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

In this paper, we show that the Bergman functions on the Siegel upper half-space enjoy the following uniqueness property: if $f\in A_t^p(\text{calU})$ and ${\text{bfL}}^{\alpha }f\equiv 0$ for some nonnegative multi-index $\alpha$, then $f\equiv 0$, where ${\text{bfL}}^{\alpha }:=({\text{bfL}}_1{)}^{{\alpha }_1}\cdots ({\text{bfL}}_n{)}^{{\alpha }_n}$ with ${\text{bfL}}_j=\frac{\mathrm{\partial }}{\mathrm{\partial }{z}_j}+2i{\bar{z}}_j\frac{\mathrm{\partial }}{\mathrm{\partial }{z}_n}$ for $j=1,\dots ,n-1$ and ${\text{bfL}}_n=\frac{\mathrm{\partial }}{\mathrm{\partial }{z}_n}$. 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.