Integral Operators, Embedding Theorems and a Littlewood–Paley Formula on Weighted Fock Spaces

Abstract

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 from 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.