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The duals of Bergman spaces in Siegel domains of type II. (English) Zbl 0920.32002
The weighted Bergman space $$A^{p,r}(D)$$ is the space of holomorphic functions that are $$p$$th power integrable with respect to Lebesgue measure weighted by $$B(z,z)^{-r}$$, where $$B$$ denotes the Bergman kernel function.
This paper proves several results about the dual space of $$A^{p,r}(D)$$ when $$D$$ is a homogeneous Siegel domain of type II. In particular, if $$D$$ is symmetric, then there exist numbers $$p_0$$ and $$p_1$$ such that when $$p_0'<p<p_0$$ (a prime denotes the conjugate index), the dual of $$A^{p,r}(D)$$ is $$A^{p',r}(D)$$, and when $$p_1<p<1$$, the dual of $$A^{p,r}(D)$$ is the Bloch space, suitably defined.

##### MSC:
 32A07 Special domains in $${\mathbb C}^n$$ (Reinhardt, Hartogs, circular, tube) (MSC2010) 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)