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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
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$\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$ 和$\unicode[STIX]{x1D6FC}>-1$ . 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
中文翻译:
FOCK-BARGMANN-HATOGS 域上加权伯格曼投影的正则性
Fock–Bargmann–Hartogs 域