\le n\) and \(\varphi \in H(\mathbb {C}).\) The linear combination of weighted composition–differentiation operators is defined by \(\begin{aligned}L_{{\textbf{u}},\varphi }^{(n)}=\sum _{i=0}^n u_iC_{\varphi }^{(i)},\end{aligned}\) where \(\begin{aligned}uC_{\varphi }^{(i)}f= u\cdot f^{(i)}\circ \varphi .\end{aligned}\) Our approach involves the study of Sobolev Carleson measures for classical Fock spaces.
