On an Integral-Type Operator Acting between Bloch-Type Spaces on the Unit Ball

and Applied Analysis 3 Here H∞ ω denotes the weighted-type space consisting of all f ∈ H B with ∥ ∥f ∥ ∥ H∞ ω sup z∈B ω z ∣ ∣f z ∣ ∣ < ∞ 1.9 see, e.g., 23, 24 . Associated weights assist us in studying of weighted-type spaces of holomorphic functions. It is known that associated weights are also continuous, 0 < ω ≤ ω̃, and for each z ∈ B, we can find an fz ∈ H∞ ω , ‖fz‖H∞ ω ≤ 1 such that fz z 1/ω̃ z . Let H∞ ω,0 be the little weighted-type space, that is, the space of all f ∈ H B such that ω z |f z | → 0 as |z| → 1−. If ω is typical, then the unit ball BH∞ ω is the closure of BH∞ ω,0 for the compact open topology. Hence we have ω̃ z 1 sup {∣ ∣f z ∣ ∣ : f ∈ H∞ ω,0, ∥ ∥f ∥ ∥ H∞ ω ≤ 1 } 1.10 and so for each z ∈ B, we can choose an fz ∈ BH∞ ω,0 such that fz z 1/ω̃ z . A weight ω is called essential if it satisfies that ω̃ ≤ Cω for some positive constant C. By the arguments in 25 , we see that a normal weight function is also essential. For some examples of essential weights, see, for example, 25 . Related results can also be found in 22, 26 . The Bloch-type space Bω is the space of all holomorphic functions f on B such that bω ( f ) sup z∈B ω z ∣Rf z ∣ < ∞, 1.11 where ω is a weight see, e.g., 20 . The little Bloch-type space Bω,0 consists of all f ∈ H B such that lim |z|→ 1− ω z ∣Rf z ∣ 0. 1.12 Both spaces Bω and Bω,0 are Banach spaces with the norm ∥f ∥ Bω ∣f 0 ∣ bω ( f ) , 1.13 and Bω,0 is a closed subspace of Bω. When ω r 1 − r2, the space Bω is a classical Bloch space. The purpose of this paper is to characterize the boundedness and compactness of the operators I φ : Bω → Bμ and I φ : Bω,0 → Bμ,0. Throughout this paper, we assume that φ is a holomorphic self-map of B and g ∈ H B with g 0 0. Furthermore, some constants are denoted by C; they are positive and may differ from one occurrence to the other. The notation a b means that there exists a positive constant C such that a ≤ Cb. Moreover, if both a b and b a hold, then one says that a b. 2. Auxiliary Results Here we formulate and prove some auxiliary results which are used in the proofs of the main ones. 4 Abstract and Applied Analysis The following lemma was proved in 20, Theorem 2.1 . Lemma 2.1. Let ω be a normal weight function and f ∈ H B . Then f ∈ Bω if and only if supz∈B ω z |∇f z | < ∞ and it holds that ∥ ∥f ∥ ∥ Bω ∣ ∣f 0 ∣ ∣ sup z∈B ω z ∣ ∣∇f z ∣. 2.1 Moreover, f ∈ Bω,0 if and only if lim|z|→ 1− ω z |∇f z | 0. As an application of Lemma 2.1, we have the following result. Lemma 2.2. Let ω be a normal weight function and f ∈ Bω. Then f ∈ Bω,0 if and only if it holds that limr→ 1− ‖fr − f‖Bω 0, where fr z f rz . Proof. Take an f ∈ Bω,0. For a fixed ε > 0, by Lemma 2.1, we can choose a δ0 ∈ 0, 1 such that ω z ∣∇f z ∣ < ε 2 2.2 for any z ∈ B \ δ0B. Since ∂fr/∂zj z r ∂f/∂zj rz for j ∈ {1, . . . , n}, r ∈ 0, 1 , and z ∈ B, we have ∥fr − f ∥ Bω sup z∈B ω z ∣r∇f rz − ∇f z ∣ ≤ sup z∈B\δ0B ω z ∣r∇f rz − ∇f z ∣ sup z∈δ0B ω z ∣r∇f rz − ∇f z ∣. 2.3