Möbius invariant \(Q_p\) spaces associated with the Green's function on the unit ball of \(C^n\)
Function spaces \(Q_p(B)\) and \(Q_{p,0}(B)\), associated with the Green's function, are defined and studied for the unit ball of \(\mathbb{C}^n\). We prove that \(Q_p(B)\) and \(Q_{p,0}(B)\) are Möbius invariant Banach spaces and that \(Q_p(B)= \text{Bloch}(B)\), \(Q_{p,0}(B)= {\mathcal B}_0(B)\) (the little Bloch space) when \(1<p<n/ (n-1)\), \(Q_1=\text{BMOA} (\partial B)\) and \(Q_{1,0}(B)= \text{VMOA}(\partial B)\). 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 \(|\varphi_z(w)|\), \(\varphi_z(w)\in \Aut(B)\), are obtained.
scientific article; zbMATH DE number 1060840