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Characterizations of Bounded Singular Integral Operators on the Fock Space and Their Berezin Transforms
Xingtang Dong and Li Feng

Anal. Theory Appl. DOI: 10.4208/ata.OA-2021-0034

Publication Date : 2022-11-10

  • Abstract

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.

  • Copyright

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