TY - JOUR T1 - Characterizations of Bounded Singular Integral Operators on the Fock Space and Their Berezin Transforms AU - Dong , Xingtang AU - Feng , Li JO - Analysis in Theory and Applications VL - 2 SP - 105 EP - 119 PY - 2023 DA - 2023/06 SN - 39 DO - http://doi.org/10.4208/ata.OA-2021-0034 UR - https://global-sci.org/intro/article_detail/ata/21818.html KW - Fock space, singular integral operators, boundedness, Berezin transform. AB -

There is a singular integral operators $S_{\varphi}$ 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 $\varphi$ with finite zeros such that the integral operator $S_{\varphi}$ 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 $\varphi$ such that the Berezin transform of $S_{\varphi}$ is bounded while the operator $S_{\varphi}$ is not. Also, the corresponding problem in higher dimensions is considered.