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 C n . Let φ = ( φ 1 , … , φ n ) 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 author I φ g f ( z ) = ∫ 0 1 R f ( φ ( tz ) ) g ( tz ) 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.