I know that it's true that if $\Omega_1$ and $\Omega_2$ are two holomorphy domains in $\mathbb{C}^n$, then the cartesian product $\Omega_1 \times \Omega_2$ is also a holomorphy domain in $\mathbb{C}^{2n}$, but I don't know how to prove this fact; in particular I can't figure out which characterization of the holomorphy I should use in this proof.
A "holomorphy domain" for me (see Krantz "Function theory of several complex variables") is an open set $\Omega \subset \mathbb{C}^{n}$ such that there NOT exist two open sets $U_1 \subset \Omega$ and $U_2 \nsubseteq \Omega$, with $U_1 \subset U_2 \cap \Omega$ such that for every holomorphic funtion $h$ on $\Omega$ there is a holomorphic function $h_2$ on $U_2$ such that $h_2|_{U_1}=h$.
It turns out that this definition is equivalent to require that $\Omega$ is pseudoconvex, or that it is convex with respect to the family of the holomorphic functions on $\Omega$, or many other things.