A Bloch-type space and the predual of $A_λ^1$ on the Siegel upper half-space

This paper aims to determine the predual of the Bergman space $A_λ^1$ on the Siegel upper half-space. To achieve this, a Bloch-type space $\widetilde{\calB}$ is introduced and studied, and some of its essential properties are established. We identify the little Bloch-type space $\widetilde{\calB}_0$ with the predual of $A_λ^1$ via a duality pairing.

💡 Research Summary

The paper investigates the predual of the weighted Bergman space (A^1_\lambda) on the Siegel upper half‑space (\mathcal{U}={z\in\mathbb{C}^n:\rho(z)>0}), where (\rho(z)=\operatorname{Im}z_n-|z’|^2). For (\lambda>-1) the weighted volume measure is (dV_\lambda(z)=c_\lambda\rho(z)^\lambda dV(z)) with a normalising constant expressed through Gamma functions. The Bergman kernel is (K_\lambda(z,w)=\rho(z,w)^{-(n+1+\lambda)}) where (\rho(z,w)=i^2(w_n-z_n)-z’\cdot w’). The associated projection operator (P_\lambda) defined by \