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

TL;DR

This paper establishes a uniqueness property for Bergman functions on the Siegel upper half-space, leading to new integral representations and insights into related function spaces like Bloch and Besov spaces.

Contribution

It proves a novel uniqueness property for Bergman functions involving differential operators and introduces new integral representations and space characterizations.

Findings

Uniqueness property for Bergman functions with differential operators

New integral representation for Bergman functions

Relation between Bergman norm and derivative norm

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.