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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.

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)