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Let $\Omega$ be a domain in the complex plane, and $A^1(\Omega)$ be the space of integrable holomorphic functions on $\Omega$ equipped with the $L^1$ norm (it is called the Bergman space).

If $\Delta = \Omega$ is the unit disk, it is proved in the book of Duren and Schuster ("Bergman spaces") that the dual and a predual of $A^1$ are respectively the Bloch space and little Bloch space. Is there a similar description for non-simply connected domains $\Omega$ ?

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This probably is not what you wish (and does not apply to p=1), but I'll mention Hedenmalm The dual of a Bergman space on simply connected domains. J. Anal. Math. 88 (2002), 311–335; by looking in mathscinet you will find other related results.

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