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$L^{p}$ REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN

Part of: Holomorphic functions of several complex variables

Published online by Cambridge University Press:  08 January 2020

LE HE Open the ORCID record for LE HE [Opens in a new window] ,
YANYAN TANG Open the ORCID record for YANYAN TANG [Opens in a new window]  and
ZHENHAN TU Open the ORCID record for ZHENHAN TU [Opens in a new window]
LE HE
Affiliation:
School of Mathematics and Statistics,Wuhan University, Wuhan, Hubei430072, PR China email hele2014@whu.edu.cn
YANYAN TANG
Affiliation:
School of Mathematics and Statistics,Wuhan University, Wuhan, Hubei430072, PR China email yanyantang@whu.edu.cn
ZHENHAN TU*
Affiliation:
School of Mathematics and Statistics,Wuhan University, Wuhan, Hubei430072, PR China email zhhtu.math@whu.edu.cn
*
✉zhhtu.math@whu.edu.cn
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Abstract

The Fock–Bargmann–Hartogs domain $D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, where $\unicode[STIX]{x1D707}>0$, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of $D_{n,m}(\,\unicode[STIX]{x1D707})$ with respect to the weight $(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, where $\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ and $\unicode[STIX]{x1D6FC}>-1$. Then, for $p\in [1,\infty ),$ we show that the corresponding weighted Bergman projection $P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$ is unbounded on $L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, except for the trivial case $p=2$. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is $L^{p}$ irregular when $p\in [1,\infty )\setminus \{2\}$, in contrast to the well-known positive $L^{p}$ regularity result on a bounded strongly pseudoconvex domain.

Keywords

Fock–Bargmann–Hartogs domainLp regularityweighted Bergman kernelweighted Bergman projection

MSC classification

Primary: 32A36: Bergman spaces
Secondary: 32A25: Integral representations; canonical kernels (Szegő, Bergman, etc.) 32A07: Special domains (Reinhardt, Hartogs, circular, tube)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 102 , Issue 2 , October 2020 , pp. 282 - 292
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The project is supported by the National Natural Science Foundation of China (No. 11671306).

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