Mobius invariant vector-valued BMOA and H-1-BMOA duality of the complex ball

Two classes of vector-valued BMOA spaces are defined, in the complex ball a nd on the complex sphere, respectively, in the case of the complex sphere, vector measures are involved, since the argument in the scalar setting is n ot appropriate. Several properties (the L-p-equivalent norm theorem, expone ntial decay, the Baernstein theorem, and so on) of BMOA in the complex ball are extended to the Banach space setting. The two classes of BMOA spaces a re proved to be isomorphic; in particular, the corresponding John-Nirenberg exponential decay is shown. Finally, the vector-valued H-1-BMOA duality th eorem is proved.