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We obtain a complete characterization of the entire functions $g$ such thatthe integral operator $(T_ g f)(z)=\int_{0}^{z}f(\zeta)\,g'(\zeta)\,d\zeta$ isbounded or compact, on a large class of Fock spaces $\mathcal{F}^\phi_p$,induced by smooth radial weights that decay faster than the classical Gaussianone. In some respects, these spaces turn out to be significantly different thanthe classical Fock spaces. Descriptions of Schatten class integral operatorsare also provided. En route, we prove a Littlewood-Paley formula for$||\cdot||_{\mathcal{F}^\phi_p}$ and we characterize the positive Borelmeasures for which $\mathcal{F}^\phi_p\subset L^q(\mu)$, $0<p,q<\infty$. Inaddition, we also address the question of describing the subspaces of$\mathcal{F}^\phi_p$ that are invariant under the classical Volterra integraloperator.

Integral operators, embedding theorems and a Littlewood-Paley formula on weighted Fock spaces