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

Abstract

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if $f\in A_t^p(\calU)$ and $\bfL^{\alpha} f\equiv 0$ for some nonnegative multi-index $\alpha$, then $f\equiv 0$, where $\bfL^{\alpha}:=(\bfL_1)^{\alpha_1} \cdots (\bfL_n)^{\alpha_n}$ with $\bfL_j = \frac{\partial }{\partial z_j} + 2i \bar{z}_j \frac{\partial }{\partial z_n}$ for $j=1,\ldots, n-1$ and $\bfL_n = \frac{\partial }{\partial z_n}$. 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.