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

Caiheng Ouyang, Weisheng Yang and Ruhan Zhao

Vol. 182 (1998), No. 1, 69–99

DOI: 10.2140/pjm.1998.182.69

Abstract

In this paper, function spaces Q_p(B) and Q_p,0(B), associated with the Green’s function, are defined and studied for the unit ball B of Cⁿ. We prove that Q_p(B) and Q_p,0(B) are Möbius invariant Banach spaces and that Q_p(B) = Bloch(B),Q_p,0(B) = ℬ₀(B) (the little Bloch space) when 1 < p < n∕(n− 1),Q₁ = BMOA(∂B) and Q_1,0(B) = VMOA(∂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,0(B)). Moreover, other properties about the above spaces and |φ_z(w)|,φ_z(w) ∈ Aut(B), are obtained.

Milestones

Received: 18 April 1996

Published: 1 January 1998