A uniqueness property for Bergman functions on the Siegel upper half-space

Congwen Liu; Jiajia Si; Heng Xu

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A uniqueness property for Bergman functions on the Siegel upper half-space

机译：A uniqueness property for Bergman functions on the Siegel upper half-space

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摘要

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if $fin A_t^p(mathcal {U})$ and $mathcal {L}^{alpha } fequiv 0$ for some nonnegative multi-index $alpha$, then $fequiv 0$, where $mathcal {L}^{alpha }≔(mathcal {L}_1)^{alpha _1} cdots (mathcal {L}_n)^{alpha _n}$ with $mathcal {L}_j = frac {partial }{partial z_j} + 2i bar {z}_j frac {partial }{partial z_n}$ for $j=1,ldots , n-1$ and $mathcal {L}_n = frac {partial }{partial z_n}$. As a consequence, we obtain a new integral representation for the Bergman functions on the Siegel upper half-space. In the end, as an application, we derive a result that relates the Bergman norm to a “derivative norm”, which suggests an alternative definition of the Bloch space and a notion of the Besov spaces over the Siegel upper half-space.

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《Proceedings of the American Mathematical Society》 |2023年第4期|1717-1728|共12页
作者
Congwen Liu; Jiajia Si; Heng Xu;
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作者单位

CAS Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China;

School of Science, Hainan University;

School of Mathematical Sciences, University of Science and Technology of China;

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正文语种英语
中图分类数学;
关键词
Siegel upper half-space; Bergman functions; uniqueness property; integral representation; derivative norm;

A uniqueness property for Bergman functions on the Siegel upper half-space

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