On an integral-type operator from iterated logarithmic Bloch spaces into Bloch-type spaces

Let H(B)H(B) denote the space of all holomorphic functions on the open unit ball BB of CnCn. Let φ=(φ1,…,φn)φ=(φ1,…,φn) be a holomorphic self-map of BB and g∈H(B)g∈H(B) such that g(0)=0g(0)=0. In this paper we study the boundedness and compactness of the following integral-type operator, recently introduced by Xiangling Zhu and the second authorIφgf(z)=∫01Rf(φ(tz))g(tz)dtt,z∈B,from the iterated logarithmic Bloch spaces into the Bloch-type spaces. For the case when φ(z)≡zφ(z)≡z we also obtain a sufficient and necessary condition for the boundedness of this operator from the iterated logarithmic Bloch space into the little Bloch-type space.