A uniqueness property for Bergman functions on the Siegel upper half-space
A uniqueness property for Bergman functions on the Siegel upper half-space
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DOI：
10.1090/proc/16290
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Heng Xu
中科院分区：
文献类型：
--
作者：
Congwen Liu;Jiajia Si;Heng Xu
In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if . f. ∈. A. t. p. (. U. ). fin A_t^p(mathcal {U}). and . L. α. f. ≡. 0. mathcal {L}^{alpha } fequiv 0. for some nonnegative multi-index . α. alpha., then . f. ≡. 0. fequiv 0., where . L. α. ≔. (. L. 1. ). α. 1. ⋯. (. L. n. ). α. n. mathcal {L}^{alpha }≔(mathcal {L}_1)^{alpha _1} cdots (mathcal {L}_n)^{alpha _n}. with . L. j. =. ∂. ∂. z. j. +. 2. i. z. ¯. j. ∂. ∂. z. n. mathcal {L}_j = frac {partial }{partial z_j} + 2i bar {z}_j frac {partial }{partial z_n}. for . j. =. 1. ,. …. ,. n. −. 1. j=1,ldots , n-1. and . L. n. =. ∂. ∂. z. n. 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.
