Generalized Volterra‐type operators on generalized Fock spaces

Let φ and g be entire functions on the complex plane C$\mathbb {C}$. The generalized Volterra‐type operators Cφg$C_\varphi ^g$ and Tφg$T_\varphi ^g$ induced by φ and g are defined by Cφgf(z)=∫0zf′(φ(ζ))g(ζ)dζ\begin{equation*} \hspace*{104pt}C_\varphi ^g f(z)=\int _0^z f^{\prime }(\varphi (\zeta ))g(\zeta )\,d\zeta \end{equation*}and Tφgf(z)=∫0zf(φ(ζ))g(ζ)dζ,\begin{equation*} \hspace*{105pt}T_\varphi ^g f(z)=\int _0^z f(\varphi (\zeta ))g(\zeta )\,d\zeta , \end{equation*}where f is an entire function and z∈C$z\in \mathbb {C}$. In this paper, we characterize the boundedness and compactness of the generalized Volterra‐type operators Cφg$C_\varphi ^g$ and Tφg$T_\varphi ^g$ acting between the generalized Fock spaces Fpϕ$\mathcal {F}_p^\phi$, induced by smooth radial weights ϕ that decay faster than the classical Gaussian ones. In addition, we obtain a upper pointwise estimate for the Bergman kernel for F2ϕ$\mathcal {F}_2^\phi$.