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 ∈ A t p ( U ) f\in A_t^p(\mathcal {U}) and L α f ≡ 0 \mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index α \alpha , then f ≡ 0 f\equiv 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.