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

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