Generalized Volterra-type operators on generalized Fock spaces

Let phi and g be entire functions on the complex plane C. The generalized Volterra-type operators C-phi(g) and T-phi(g) induced by phi and g are defined by C-phi(g) f(z) = integral(z)(0) f'(phi(zeta))g(zeta) d zeta and T-phi(g) f(z) = integral(z)(0) f(phi(zeta))g(zeta) d zeta, where f is an entire function and z is an element of C. In this paper, we characterize the boundedness and compactness of the generalized Volterra-type operators C-phi(g) and T-phi(g) acting between the generalized Fock spaces F-p(phi), induced by smooth radial weights phi that decay faster than the classical Gaussian ones. In addition, we obtain a upper pointwise estimate for the Bergman kernel for F-2(phi).