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REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN
Bulletin of the Australian Mathematical Society ( IF 0.7 ) Pub Date : 2020-01-08 , DOI: 10.1017/s0004972719001424
LE HE , YANYAN TANG , ZHENHAN TU

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.

中文翻译:

FOCK-BARGMANN-HATOGS 域上加权伯格曼投影的正则性

Fock–Bargmann–Hartogs 域$D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m} :\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$, 在哪里$\unicode[STIX]{x1D707}>0$, 是一个具有平滑实解析边界的无界强伪凸域。我们计算加权伯格曼核$D_{n,m}(\,\unicode[STIX]{x1D707})$关于重量$(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$, 在哪里$\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$$\unicode[STIX]{x1D6FC}>-1$. 那么,对于$p\in [1,\infty),$我们证明了相应的加权伯格曼投影$P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$是无限的$L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$, 除了平凡的情况$p=2$. 这给出了一个无界强伪凸域的示例,其普通伯格曼投影为$L^{p}$不规则时$p\in [1,\infty )\setminus \{2\}$,与众所周知的正相反$L^{p}$有界强伪凸域上的正则性结果。
更新日期:2020-01-08
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