Bloch-to-BMOA compositions in several complex variables

Abstract

We study analytic mappings φ : Bn → Bm and the corresponding analytic composition operators Cφ : f → f ◦ φ. Here n,m ∈ N and Bn is the unit ball of C. In the one complex variable case n = m = 1, D := B1, the investigation of composition operators from the Bloch space B(D) into BMOA(D) has only recently taken place. Boundedness and compactness of Cφ : B(D) → BMOA(D), Cφ : B0(D) → VMOA(D) and Cφ : B(D) → VMOA(D) has been studied in [SZ] by Smith and Zhao and by Makhmutov and Tjani in [MT]. Madigan and Matheson [MM] proved that Cφ is always bounded on B(D). Moreover, [MM] contains a characterization of symbols φ inducing compact composition operators on B(D) and B0(D). The essential norm of a composition operator from B(D) into Qp(D) was computed in [LMT]. In the case of several complex variables, Ramey and Ullrich [RU] have studied the case mentioned in the beginning: their result states that if φ : Bn → D is Lipschitz, then Cφ : B(D) → BMOA(Bn) is well defined, and consequently bounded by the closed graph theorem. Our results below are, of course, more general. The case of Cφ : B(Bn) → B(Bn) was considered by Shi and Luo [SL], where they proved that Cφ is always bounded and gave a necessary and sufficient condition for Cφ to be compact. Our main result states that if φ : Bn → Bm satisfies a very mild regularity condition, then the boundedness of Cφ : B(Bm) → BMOA(Bn) is characterized by the fact that dμφ(z) = (1−|z|2)|Rφ(z)|2 (1−|φ(z)|2)2 dA(z) is a Carleson measure (see notations below). Similarly, a corresponding o–growth condition characterizes the compactness. Let N := {1, 2, 3, . . . }. For z, w ∈ C let 〈z, w〉 = ∑n i=1 ziwi denote the complex inner product on C and |z| = 〈z, z〉1/2. The radial derivative operator is denoted by R; so, if f : Bn → C is analytic, then