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

In this paper, we show that the Bergman functions on Siegel upper half-space enjoy following uniqueness property: if f ∈ A t p ( U stretchy="false">) f\in A_t^p(\mathcal {U}) and alttext="script L alpha f identical-to 0"> mathvariant="script">L α ≡<!-- ≡ <mml:mn>0 encoding="application/x-tex">\mathcal {L}^{\alpha } f\equiv 0 for some nonnegative multi-index alttext="alpha"> encoding="application/x-tex">\alpha , then encoding="application/x-tex">f\equiv where colon-equal 1 right-parenthesis midline-horizontal-ellipsis n Super n"> ≔ 1 ⋯<!-- ⋯ <mml:mi>n }≔(\mathcal {L}_1)^{\alpha _1} \cdots (\mathcal {L}_n)^{\alpha _n} with j equals StartFraction partial-differential Over z EndFraction plus 2 i overbar EndFraction"> j = mathvariant="normal">∂<!-- ∂ <mml:mi>z + 2 i stretchy="false">¯<!-- ¯ </mml:mover> {L}_j = \frac {\partial }{\partial z_j} + 2i \bar {z}_j z_n} alttext="j comma ellipsis minus 1"> , …<!-- … <mml:mo>−<!-- − encoding="application/x-tex">j=1,\ldots , n-1 {L}_n . As a consequence, obtain new integral representation half-space. end, as an application, derive result relates norm to “derivative norm”, which suggests alternative definition of Bloch space notion Besov spaces over