Essential Norms of Weighted Composition Operators from the 𝛼-Bloch Space to a Weighted-Type Space on the Unit Ball

Abstract

This paper finds some lower and upper bounds for the essential norm of the weighted composition operator from 𝛼-Bloch spaces to the weighted-type space 𝐻^∞_𝜇 on the unit ball for the case 𝛼≥1.

1. Introduction

Let 𝔹={𝑧∈ℂ^𝑛∶|𝑧|<1} be the open unit ball in ℂ^𝑛, 𝐻(𝔹) be the class of all holomorphic functions on the unit ball, and let 𝐻^∞(𝔹) be the class of all bounded holomorphic functions on 𝔹 with the norm‖𝑓‖_∞=sup_𝑧∈𝔹|𝑓(𝑧)|. (1.1)Let 𝑧=(𝑧₁,…,𝑧_𝑛) and 𝑤=(𝑤₁,…,𝑤_𝑛) be points in ℂ^𝑛 and ⟨𝑧,𝑤⟩=∑^𝑛_𝑘=1𝑧_𝑘𝑤_𝑘. For a holomorphic function 𝑓, we denote∇𝑓=(𝜕𝑓𝜕𝑧₁,…,𝜕𝑓𝜕𝑧_𝑛). (1.2)

A positive continuous function 𝜙 on the interval [0,1) is called normal (see [1]) if there is 𝛿∈[0,1) and 𝑎 and 𝑏, 0<𝑎<𝑏 such that𝜙(𝑟)(1−𝑟)^𝑎isdecreasingon[𝛿,1)andlim_𝑟→1𝜙(𝑟)(1−𝑟)^𝑎=0,𝜙(𝑟)(1−𝑟)^𝑏isincreasingon[𝛿,1)andlim_𝑟→1𝜙(𝑟)(1−𝑟)^𝑏=∞. (1.3)From now on, if we say that a function 𝜙∶𝔹→[0,∞) is normal, we will also assume that it is radial, that is, 𝜙(𝑧)=𝜙(|𝑧|),𝑧∈𝔹.

The weighted space 𝐻^∞_𝜇=𝐻^∞_𝜇(𝔹) consists of all 𝑓∈𝐻(𝔹) such thatsup_𝑧∈𝔹𝜇(𝑧)|𝑓(𝑧)|<∞, (1.4)where 𝜇 is normal on the interval [0,1). For 𝜇(𝑧)=(1−|𝑧|²)^𝛽, 𝛽>0, we obtain the weighted space 𝐻^∞_𝛽=𝐻^∞_𝛽(𝔹) (see, e.g., [2, 3]).

The 𝛼-Bloch space ℬ^𝛼=ℬ^𝛼(𝔹),𝛼>0, is the space of all 𝑓∈𝐻(𝔹) such that𝑏_𝛼(𝑓)=sup_𝑧∈𝔹(1−|𝑧|²)^𝛼|∇𝑓(𝑧)|<∞. (1.5)With the norm‖𝑓‖_ℬ^𝛼=|𝑓(0)|+𝑏_𝛼(𝑓), (1.6)the space ℬ^𝛼 is a Banach space ([4–6]).

The little 𝛼-Bloch space ℬ^𝛼₀=ℬ^𝛼₀(𝔹) is the subspace of ℬ^𝛼 consisting of all 𝑓∈𝐻(𝔹) such thatlim_|𝑧|→1(1−|𝑧|²)^𝛼|∇𝑓(𝑧)|=0. (1.7)

Let 𝑢∈𝐻(𝔹) and 𝜑 be a holomorphic self-map of the unit ball. Weighted composition operator on 𝐻(𝔹), induced by 𝑢 and 𝜑 is defined by(𝑢𝐶_𝜑𝑓)(𝑧)=𝑢(𝑧)𝑓(𝜑(𝑧)),𝑧∈𝔹. (1.8)This operator can be regarded as a generalization of a multiplication operator and a composition operator. It is interesting to provide a function theoretic characterization when 𝑢 and 𝜑 induce a bounded or compact weighted composition operator between some spaces of holomorphic functions on 𝔹. (For some classical results in the topic see, e.g., [5]. For some recent results on this and related operators, see, e.g., [2–4, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25] and the references therein.)

In [18], Ohno has characterized the boundedness and compactness of weighted composition operators between 𝐻^∞ and the Bloch space ℬ on the unit disk. In the setting of the unit polydisk 𝔻^𝑛, we have given some necessary and sufficient conditions for a weighted composition operator to be bounded or compact from 𝐻^∞(𝔻^𝑛) to the Bloch space ℬ(𝔻^𝑛) in [12] (see, also [21]). Corresponding results for the case of the unit ball are given in [14]. Among other results, in [14], we have given some necessary and sufficient conditions for the compactness of the operator 𝑢𝐶_𝜑∶ℬ^𝛼(𝔹)→𝐻^∞(𝔹), which we incorporate in the following theorem.

Theorem 1. Let 𝜑=(𝜑₁,…,𝜑_𝑛) be a holomorphic self-map of 𝔹 and 𝑢∈𝐻(𝔹).(a)If 𝛼>1, then the following statements are equivalent:(a1)𝑢𝐶_𝜑∶ℬ^𝛼₀→𝐻^∞ is a compact operator,(a2)𝑢𝐶_𝜑∶ℬ^𝛼→𝐻^∞ is a compact operator,(a3)𝑢∈𝐻^∞, andlim_{|𝜑(𝑧)|→1}|𝑢(𝑧)|(1−|𝜑(𝑧)|²)^𝛼−1=0. (1.9)(b)If 𝛼=1, then the following statements are equivalent:(b1)𝑢𝐶_𝜑∶ℬ₀→𝐻^∞ is a compact operator,(b2)𝑢𝐶_𝜑∶ℬ→𝐻^∞ is a compact operator,(b3)𝑢∈𝐻^∞, and lim_{|𝜑(𝑧)|→1}2|𝑢(𝑧)|ln1−|𝜑(𝑧)|²=0. (1.10)

We would also like to point out that if 𝛼≥1, then the boundedness of 𝑢𝐶_𝜑∶ℬ^𝛼→𝐻^∞_𝜇 and 𝑢𝐶_𝜑∶ℬ^𝛼₀→𝐻^∞_𝜇 are equivalent (see [22] for the case 𝛼=1, and the proof of Theorem 3 in [14]).

The essential norm of an operator is its distance in the operator norm from the compact operators. More precisely, assume that 𝑋₁ and 𝑋₂ are Banach spaces and 𝐴∶𝑋₁→𝑋₂ is a bounded linear operator, then the essential norm of 𝐴, denoted by ‖𝐴‖_{𝑒,𝑋1→𝑋2}, is defined as follows:‖𝐴‖_{𝑒,𝑋1→𝑋2}=inf{‖𝐴+𝐿‖_{𝑋1→𝑋2}∶𝐿∶𝑋₁→𝑋₂,𝐿iscompact}, (1.11)where ‖⋅‖_{𝑋1→𝑋2} denotes the operator norm. If 𝑋₁=𝑋₂, it is simply denoted by ‖⋅‖_𝑒 (see, e.g., [5, page 132]). If 𝐴∶𝑋₁→𝑋₂ is an unbounded linear operator, then clearly ‖𝐴‖_{𝑒,𝑋1→𝑋2}=∞.

Since the set of all compact operators is a closed subset of the set of bounded operators, it follows that an operator 𝐴 is compact if and only if ‖𝐴‖_{𝑒,𝑋1→𝑋2}=0.

Motivated by Theorem A, in this paper, we find some lower and upper bounds for the essential norm of the weighted composition operator 𝑢𝐶_𝜑∶ℬ^𝛼(𝔹)(orℬ^𝛼₀(𝔹))→𝐻^∞_𝜇(𝔹), when 𝛼≥1.

Throughout this paper, constants are denoted by 𝐶, they are positive and may differ from one occurrence to another. The notation 𝑎≼𝑏 means that there is a positive constant 𝐶 such that 𝑎≤𝐶𝑏. If both 𝑎≼𝑏 and 𝑏≼𝑎 hold, then we say that 𝑎≍𝑏.

2. Auxiliary Results

In this section, we quote several auxiliary results which we need in the proofs of the main results in this paper. The following lemma should be folklore.

Lemma 2.1. Let 𝑓∈ℬ^𝛼(𝔹),0<𝛼<∞. Then, |𝑓(𝑧)|≤𝐶‖𝑓‖_ℬ^𝛼,𝛼∈(0,1),|𝑓(0)|+𝑏₁1(𝑓)2ln1+|𝑧|𝐶1−|𝑧|,𝛼=1,‖𝑓‖_ℬ^𝛼(1−|𝑧|²)^𝛼−1,𝛼>1, (2.1) for some 𝐶>0 independent of 𝑓.

The proof of the lemma for the case 𝛼≠1 can be found, for example, in [26]. The formulation of the corresponding estimate in [26], for the case 𝛼=1, is slightly different. In this case, Lemma 2.1 follows from the following estimate:|𝑓(𝑧)−𝑓(0)|=|¹₀⟨∇𝑓(𝑡𝑧),𝑧⟩|≤𝑏₁(𝑓)¹₀|𝑧|𝑑𝑡1−|𝑧|²𝑡²=𝑏₁1(𝑓)2ln1+|𝑧|.1−|𝑧| (2.2)

The next lemma can be proved in a standard way (see, e.g., the proofs of the corresponding results in [5, 27–29]).

Lemma 2.2. Assume 𝛼>0, 𝑔∈𝐻(𝔹), 𝜇 is normal, and 𝜑 is an analytic self-map of 𝔹. Then, 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is compact if and only if 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is bounded and for any bounded sequence (𝑓_𝑘)_𝑘∈ℕ in ℬ^𝛼(orℬ^𝛼₀) converging to zero uniformly on compacts of 𝔹 as 𝑘→∞, one has‖𝑢𝐶_𝜑𝑓_𝑘‖_𝐻^∞_𝜇→0 as 𝑘→∞.

Lemma 2.3. Let 𝑓_𝑤(𝑧)=(1−|𝑤|²)^𝜀(1−⟨𝑧,𝑤⟩)^{𝛼+𝜀−1},𝑤∈𝔹, (2.3)where 𝛼>1 and 𝜀∈(0,1]. Then,‖𝑓_𝑤‖_ℬ^𝛼=(1−|𝑤|²)^𝜀+(𝛼+𝜀−1)(2𝛼)^𝛼|𝑤|(𝛼²+|𝑤|²𝜀²−|𝑤|²𝛼²+𝜀)^𝜀(𝛼+𝜀)^𝜀(𝛼²+|𝑤|²𝜀²−|𝑤|²𝛼²+𝛼)^𝛼. (2.4)

Proof. We have (1−|𝑧|²)^𝛼|∇𝑓_𝑤(𝑧)|=(𝛼+𝜀−1)(1−|𝑧|²)^𝛼(1−|𝑤|²)^𝜀|𝑤||1−⟨𝑧,𝑤⟩|^𝛼+𝜀, (2.5) from which it easily follows that 𝑏_𝛼(𝑓_𝑤)≤(𝛼+𝜀−1)2^𝛼+𝜀. (2.6) Set 𝑔_𝑠(𝑥)=(𝛼+𝜀−1)𝑠(1−𝑠²)^𝜀(1−𝑥²)^𝛼(1−𝑠𝑥)^𝛼+𝜀,𝑥∈[0,1],𝑠∈[0,1). (2.7) Then, 𝑔^′_𝑠(𝑥)=(𝛼+𝜀−1)𝑠(1−𝑠²)^𝜀(1−𝑥²)^𝛼−1𝑠(𝛼−𝜀)𝑥²−2𝛼𝑥+𝑠(𝛼+𝜀)(1−𝑠𝑥)^𝛼+𝜀+1. (2.8) Hence, the points 𝑥_𝑀,𝑚=𝛼±√𝛼²+𝑠²𝜀²−𝑠²𝛼²𝑠(𝛼−𝜀) (2.9) are stationary for the function 𝑔_𝑠(𝑥). Since 𝑥_𝑀>1, it follows that 𝑔_𝑠(𝑥) attains its maximum on the interval [0,1] at the point 𝑥_𝑚=𝛼−√𝛼²+𝑠²𝜀²−𝑠²𝛼²=𝑠(𝛼−𝜀)𝑠(𝛼+𝜀)𝛼+√𝛼²+𝑠²𝜀²−𝑠²𝛼²∈(0,1). (2.10) By some long but elementary calculations, it follows that 𝑔_𝑠(𝑥_𝑚)=(𝛼+𝜀−1)(2𝛼)^𝛼𝑠(√𝛼²+𝑠²𝜀²−𝑠²𝛼²+𝜀)^𝜀(𝛼+𝜀)^𝜀(√𝛼²+𝑠²𝜀²−𝑠²𝛼²+𝛼)^𝛼. (2.11)From this and since 𝑓_𝑤(0)=(1−|𝑤|²)^𝜀, (2.4) follows.

Remark 2.4. Note thatlim_|𝑤|→1‖𝑓_𝑤‖_ℬ^𝛼=lim_𝑠→1−0[(1−𝑠²)^𝜀+𝑔_𝑠(𝑥_𝑚)]=(𝛼+𝜀−1)2^𝛼+𝜀𝜀^𝜀𝛼^𝛼(𝛼+𝜀)^𝛼+𝜀,lim_𝜀→0+(𝛼+𝜀−1)2^𝛼+𝜀𝜀^𝜀𝛼^𝛼(𝛼+𝜀)^𝛼+𝜀=(𝛼−1)2^𝛼. (2.12)(2.13)

3. Estimates of the Essential Norm of 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇

In this section, we prove the main results in this paper. Before we formulate and prove these results, we prove another auxiliary result.

Lemma 3.1. Assume 𝛼∈(0,∞), 𝑢∈𝐻(𝔹),𝜇 is normal, 𝜑 is a holomorphic self-map of 𝔹 such that ‖𝜑‖_∞<1, and the operator 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is bounded. Then, 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is compact.

Proof. First note that since 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is bounded and 𝑓₀(𝑧)≡1∈ℬ^𝛼₀⊂ℬ^𝛼, it follows that 𝑢𝐶_𝜑(𝑓₀)=𝑢∈𝐻^∞_𝜇. Now, assume that (𝑓_𝑘)_𝑘∈ℕ is a bounded sequence in ℬ^𝛼(orℬ^𝛼₀) converging to zero on compacts of 𝔹 as 𝑘→∞. Then, we have ‖𝑢𝐶_𝜑(𝑓_𝑘)‖_𝐻^∞_𝜇≤‖𝑢‖_𝐻^∞_𝜇sup_{𝑤∈𝜑(𝔹)}|𝑓_𝑘(𝑤)|→0, (3.1) as 𝑘→∞, since 𝜑(𝔹) is contained in the ball |𝑤|≤‖𝜑‖_∞ which is a compact subset of 𝔹, according to the assumption, ‖𝜑‖_∞<1. Hence, by Lemma 2.2 , the operator 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is compact.

Theorem 3.2. Assume 𝛼>1, 𝜇 is normal, 𝑢∈𝐻(𝔹), 𝜑=(𝜑₁,…,𝜑_𝑛) is a holomorphic self-map of 𝔹, and 𝑢𝐶_𝜑∶ℬ^𝛼→𝐻^∞_𝜇 is bounded. Then, 2^−𝛼𝛼−1limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|(1−|𝜑(𝑧)|²)^𝛼−1≤‖𝑢𝐶_𝜑‖_𝑒,ℬ^𝛼_0→𝐻^∞_𝜇≤‖𝑢𝐶_𝜑‖_𝑒,ℬ^𝛼_→𝐻^∞_𝜇≤𝐶limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|(1−|𝜑(𝑧)|²)^𝛼−1, (3.2) for some positive constant 𝐶.

Proof. Since 𝑢𝐶_𝜑∶ℬ^𝛼→𝐻^∞_𝜇 is bounded, recall that 𝑢∈𝐻^∞_𝜇. If ‖𝜑‖_∞<1, then, from Lemma 3.1, it follows that 𝑢𝐶_𝜑∶ℬ^𝛼→𝐻^∞_𝜇 is compact which is equivalent with ‖𝑢𝐶_𝜑‖_𝑒,ℬ^𝛼_→𝐻^∞_𝜇=0, and, consequently, ‖𝑢𝐶_𝜑‖_𝑒,ℬ^𝛼_0→𝐻^∞_𝜇=0. On the other hand, it is clear that in this case the condition |𝜑(𝑧)|→1 is vacuous, so that inequalities in (3.2) are vacuously satisfied.
Hence, assume ‖𝜑‖_∞=1. Let (𝑧_𝑘)_𝑘∈ℕ be a sequence in 𝔹 such that lim_𝑘→∞|𝜑(𝑧_𝑘)|=1 and 𝜀∈(0,1) be fixed. Set 𝑓^𝜀_𝑘(𝑧)=(1−|𝜑(𝑧_𝑘)|²)^𝜀(1−⟨𝑧,𝜑(𝑧_𝑘)⟩)^{𝛼+𝜀−1},𝑘∈ℕ. (3.3) By Lemma 2.3, it follows that sup_𝑘∈ℕ‖𝑓^𝜀_𝑘‖_ℬ^𝛼<∞, moreover, it is easy to see that 𝑓^𝜀_𝑘∈ℬ^𝛼₀ for every 𝑘∈ℕ and 𝑓^𝜀_𝑘→0 uniformly on compacts of 𝔹 as 𝑘→∞. Then, by [6, Theorem 7.5], it follows that 𝑓^𝜀_𝑘 converges to zero weakly as 𝑘→∞. Hence, for every compact operator 𝐿∶ℬ^𝛼₀→𝐻^∞_𝜇, we have ‖𝐿𝑓^𝜀_𝑘‖_𝐻^∞_𝜇→0 as 𝑘→∞.
We have ‖𝑓^𝜀_𝑘‖_ℬ^𝛼‖𝑢𝐶_𝜑+𝐿‖_ℬ^𝛼_0→𝐻^∞_𝜇≥‖(𝑢𝐶_𝜑+𝐿)(𝑓^𝜀_𝑘)‖_𝐻^∞_𝜇≥‖𝑢𝐶_𝜑𝑓^𝜀_𝑘‖_𝐻^∞_𝜇−‖𝐿𝑓^𝜀_𝑘‖_𝐻^∞_𝜇 (3.4) for every compact operator 𝐿∶ℬ^𝛼₀→𝐻^∞_𝜇.
Letting 𝑘→∞ in (3.4) and using the definition of 𝑓^𝜀_𝑘, we obtain 𝐶(𝜀,𝛼)‖𝑢𝐶_𝜑+𝐿‖_ℬ^𝛼_0→𝐻^∞_𝜇=limsup_𝑘→∞‖𝑢𝐶_𝜑𝑓^𝜀_𝑘‖_𝐻^∞_𝜇≥limsup_𝑘→∞𝜇(𝑧_𝑘)|𝑢(𝑧_𝑘)|(1−|𝜑(𝑧_𝑘)|²)^𝛼−1, (3.5)where 𝐶(𝜀,𝛼) is the quantity in (2.12).
Taking in (3.5) the infimum over the set of all compact operators 𝐿∶ℬ^𝛼₀→𝐻^∞_𝜇, then letting 𝜀→0+ in such obtained inequality, and using (2.13), we obtain 2^𝛼(𝛼−1)‖𝑢𝐶_𝜑‖_𝑒,ℬ^𝛼_0→𝐻^∞_𝜇≥limsup_𝑘→∞𝜇(𝑧_𝑘)|𝑢(𝑧_𝑘)|(1−|𝜑(𝑧_𝑘)|²)^𝛼−1, (3.6) from which the first inequality in (3.2) follows.
Since the second inequality in (3.2) is obvious, we only have to prove the third one. By Lemma 3.1, we have that for each fixed 𝜌∈(0,1) the operator 𝑢𝐶_𝜌𝜑∶ℬ^𝛼→𝐻^∞_𝜇 is compact.
Let 𝛿∈(0,1) be fixed, and let (𝜌_𝑚)_𝑚∈ℕ be a sequence of positive numbers which increasingly converges to 1, then for each fixed 𝑚∈ℕ, we have ‖𝑢𝐶_𝜑‖_𝑒,ℬ^𝛼_→𝐻^∞_𝜇≤‖𝑢𝐶_𝜑−𝑢𝐶_𝜌𝑚𝜑‖_ℬ^𝛼_→𝐻^∞_𝜇=sup_{‖𝑓‖ℬ𝛼≤1}‖(𝑢𝐶_𝜑−𝑢𝐶_𝜌𝑚𝜑)(𝑓)‖_𝐻^∞_𝜇=sup_𝑧∈𝔹sup_{‖𝑓‖ℬ𝛼≤1}𝜇(𝑧)|𝑢(𝑧)(𝑓(𝜑(𝑧))−𝑓(𝜌_𝑚𝜑(𝑧)))|≤sup_{|𝜑(𝑧)|≤𝛿}sup_{‖𝑓‖ℬ𝛼≤1}𝜇(𝑧)|𝑢(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝜌_𝑚𝜑(𝑧))|+sup_{|𝜑(𝑧)|>𝛿}sup_{‖𝑓‖ℬ𝛼≤1}𝜇(𝑧)|𝑢(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝜌_𝑚𝜑(𝑧))|. (3.7)
By the mean-value theorem, we have sup_{|𝜑(𝑧)|≤𝛿}sup_{‖𝑓‖ℬ𝛼≤1}𝜇(𝑧)|𝑢(𝑧)(𝑓(𝜑(𝑧))−𝑓(𝜌_𝑚𝜑(𝑧)))|≤(1−𝜌_𝑚)sup_{|𝜑(𝑧)|≤𝛿}sup_{‖𝑓‖ℬ𝛼≤1}𝜇(𝑧)|𝑢(𝑧)||𝜑(𝑧)|sup_{|𝑤|≤𝛿}|∇𝑓(𝑤)|≤(1−𝜌_𝑚)𝛿(1−𝛿²)^𝛼‖𝑢‖_𝐻^∞_𝜇sup_{‖𝑓‖ℬ𝛼≤1}sup_{|𝑤|≤𝛿}(1−|𝑤|²)^𝛼|∇𝑓(𝑤)|≤(1−𝜌_𝑚)𝛿(1−𝛿²)^𝛼‖𝑢‖_𝐻^∞_𝜇→0, (3.8) as 𝑚→∞.
Moreover, by Lemma 2.1 (case 𝛼>1), and known inequality ‖𝑓_𝑟‖_ℬ^𝛼≤‖𝑓‖_ℬ^𝛼, (3.9) where 𝑓_𝑟(𝑧)=𝑓(𝑟𝑧), 𝑟∈[0,1), we have 𝜇(𝑧)|𝑢(𝑧)(𝑓(𝜑(𝑧))−𝑓(𝜌_𝑚𝜑(𝑧)))|≤𝐶‖𝑓−𝑓_𝜌𝑚‖_ℬ^𝛼𝜇(𝑧)|𝑢(𝑧)|(1−|𝜑(𝑧)|²)^𝛼−1≤2𝐶‖𝑓‖_ℬ^𝛼𝜇(𝑧)|𝑢(𝑧)|(1−|𝜑(𝑧)|²)^𝛼−1 (3.10) for some positive constant 𝐶. Replacing (3.10) in (3.7), letting in such obtained inequality 𝑚→∞, employing (3.8), and then letting 𝛿→1, the third inequality in (3.2) follows, finishing the proof of the theorem.

Corollary 3.3. Assume 𝛼>1, 𝜇 is normal, 𝑢∈𝐻(𝔹), 𝜑=(𝜑₁,…,𝜑_𝑛) is a holomorphic self-map of 𝔹, and the operator 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is bounded. Then, 𝑢𝐶_𝜑∶ℬ^𝛼(orℬ^𝛼₀)→𝐻^∞_𝜇 is compact if and only if limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|(1−|𝜑(𝑧)|²)^𝛼−1=0. (3.11)

Theorem 3.4. Assume 𝑢∈𝐻(𝔹),𝜇 is normal, 𝜑=(𝜑₁,…,𝜑_𝑛) is a holomorphic self-map of 𝔹, and 𝑢𝐶_𝜑∶ℬ→𝐻^∞_𝜇 is bounded. Then, 𝐶limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|ln1+|𝜑(𝑧)|1−|𝜑(𝑧)|≤‖𝑢𝐶_𝜑‖_{𝑒,ℬ0→𝐻}^∞_𝜇≤‖𝑢𝐶_𝜑‖_{𝑒,ℬ→𝐻}^∞_𝜇≤limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|ln1+|𝜑(𝑧)|1−|𝜑(𝑧)| (3.12) for a positive constant 𝐶.

Proof. Clearly, 𝑢=(𝑢𝐶_𝜑)(1)∈𝐻^∞_𝜇. If ‖𝜑‖_∞<1, then the result follows as in Theorem 3.2.
Hence, assume ‖𝜑‖_∞=1. We use the following family of test functionsℎ_𝑤(𝑧)=(ln(1+|𝑤|)²)1−⟨𝑧,𝑤⟩²(ln1+|𝑤|)1−|𝑤|⁻¹,𝑤∈𝔹. (3.13)
We have sup_𝑧∈𝔹(1−|𝑧|²)|∇ℎ_𝑤(𝑧)|=sup_𝑧∈𝔹2(1−|𝑧|²)|𝑤||1−⟨𝑧,𝑤⟩||ln(1+|𝑤|)²1−⟨𝑧,𝑤⟩|(ln1+|𝑤|)1−|𝑤|⁻¹≤4|𝑤|(2ln1+|𝑤|1−|𝑤|+2𝜋)(ln1+|𝑤|)1−|𝑤|⁻¹≤4(2+2𝜋ln3), (3.14) when |𝑤|≥1/2.
From this and since |ℎ_𝑤(0)|≤4(ln2)²/ln3, for |𝑤|≥1/2, we have that ‖ℎ_𝑤‖_ℬ≤4(2+2𝜋+ln3(ln2)²ln3)=𝐶₀. (3.15)
Assume (𝜑(𝑧_𝑘))_𝑘∈ℕ is a sequence in 𝔹 such that |𝜑(𝑧_𝑘)|→1 as 𝑘→∞. Note that (ℎ_{𝜑(𝑧𝑘)})_𝑘∈ℕ is a bounded sequence in ℬ (moreover in ℬ₀) converging to zero uniformly on compacts of 𝔹. Then, by [6, Theorem 7.5], it follows that ℎ_{𝜑(𝑧𝑘)} converges to zero weakly as 𝑘→∞. Hence, for every compact operator 𝐿∶ℬ₀→𝐻^∞_𝜇, we have lim_𝑘→∞‖𝐿ℎ_{𝜑(𝑧𝑘)}‖_𝐻^∞_𝜇=0. (3.16)
On the other hand, for every compact operator 𝐿∶ℬ₀→𝐻^∞_𝜇, we have‖ℎ_{𝜑(𝑧𝑘)}‖_ℬ‖𝑢𝐶_𝜑+𝐿‖_ℬ0→𝐻^∞_𝜇≥‖(𝑢𝐶_𝜑+𝐿)ℎ_{𝜑(𝑧𝑘)}‖_𝐻^∞_𝜇≥‖𝑢𝐶_𝜑ℎ_{𝜑(𝑧𝑘)}‖_𝐻^∞_𝜇−‖𝐿ℎ_{𝜑(𝑧𝑘)}‖_𝐻^∞_𝜇. (3.17) Using (3.15), letting 𝑘→∞ in (3.17), and applying (3.16), it follows that
𝐶₀‖𝑢𝐶_𝜑+𝐿‖_ℬ0→𝐻^∞_𝜇≥limsup_𝑘→∞‖𝑢𝐶_𝜑ℎ_{𝜑(𝑧𝑘)}‖_𝐻^∞_𝜇≥limsup_𝑘→∞𝜇(𝑧_𝑘)|𝑢(𝑧_𝑘)ℎ_{𝜑(𝑧𝑘)}(𝜑(𝑧_𝑘))|=limsup_𝑘→∞𝜇(𝑧_𝑘)|𝑢(𝑧_𝑘)|ln1+|𝜑(𝑧_𝑘)|1−|𝜑(𝑧_𝑘.)| (3.18)
Taking in (3.18) the infimum over the set of all compact operators 𝐿∶ℬ₀→𝐻^∞_𝜇, we obtain ‖𝑢𝐶_𝜑‖_{𝑒,ℬ0→𝐻}^∞_𝜇≥1𝐶₀limsup_𝑘→∞𝜇(𝑧_𝑘)|𝑢(𝑧_𝑘)|ln1+|𝜑(𝑧_𝑘)|1−|𝜑(𝑧_𝑘,)| (3.19) from which the first inequality in (3.12) follows.
As in Theorem 3.2, we need only to prove the third inequality in (3.12).
Recall that for each 𝜌∈(0,1), the operator 𝑢𝐶_𝜌𝜑∶ℬ→𝐻^∞_𝜇 is compact. Let the sequence (𝜌_𝑚)_𝑚∈ℕ be as in Theorem 3.2. Note that inequality (3.7) and relationship (3.8) also hold for 𝛼=1. Hence, we should only estimate the quantity 𝐼^𝛿_𝑚=sup_{|𝜑(𝑧)|>𝛿}sup_{‖𝑓‖ℬ≤1}𝜇(𝑧)|𝑢(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝜌_𝑚𝜑(𝑧))|. (3.20) On the other hand, by Lemma 2.1 (case 𝛼=1) applied to the function 𝑓−𝑓_𝜌𝑚, which belongs to the Bloch space for each 𝑚∈ℕ, and inequality (3.9) with 𝛼=1, we have 𝜇(𝑧)|𝑢(𝑧)(𝑓(𝜑(𝑧))−𝑓(𝜌_𝑚1𝜑(𝑧)))|≤2‖𝑓−𝑓_𝜌𝑚‖_ℬ𝜇(𝑧)|𝑢(𝑧)|ln1+|𝜑(𝑧)|1−|𝜑(𝑧)|≤‖𝑓‖_ℬ𝜇(𝑧)|𝑢(𝑧)|ln1+|𝜑(𝑧)|.1−|𝜑(𝑧)| (3.21)
From (3.21), by letting 𝑚→∞ in (3.7) and using (3.8) (with 𝛼=1), and letting 𝛿→1 in such obtained inequality, we obtain‖𝑢𝐶_𝜑‖_{𝑒,ℬ→𝐻}^∞_𝜇≤limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|ln1+|𝜑(𝑧)|,1−|𝜑(𝑧)| (3.22) as desired.

Corollary 3.5. Assume 𝑢∈𝐻(𝔹),𝜇 is normal, 𝜑=(𝜑₁,…,𝜑_𝑛) is a holomorphic self-map of 𝔹, and the operator 𝑢𝐶_𝜑∶ℬ(orℬ^𝛼₀)→𝐻^∞_𝜇 is bounded. Then, 𝑢𝐶_𝜑∶ℬ(orℬ^𝛼₀)→𝐻^∞_𝜇 is compact if and only if limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|ln1+|𝜑(𝑧)|1−|𝜑(𝑧)|=0. (3.23)