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Volume 39, Issue 2
Characterizations of Bounded Singular Integral Operators on the Fock Space and Their Berezin Transforms

Xingtang Dong & Li Feng

Anal. Theory Appl., 39 (2023), pp. 105-119.

Published online: 2023-06

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  • 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.

  • AMS Subject Headings

30H20, 47G10

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{ATA-39-105, author = {Dong , Xingtang and Feng , Li}, title = {Characterizations of Bounded Singular Integral Operators on the Fock Space and Their Berezin Transforms}, journal = {Analysis in Theory and Applications}, year = {2023}, volume = {39}, number = {2}, pages = {105--119}, 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.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.OA-2021-0034}, url = {http://global-sci.org/intro/article_detail/ata/21818.html} }
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.

Dong , Xingtang and Feng , Li. (2023). Characterizations of Bounded Singular Integral Operators on the Fock Space and Their Berezin Transforms. Analysis in Theory and Applications. 39 (2). 105-119. doi:10.4208/ata.OA-2021-0034
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