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

Abstract

Let H (B) denote the space of all holomorphic functions on the open unit ball B of Cn. Let φ = (φ1, ..., φn) be a holomorphic self-map of B and g ∈ H (B) such that g (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 authorIφg f (z) = ∫01 R f (φ (tz)) g (tz) frac(dt, t), z ∈ B,from the iterated logarithmic Bloch spaces into the Bloch-type spaces. For the case when φ (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.