Characterizations of Bounded Singular Integral Operators on the Fock Space and Their Berezin Transforms (Analysis in Theory and Applications, 2023)
There is a singular integral operators $S_{φ}$ on the Fock space $\mathcal{F}^2(\mathbb{C}),$ which originated from the unitarily equivalent version of the Hilbert transform on $L^2(\mathbb{R}).$ In this paper, we give an analytic characterization of functions $φ$ with finite zeros such that the integral operator $S_{φ}$ is bounded on $\mathcal{F}^2(\mathbb{C})$ using Hadamard’s factorization theorem. As an application, we obtain a complete characterization for such symbol functions $φ$ such that the Berezin transform of $S_{φ}$ is bounded while the operator $S_{φ}$ is not. Also, the corresponding problem in higher dimensions is considered.
Keywords: Fock space; singular integral operators; boundedness; Berezin transform.