Weak Definitions of the Hyperbolic BMOA Class

We prove that weak Möbius-invariant conditions of the BMOA type on the unit ball $$B_n$$ in $$\mathbb{C}^n$$ define the standard space $$\mathrm{BMOA}(B_n)$$ . We also obtain descriptions of the hyperbolic class $$\mathrm{BMOA}_h$$ of holomorphic mappings of $$B_n$$ into $$B_m$$ .