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

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