Extensions of the Laurent Decomposition and the spaces $A^p(\Omega)$

We generalize the classical Laurent decomposition in the setting of domains $\Omega\subseteq \mathbb C$ bounded by Jordan curves. This leads us to study the Fr\'echet spaces $A^p(\Omega)$, and their relation to the spaces $C^p(\partial \Omega)$. In the final section, we examine the case of a non Jordan domain $\Omega$.