Möbius invariant Qpspaces associated with the Green’s function on the unit ball of Cn

Abstract

In this paper, function spaces Q(p)(B) and Q(p,o)(B), associated with the Green's function, are defined and studied for the unit ball B of C-n. We prove that Q(p)(B) and Q(p,o)(B) are Mobius invariant Banach spaces and that Q(p)(B) = Bloch(B), Q(p,o)(B) = B-o(B) (the little Bloch space) when 1 < p < n/(n - 1),Q(1) = BMOA(partial derivative B) and Q(1,0)(B) = VMOA(partial derivative B). This fact makes it possible for us to deal with BMOA and Bloch space in the same way. And we give necessary and sufficient conditions on boundedness (and compactness) of the Hankel operator with antiholomorphic symbols relative to Q(p)(B) (and Q(p,o)(B)). Moreover, other properties about the above spaces and \phi(z)(omega)\,phi(z)(omega) is an element of Aut(B), are obtained.