Small Bergman-Orlicz and Hardy-Orlicz spaces, and their composition operators

We show that the weighted Bergman-Orlicz space $A\_{\alpha}^{\psi}$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $\psi$ satisfies the so-called $\Delta^{2}$--condition. In addition we prove that this condition characterizes those $A\_{\alpha}^{\psi}$ on which every composition operator is bounded or order bounded into the Orlicz space $L\_{\alpha}^{\psi}$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $\psi$ satisfies the $\Delta^{2}$--condition, a composition operator is compact on $A\_{\alpha}^{\psi}$ if and only if it is order bounded into the so-called Morse-Transue space $M\_{\alpha}^{\psi}$. Our results stand in the unit ball of $\mathbb{C}^{N}$.