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

Abstract

Let 𝔹 denote the open unit ball of ℂ^𝑛. For a holomorphic self-map 𝜑 of 𝔹 and a holomorphic function 𝑔 in 𝔹 with 𝑔(0)=0, we define the following integral-type operator: 𝐼^𝑔_𝜑𝑓(𝑧)=∫¹₀ℜ𝑓(𝜑(𝑡𝑧))𝑔(𝑡𝑧)(𝑑𝑡/𝑡), 𝑧∈𝔹. Here ℜ𝑓 denotes the radial derivative of a holomorphic function 𝑓 in 𝔹. We study the boundedness and compactness of the operator between Bloch-type spaces ℬ_𝜔 and ℬ_𝜇, where 𝜔 is a normal weight function and 𝜇 is a weight function. Also we consider the operator between the little Bloch-type spaces ℬ_𝜔,0 and ℬ_𝜇,0.

1. Introduction

Let 𝔹 denote the open unit ball of the 𝑛-dimensional complex vector space ℂ^𝑛 and 𝐻(𝔹) the space of all holomorphic functions on 𝔹. For 𝑓∈𝐻(𝔹) with the Taylor expansion 𝑓(𝑧)=∑_|𝛾|≥0𝑎_𝛾𝑧^𝛾, letℜ𝑓(𝑧)=_|𝛾|≥0|𝛾|𝑎_𝛾𝑧^𝛾(1.1) be the radial derivative of 𝑓, where 𝛾=(𝛾₁,…,𝛾_𝑛) is a multi-index, |𝛾|=𝛾₁+⋯+𝛾_𝑛, and 𝑧^𝛾=𝑧₁^𝛾1…𝑧_𝑛^𝛾𝑛. It is well known thatℜ𝑓(𝑧)=^𝑛_𝑗=1𝑧_𝑗𝜕𝑓𝜕𝑧_𝑗(𝑧)=⟨∇𝑓(𝑧),𝑧⟩,(1.2) where ∇ is the usual gradient on ℂ^𝑛.

Let 𝜑 be a holomorphic self-map of 𝔹 and 𝑔∈𝐻(𝔹) with 𝑔(0)=0. Then 𝜑 and 𝑔 define an operator 𝐼^𝑔_𝜑 on 𝐻(𝔹) as follows:𝐼^𝑔_𝜑𝑓(𝑧)=¹₀ℜ𝑓(𝜑(𝑡𝑧))𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹),𝑧∈𝔹.(1.3)

The following important formula involving ℜ and 𝐼^𝑔_𝜑𝑓 was proved, for example, in [1]ℜ[𝐼^𝑔_𝜑𝑓](𝑧)=ℜ𝑓(𝜑(𝑧))𝑔(𝑧),𝑧∈𝔹.(1.4)

Motivated by papers [2, 3], operators 𝐼^𝑔_𝜑 were introduced by the first author of the present paper and Zhu in [1, 4–6], where its boundedness and compactness from the 𝛼-Bloch space, the Zygmund space, the mixed-norm space, and the generalized weighted Bergman space into the Bloch-type space on the unit ball are studied. In our previous work [7], we studied the boundedness and compactness of 𝐼^𝑔_𝜑 acting between weighted-type spaces. For related operators on ℂ^𝑛 see, for example, [8–21] and the references therein.

Let 𝜔 be a strictly positive continuous function on 𝔹 (weight). If 𝜔(𝑧)=𝜔(|𝑧|) for every 𝑧∈𝔹, we call it radial weight. A weight 𝜔 is called normal ([9, 22]) if it is radial and there are 𝑎 and 𝑏, 0<𝑎<𝑏<∞ such that 𝜔(𝑟)/(1−𝑟)^𝑎 is decreasing on [0,1), 𝜔(𝑟)/(1−𝑟)^𝑏 is increasing on [0,1),lim_𝑟→1𝜔(𝑟)(1−𝑟)^𝑎=0,lim_𝑟→1𝜔(𝑟)(1−𝑟)^𝑏=∞.(1.5)

A radial weight 𝜔 is called typical if it is nonincreasing with respect to |𝑧| and 𝜔(𝑧)→0 as |𝑧|→1⁻. If 𝜔 is normal, then by the monotonicity of 𝜔(𝑟)/(1−𝑟)^𝑎, for 0≤𝑟₁<𝑟<1, we have that𝜔(𝑟)=(1−𝑟)^𝑎𝜔(𝑟)(1−𝑟)^𝑎<(1−𝑟)^𝑎𝜔(𝑟₁)(1−𝑟₁)^𝑎<𝜔(𝑟₁),(1.6) that is, 𝜔 is decreasing on [0,1). On the other hand, from the first equality in (1.5), we have that for any 𝜀>0, there is a 𝛿>0 such that0<𝜔(𝑟)<𝜀(1−𝑟)^𝑎,(𝛿<𝑟<1),(1.7) which implies lim_𝑟→1⁻𝜔(𝑟)=0. Hence every normal weight 𝜔 is also typical.

For a weight 𝜔, the associated weight 𝜔 ([23]) is defined by1𝜔(𝑧)∶=sup{|𝑓(𝑧)|∶𝑓∈𝐻^∞_𝜔,‖𝑓‖_𝐻∞𝜔≤1},𝑧∈𝔹.(1.8)

Here 𝐻^∞_𝜔 denotes the weighted-type space consisting of all 𝑓∈𝐻(𝔹) with ‖𝑓‖_𝐻∞𝜔=sup_𝑧∈𝔹𝜔(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≤𝜔, and for each 𝑧∈𝔹, we can find an 𝑓_𝑧∈𝐻^∞_𝜔,‖𝑓_𝑧‖_𝐻^∞_𝜔≤1 such that 𝑓_𝑧(𝑧)=1/𝜔(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 𝐵_𝐻^∞_𝜔 is the closure of 𝐵_𝐻^∞_𝜔,0 for the compact open topology. Hence we have1𝜔(𝑧)=sup{|𝑓(𝑧)|∶𝑓∈𝐻^∞_𝜔,0,‖𝑓‖_𝐻∞𝜔≤1}(1.10) and so for each 𝑧∈𝔹, we can choose an 𝑓_𝑧∈𝐵_𝐻^∞_𝜔,0 such that 𝑓_𝑧(𝑧)=1/𝜔(𝑧). A weight 𝜔 is called essential if it satisfies that 𝜔≤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 ℬ_𝜔 is the space of all holomorphic functions 𝑓 on 𝔹 such that𝑏_𝜔(𝑓)=sup_𝑧∈𝔹𝜔(𝑧)|ℜ𝑓(𝑧)|<∞,(1.11) where 𝜔 is a weight (see, e.g., [20]). The little Bloch-type space ℬ_𝜔,0 consists of all 𝑓∈𝐻(𝔹) such that lim_{|𝑧|→1−}𝜔(𝑧)|ℜ𝑓(𝑧)|=0.(1.12) Both spaces ℬ_𝜔 and ℬ_𝜔,0 are Banach spaces with the norm ‖𝑓‖_ℬ𝜔=|𝑓(0)|+𝑏_𝜔(𝑓),(1.13) and ℬ_𝜔,0 is a closed subspace of ℬ_𝜔. When 𝜔(𝑟)=1−𝑟², the space ℬ_𝜔 is a classical Bloch space.

The purpose of this paper is to characterize the boundedness and compactness of the operators 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 and 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,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.

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∈ℬ_𝜔 if and only if sup_𝑧∈𝔹𝜔(𝑧)|∇𝑓(𝑧)|<∞ and it holds that ‖𝑓‖_ℬ𝜔≍|𝑓(0)|+sup_𝑧∈𝔹𝜔(𝑧)|∇𝑓(𝑧)|.(2.1) Moreover, 𝑓∈ℬ_𝜔,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∈ℬ_𝜔. Then 𝑓∈ℬ_𝜔,0 if and only if it holds that lim_𝑟→1⁻‖𝑓_𝑟−𝑓‖_ℬ𝜔=0, where 𝑓_𝑟(𝑧)=𝑓(𝑟𝑧).

Proof. Take an 𝑓∈ℬ_𝜔,0. For a fixed 𝜀>0, by Lemma 2.1, we can choose a 𝛿₀∈(0,1) such that 𝜀𝜔(𝑧)|∇𝑓(𝑧)|<2(2.2) for any 𝑧∈𝔹⧵𝛿₀²𝔹. Since (𝜕𝑓_𝑟/𝜕𝑧_𝑗)(𝑧)=𝑟(𝜕𝑓/𝜕𝑧_𝑗)(𝑟𝑧) for 𝑗∈{1,…,𝑛}, 𝑟∈(0,1), and 𝑧∈𝔹, we have ‖𝑓_𝑟−𝑓‖_ℬ𝜔≍sup_𝑧∈𝔹𝜔(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) Since max_{|𝑧|≤𝛿0}|𝑟∇𝑓(𝑟𝑧)−∇𝑓(𝑧)|→0,as𝑟→1⁻,(2.4) we see that the second term in (2.3) converges to 0 as 𝑟→1⁻.
If 𝑟∈(𝛿₀,1) and 𝑧∈𝔹⧵𝛿₀𝔹, then by (2.2) we have 𝜀𝜔(𝑟𝑧)|∇𝑓(𝑟𝑧)|<2.(2.5) By (1.6) we have that 𝜔(𝑧)≤𝜔(𝑟𝑧) for 𝑟,|𝑧|∈[0,1).
Hence we have sup_{𝑧∈𝔹⧵𝛿0𝔹}𝜔(𝑧)|𝑟∇𝑓(𝑟𝑧)−∇𝑓(𝑧)|≤sup_{𝑧∈𝔹⧵𝛿0𝔹}𝜔(𝑟𝑧)|∇𝑓(𝑟𝑧)|+sup_{𝑧∈𝔹⧵𝛿0𝔹}𝜔(𝑧)|∇𝑓(𝑧)|<𝜀,(2.6) for all 𝑟∈(𝛿₀,1). This proves that lim_𝑟→1⁻‖𝑓_𝑟−𝑓‖_ℬ𝜔=0 whenever 𝑓∈ℬ_𝜔,0.
Conversely, the normality of 𝜔 implies that for any 𝜀>0 we have 𝜔(𝑧)|∇𝑓(𝑟𝑧)|≤𝜀(1−|𝑧|)^𝑎sup_{|𝑤|≤𝑟}|∇𝑓(𝑤)|→0,as|𝑧|→1⁻,(2.7) so that 𝑓_𝑟∈ℬ_𝜔,0 for any 𝑟∈(0,1). On the other hand, by the assumption lim_𝑟→1⁻‖𝑓_𝑟−𝑓‖_ℬ𝜔=0, we have that for every 𝜀>0 there is an 𝑟₁∈(0,1) such that ‖𝑓_𝑟−𝑓‖_ℬ𝜔<𝜀,(2.8) for 𝑟∈(𝑟₁,1).
By letting |𝑧|→1⁻ in the following inequality, which easily follows from Lemma 2.1: 𝜔(𝑧)|∇𝑓(𝑧)|≼𝜔(𝑧)|∇𝑓(𝑟𝑧)|+‖𝑓_𝑟−𝑓‖_ℬ𝜔,(2.9) then using (2.7) and (2.8), we get 𝑓∈ℬ_𝜔,0, as claimed.

Corollary 2.3. Let 𝜔 be a normal weight function. Then the set of all holomorphic polynomials is dense in ℬ_𝜔,0.

Proof. For the homogeneous expansion 𝑓=∑^∞_𝑘=0𝐹_𝑘 of an 𝑓∈ℬ_𝜔,0, we set 𝑃^𝑗=∑^𝑗_𝑘=0𝐹_𝑘 for each 𝑗∈ℕ. Since 𝑃^𝑗→𝑓 uniformly on compact subsets of 𝔹 as 𝑗→∞, we see that ℜ[𝑃^𝑗_𝑟]→ℜ[𝑓_𝑟] uniformly on 𝔹 for any 𝑟∈(0,1). Moreover, we have ‖𝑃^𝑗_𝑟−𝑓‖_ℬ𝜔≤‖𝑃^𝑗_𝑟−𝑓_𝑟‖_ℬ𝜔+‖𝑓_𝑟−𝑓‖_ℬ𝜔≤sup_{𝑧∈𝑟𝔹}𝜔(𝑧)sup_𝑧∈𝔹|ℜ[𝑃^𝑗_𝑟](𝑧)−ℜ[𝑓_𝑟](𝑧)|+‖𝑓_𝑟−𝑓‖_ℬ𝜔.(2.10) Combining this with Lemma 2.2, we get the desired result.

The following lemma can be found in [1, Lemma  3]. Its proof is similar to the proof of the corresponding one-dimensional result in [27], for the case of the little Bloch space ℬ_(1−𝑟),0. Hence we omit the proof.

Lemma 2.4. A closed subset 𝐾 in ℬ_𝜔,0 is compact if and only if it is bounded and lim_{|𝑧|→1−}sup_𝑓∈𝐾𝜔(𝑧)|ℜ𝑓(𝑧)|=0.(2.11)

The following lemma is very useful for estimating the norm of the Bloch-type space.

Lemma 2.5. Assume that 𝑚 is a positive integer and 𝜔 is normal. Then for every 𝑓∈𝐻(𝔹),sup_𝑧∈𝔹𝜔(𝑧)|𝑓(𝑧)|≍|𝑓(0)|+sup_𝑧∈𝔹(1−|𝑧|)^𝑚𝜔(𝑧)|ℜ^𝑚𝑓(𝑧)|.(2.12)

Proof. For the details of the proof, we can refer [9] or [28].

3. The Boundedness of Operator 𝐼^𝑔_𝜑

In this section we consider the boundedness of the operator 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 or 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0.

Theorem 3.1. Let 𝜔 be a normal weight function and 𝜇 a weight function. Then the following conditions are equivalent: (a)𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is bounded;(b)𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇 is bounded;(c)𝜑 and 𝑔 satisfysup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))<∞.(3.1) Moreover, if 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is bounded, then ‖𝐼^𝑔_𝜑‖_{ℬ𝜔→ℬ𝜇}≍sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|.𝜔(𝜑(𝑧))(3.2)

Proof. The implication (a)⇒(b) is clear, so we only prove (b)⇒(c) and (c)⇒(a).
(b)⇒(c): assume that 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇 is bounded and fix 𝑧∈𝔹. We may assume that 𝜑(𝑧)≠0. For 𝑤∶=𝜑(𝑧), there exists ℎ_𝑤∈𝐻^∞_𝜔,0 such that ‖ℎ_𝑤‖_𝐻^∞_𝜔≤1 and ℎ_𝑤(𝑤)=1/𝜔(𝑤). We define the function 𝑓_𝑤 as follows: 𝑓_𝑤(𝑣)=¹₀ℎ_𝑤(𝑡𝑣)⟨𝑡𝑣,𝑤⟩|𝑤|𝑑𝑡𝑡,𝑣∈𝔹.(3.3) Since ℜ𝑓_𝑤(𝑣)=ℎ_𝑤(𝑣)(⟨𝑣,𝑤⟩/|𝑤|), we see that 𝑓_𝑤∈ℬ_𝜔,0 and ‖𝑓_𝑤‖_ℬ𝜔≤1. Hence, by (1.4), we have ‖𝐼^𝑔_𝜑‖_{ℬ𝜔→ℬ𝜇}≥‖𝐼^𝑔_𝜑𝑓_𝑤‖_ℬ𝜇≥𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓_𝑤(𝜑(𝑧))|=𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|,𝜔(𝜑(𝑧))(3.4) and so condition (3.1) is true.
(c)⇒(a): we assume (3.1) and take an 𝑓∈ℬ_𝜔. Since 𝜔 is an essential weight (due to its normality), (1.4) gives 𝜇(𝑧)|ℜ[𝐼^𝑔_𝜑𝑓](𝑧)|=𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))|≤𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))|∇𝑓(𝜑(𝑧))|⪯𝜔(𝜑(𝑧))𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))sup_𝑤∈𝔹𝜔(𝑤)|∇𝑓(𝑤)|,(3.5) for any 𝑧∈𝔹. By Lemma 2.1, we have sup_𝑤∈𝔹𝜔(𝑤)|∇𝑓(𝑤)|⪯‖𝑓‖_ℬ𝜔, and so we obtain ‖𝐼^𝑔_𝜑𝑓‖_ℬ𝜇⪯sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))‖𝑓‖_𝐵𝜔.(3.6) This implies that 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is bounded. The relation (3.2) follows from (3.4) and (3.6). This completes the proof.

Theorem 3.2. Let 𝜔 be a normal weight function and 𝜇 a weight function. Then the following conditions are equivalent: (a)𝐼^𝑔_𝜑 : ℬ_𝜔,0→ℬ_𝜇,0 is bounded; (b)𝜑 and 𝑔 satisfylim_{|𝑧|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|=0,sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))<∞.(3.7)

Proof. (a)⇒(b): as in the proof of Theorem 3.1, for fixed 𝑧∈𝔹 and 𝑤=𝜑(𝑧), we see that 𝜑 and 𝑔 satisfy the condition sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))<∞.(3.8)
On the other hand, since the normality of 𝜔 implies that the function 𝜋_𝑗(𝑧)∶=𝑧_𝑗(1≤𝑗≤𝑛) belongs to ℬ_𝜔,0, we obtain that 𝜇(𝑧)|𝑔(𝑧)||𝜑_𝑗(𝑧)|→0 for each 𝑗, and so 𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|→0 as |𝑧|→1⁻.
 (b)⇒(a): the assumption lim_|𝑧|→1⁻𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|=0 shows that 𝐼^𝑔_𝜑𝑝∈ℬ_𝜇,0 for any polynomial 𝑝. For each 𝑓∈ℬ_𝜔,0, by Corollary 2.3, we can choose a sequence of polynomials {𝑝_𝑗}_𝑗∈ℕ such that ‖𝑓−𝑝_𝑗‖_ℬ𝜔→0 as 𝑗→∞. Furthermore, the assumption sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))<∞(3.9) shows that 𝐼^𝑔_𝜑: ℬ_𝜔,0→ℬ_𝜇 is bounded by Theorem 3.1. Thus we obtain 0≤‖𝐼^𝑔_𝜑𝑓−𝐼^𝑔_𝜑𝑝_𝑗‖_ℬ𝜇≤‖𝐼^𝑔_𝜑‖_{ℬ𝜔→ℬ𝜇}‖𝑓−𝑝_𝑗‖_ℬ𝜔→0(as𝑗→∞).(3.10) Since 𝐼^𝑔_𝜑𝑓∈ℬ_𝜇, {𝐼^𝑔_𝜑𝑝_𝑗}_𝑗∈ℕ⊂ℬ_𝜇,0, and ℬ_𝜇,0 is closed in ℬ_𝜇, we have 𝐼^𝑔_𝜑𝑓∈ℬ_𝜇,0 for any 𝑓∈ℬ_𝜔,0. Hence 𝐼^𝑔_𝜑(ℬ_𝜔,0)⊆ℬ_𝜇,0 which means that 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0 is bounded. The proof is accomplished.

The following corollary is an immediate consequence of Theorems 3.1 and 3.2.

Corollary 3.3. Let 𝜔 be a normal weight function and 𝜇 a weight function. Then 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0 is bounded if and only if lim_|𝑧|→1⁻𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|=0 and 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is bounded.

4. The Compactness of Operator 𝐼^𝑔_𝜑

In this section we characterize the compactness of 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 or 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0. To do this, we need the following standard lemma (see, e.g., [13, Lemma  3]).

Lemma 4.1. Let 𝜔 and 𝜇 be weight functions. Suppose that the operator 𝐼^𝑔_𝜑∶ℬ_𝜔(orℬ_𝜔,0)→ℬ_𝜇 is bounded. Then 𝐼^𝑔_𝜑∶ℬ_𝜔(orℬ_𝜔,0)→ℬ_𝜇 is compact if and only if for every bounded sequence {𝑓_𝑗}_𝑗∈ℕ in ℬ_𝜔(orℬ_𝜔,0) which converges to 0 uniformly on compact subsets of 𝔹, ‖𝐼^𝑔_𝜑𝑓_𝑗‖_ℬ𝜇→0 as 𝑗→∞.

Theorem 4.2. Let 𝜔 and 𝜇 be weight functions. Suppose that 𝜑 is a holomorphic self-map of 𝔹 such that ‖𝜑‖_∞<1 and the operator 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is bounded. Then 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is compact. Here ‖𝜑‖_∞ denotes the supremum sup_𝑧∈𝔹|𝜑(𝑧)|.

Proof. Since ‖𝜑‖_∞<1, we see that |𝜑(𝑧)|≤𝑟 for some 𝑟∈(0,1) and any 𝑧∈𝔹. From the proof of Theorem 3.1, we see that the boundedness of 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 implies 𝑀∶=sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))<∞.(4.1) Thus we obtain that sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|≤𝑀sup_{𝑤∈𝑟𝔹}𝜔(𝑤)<∞.(4.2) Take a bounded sequence {𝑓_𝑗}_𝑗∈ℕ in ℬ_𝜔 such that 𝑓_𝑗→0 uniformly on compact subsets of 𝔹 as 𝑗→∞. By (1.4), we have ‖𝐼^𝑔_𝜑𝑓_𝑗‖_𝐵𝜇=sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓_𝑗(𝜑(𝑧))|≤sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)||∇𝑓_𝑗(𝜑(𝑧))|≤𝑀sup_{𝑤∈𝑟𝔹}𝜔(𝑤)sup_{𝑤∈𝑟𝔹}|∇𝑓_𝑗(𝑤)|.(4.3) Since 𝜕𝑓_𝑗/𝜕𝑧_𝑘(1≤𝑘≤𝑛) also converges to 0 uniformly on 𝑟𝔹 as 𝑗→∞, (4.2) and (4.3) show that ‖𝐼^𝑔_𝜑𝑓_𝑗‖_ℬ𝜇→0 as 𝑗→∞. From Lemma 4.1, it follows that 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is compact, and so we get the assertions.

Lemma 4.3. Suppose that 𝜔 is a weight function. Then there exists a sequence {𝑓_𝑘}_𝑘∈ℕ in the closed unit ball of ℬ_𝜔 such that 𝑓_𝑘→0 uniformly on compact subsets of 𝔹 as 𝑘→∞.

Proof. Let {𝑤_𝑘}_𝑘∈ℕ⊂𝔹 with |𝑤_𝑘|→1⁻ as 𝑘→∞. For each 𝑤_𝑘, there exists ℎ_𝑘∶=ℎ_𝑤𝑘∈𝐻^∞_𝜔 such that ‖ℎ_𝑘‖_𝐻^∞_𝜔≤1 and ℎ_𝑘(𝑤_𝑘)=1/𝜔(𝑤_𝑘). We define 𝑓_𝑘 as follows: 𝑓_𝑘(𝑧)=¹₀ℎ_𝑘(𝑡𝑧){⟨𝑡𝑧,𝑤_𝑘⟩|𝑤_𝑘|}^{1/(1−|𝑤𝑘|)}𝑑𝑡𝑡,𝑧∈𝔹.(4.4) Since 𝑓_𝑘(0)=0 and |ℜ𝑓_𝑘(𝑧)|≤|ℎ_𝑘(𝑧)|, we have {𝑓_𝑘}_𝑘∈ℕ⊂ℬ_𝜔 and ‖𝑓_𝑘‖_ℬ𝜔≤1 for each 𝑘∈ℕ. For any compact subset 𝒦 of 𝔹, we can choose an 𝑟∈(0,1) such that 𝒦⊂𝑟𝔹. Hence we obtain that for any 𝑧∈𝒦|𝑓_𝑘(𝑧)|≤¹₀|ℎ_𝑘(𝑡𝑧)|𝑡^{|𝑤𝑘|/(1−|𝑤𝑘|)}𝑑𝑡≤‖ℎ_𝑘‖_𝐻∞𝜔¹₀1𝑡𝜔(𝑡𝑧)^{|𝑤𝑘|/(1−|𝑤𝑘|)}𝑑𝑡≤max_{𝑤∈𝑟𝔹}1𝜔(𝑤)(1−|𝑤_𝑘|).(4.5) From the above inequality, it follows that 𝑓_𝑘 converges to 0 uniformly on compact subsets of 𝔹 as 𝑘→∞. This completes the proof.

Remark 4.4. If we assume that 𝜔 is typical in Lemma 4.3, then we can choose ℎ_𝑘∈𝐻^∞_𝜔,0. In this case, hence, we see that 𝑓_𝑘 belongs to ℬ_𝜔,0 for each 𝑘∈ℕ.

Theorem 4.5. Let 𝜔 be a normal weight function and 𝜇 a weight function. Suppose that the operator 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is bounded and ‖𝜑‖_∞=1. Then the following conditions are equivalent: (a)𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is compact;(b)𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇 is compact; (c)𝜑 and 𝑔 satisfylim_{|𝜑(𝑧)|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))=0.(4.6)

Proof. (a)⇒(b): this implication is obvious.
(b)⇒(c): take a sequence {𝑧_𝑘}_𝑘∈ℕ in 𝔹 with |𝜑(𝑧_𝑘)|→1⁻ as 𝑘→∞ and put 𝑤_𝑘=𝜑(𝑧_𝑘) for each 𝑘. Then, by Remark 4.4 after Lemma 4.3, there exists a sequence {𝑓_𝑘}_𝑘∈ℕ in ℬ_𝜔,0 such that sup_𝑘∈ℕ‖𝑓_𝑘‖_ℬ𝜔≤1 and 𝑓_𝑘→0 uniformly on compact subsets of 𝔹 as 𝑘→∞. By Lemma 4.1, the compactness of 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇 implies that ‖𝐼^𝑔_𝜑𝑓_𝑘‖_ℬ𝜇→0 as 𝑘→∞.
On the other hand, (1.4) gives ℜ[𝐼^𝑔_𝜑𝑓_𝑘](𝑧)=ℜ𝑓_𝑘(𝜑(𝑧))𝑔(𝑧), and so we have ‖𝐼^𝑔_𝜑𝑓_𝑘‖_ℬ𝜇≥𝜇(𝑧_𝑘)|ℜ𝑓_𝑘(𝜑(𝑧_𝑘))||𝑔(𝑧_𝑘)|≥𝜇(𝑧_𝑘)|ℜ𝑓_𝑘(𝜑(𝑧_𝑘))||𝑔(𝑧_𝑘)||𝜑(𝑧_𝑘)|.(4.7) From the construction (4.4) of 𝑓_𝑘, we obtain ℜ𝑓_𝑘(𝜑(𝑧_𝑘))=|𝜑(𝑧_𝑘)|^{1/(1−|𝜑(𝑧𝑘)|)}𝜔(𝜑(𝑧_𝑘,))(4.8) for each 𝑘∈ℕ. Combining this with (4.7), we have ‖𝐼^𝑔_𝜑𝑓_𝑘‖_ℬ𝜇≥𝜇(𝑧_𝑘)|𝑔(𝑧_𝑘)||𝜑(𝑧_𝑘)|𝜔(𝜑(𝑧_𝑘))|𝜑(𝑧_𝑘)|^{1/(1−|𝜑(𝑧𝑘)|)}.(4.9) Letting 𝑘→∞, we have lim_𝑘→∞𝜇(𝑧_𝑘)|𝑔(𝑧_𝑘)||𝜑(𝑧_𝑘)|𝜔(𝜑(𝑧_𝑘))=0,(4.10) for any sequence {𝑧_𝑘}_𝑘∈ℕ with |𝜑(𝑧_𝑘)|→1⁻. This proves that (4.6) is true.
(c)⇒(a): we will prove the following estimate: ‖𝐼^𝑔_𝜑‖_𝑒⪯limsup_{|𝜑(𝑧)|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|.𝜔(𝜑(𝑧))(4.11) Here ‖𝐼^𝑔_𝜑‖_𝑒 denotes the essential norm of 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇, namely, ‖𝐼^𝑔_𝜑‖_𝑒=inf{‖𝐼^𝑔_𝜑+𝒦‖_{ℬ𝜔→ℬ𝜇}∣𝒦∶ℬ_𝜔→ℬ_𝜇iscompact}.(4.12) Now we take a sequence {𝑟_𝑙}_𝑙∈ℕ⊂(0,1) which increasingly converges to 1 and put 𝐼^𝑔_𝑟𝑙𝜑𝑓(𝑧)=¹₀ℜ𝑓(𝑟_𝑙𝜑(𝑡𝑧))𝑔(𝑡𝑧)𝑑𝑡𝑡.(4.13) Since ‖𝑟_𝑙𝜑‖_∞≤𝑟_𝑙<1, Theorem 4.2 implies that 𝐼^𝑔_𝑟𝑙𝜑∶ℬ_𝜔→ℬ_𝜇 is compact for each 𝑙∈ℕ. For any 𝑓∈ℬ_𝜔 with ‖𝑓‖_ℬ𝜔≤1, from (1.4) it follows that ‖𝐼^𝑔_𝜑𝑓−𝐼^𝑔_𝑟𝑙𝜑𝑓‖_ℬ𝜔=sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))−ℜ𝑓(𝑟_𝑙𝜑(𝑧))|≤sup_{𝑅<|𝜑(𝑧)|<1}𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))−ℜ𝑓(𝑟_𝑙𝜑(𝑧))|+sup_{|𝜑(𝑧)|≤𝑅}𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))−ℜ𝑓(𝑟_𝑙𝜑(𝑧))|,(4.14) for some fixed 𝑅∈(0,1). The essentiality of 𝜔 and Lemma 2.1 give 𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))|≤𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|⪯𝜔(𝜑(𝑧))𝜔(𝜑(𝑧))|∇𝑓(𝜑(𝑧))|𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))sup_𝑤∈𝔹⪯𝜔(𝑤)|∇𝑓(𝑤)|𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|.𝜔(𝜑(𝑧))(4.15) Similarly, we also have 𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝑟_𝑙𝜑(𝑧))|⪯𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝑟_𝑙,𝜑(𝑧))(4.16) for each 𝑙∈ℕ. The normality of 𝜔 implies that 𝜔(𝑟_𝑙𝜑(𝑧))(1−|𝑟_𝑙𝜑(𝑧)|)^𝑎≥𝜔(𝜑(𝑧))(1−|𝜑(𝑧)|)^𝑎,(4.17) for each 𝑙∈ℕ and some 𝑎>0, and so by the essentiality, 𝜔(𝑟_𝑙𝜑(𝑧))(1−|𝑟_𝑙𝜑(𝑧)|)^𝑎≽𝜔(𝜑(𝑧))(1−|𝜑(𝑧)|)^𝑎.(4.18) Thus (4.16) and (4.18) give 𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝑟_𝑙𝜑(𝑧))|⪯𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|,𝜔(𝜑(𝑧))(4.19) for each 𝑙∈ℕ. By (4.15) and (4.19), we obtain sup_{𝑅<|𝜑(𝑧)|<1}𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))−ℜ𝑓(𝑟_𝑙𝜑(𝑧))|⪯sup_{𝑅<|𝜑(𝑧)|<1}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|.𝜔(𝜑(𝑧))(4.20)
When |𝜑(𝑧)|≤𝑅, by using the mean value theorem, we have 𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))−ℜ𝑓(𝑟_𝑙𝜑(𝑧))|≤(1−𝑟_𝑙)𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|sup_{|𝑤|≤𝑅}≤|∇[ℜ𝑓](𝑤)|1−𝑟_𝑙1−𝑅max_{𝑤∈𝑅𝔹}1𝜔(𝑤)sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|×sup_𝑤∈𝔹𝜔(𝑤)(1−|𝑤|)|∇[ℜ𝑓](𝑤)|.(4.21) Since 𝜔(𝑤)(1−|𝑤|) is also normal, by Lemmas 2.1 and 2.5, we have sup_𝑤∈𝔹𝜔(𝑤)(1−|𝑤|)|∇[ℜ𝑓](𝑤)|≍sup_𝑤∈𝔹𝜔(𝑤)(1−|𝑤|)|ℜ²𝑓(𝑤)|≍sup_𝑤∈𝔹𝜔(𝑤)|ℜ𝑓(𝑤)|.(4.22) Hence we obtain sup_{|𝜑(𝑧)|≤𝑅}𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))−ℜ𝑓(𝑟_𝑙𝜑(𝑧))|⪯1−𝑟_𝑙1−𝑅max_{𝑤∈𝑟𝔹}1𝜔(𝑤)sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|.(4.23) Since the boundedness of 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 implies sup_𝑧∈𝔹𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|<∞, letting 𝑙→∞ in the above inequality, we have sup_{‖𝑓‖ℬ𝜔≤1}sup_{|𝜑(𝑧)|≤𝑅}𝜇(𝑧)|𝑔(𝑧)||ℜ𝑓(𝜑(𝑧))−ℜ𝑓(𝑟_𝑙𝜑(𝑧))|→0.(4.24) By using (4.14), (4.20), and (4.24) and letting 𝑅→1⁻, we obtain the desired estimate ‖𝐼^𝑔_𝜑‖_𝑒⪯limsup_{|𝜑(𝑧)|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|.𝜔(𝜑(𝑧))(4.25) So if condition (4.6) is true, then ‖𝐼^𝑔_𝜑‖_𝑒=0, which means that 𝐼^𝑔_𝜑∶ℬ_𝜔→ℬ_𝜇 is compact. Our proof is accomplished.

Theorem 4.6. Let 𝜔 be a normal weight function and 𝜇 a weight function. Suppose that the operator 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0 is bounded. Then 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0 is compact if and only if lim_{|𝑧|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))=0.(4.26)

Proof. Suppose that (4.26) holds. For any 𝑓∈ℬ_𝜔,0, by Lemma 2.1 and (1.4), we have 𝜇(𝑧)|ℜ[𝐼^𝑔_𝜑⪯𝑓](𝑧)|=𝜇(𝑧)|ℜ𝑓(𝜑(𝑧))||𝑔(𝑧)|≤𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)||∇𝑓(𝜑(𝑧))|𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|⪯𝜔(𝜑(𝑧))𝜔(𝜑(𝑧))|∇𝑓(𝜑(𝑧))|𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))‖𝑓‖_ℬ𝜔.(4.27) Combining this with (4.26), we obtain lim_{|𝑧|→1−}sup_{‖𝑓‖ℬ𝜔≤1}𝜇(𝑧)|ℜ[𝐼^𝑔_𝜑𝑓](𝑧)|=0.(4.28) Hence it follows from Lemma 2.4 that 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0 is compact.
Conversely, we assume that 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇,0 is compact. By Theorem 3.2, we see thatlim_{|𝑧|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|=0.(4.29) Thus this implies (4.26) if ‖𝜑‖_∞<1.
Now assume ‖𝜑‖_∞=1. We claim that limsup_{|𝑧|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))=limsup_{|𝜑(𝑧)|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|.𝜔(𝜑(𝑧))(4.30) Further assume that {𝑧_𝑘}_𝑘∈ℕ is a sequence in 𝔹 such that limsup_{|𝑧|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))=lim_𝑘→∞𝜇(𝑧_𝑘)|𝑔(𝑧_𝑘)||𝜑(𝑧_𝑘)|𝜔(𝜑(𝑧_𝑘.))(4.31) If sup_𝑘∈ℕ|𝜑(𝑧_𝑘)|<1, then from this and (4.29) we have that both limits in (4.30) are equal to zero. If sup_𝑘∈ℕ|𝜑(𝑧_𝑘)|=1, then there is a subsequence {𝜑(𝑧_𝑘𝑙)}_𝑙∈ℕ such that |𝜑(𝑧_𝑘𝑙)|→1⁻ as 𝑙→∞. Hence we have limsup_{|𝑧|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|𝜔(𝜑(𝑧))=lim_𝑙→∞𝜇(𝑧_𝑘𝑙)|𝑔(𝑧_𝑘𝑙)||𝜑(𝑧_𝑘𝑙)|𝜔(𝜑(𝑧_𝑘𝑙))≤limsup_{|𝜑(𝑧)|→1−}𝜇(𝑧)|𝑔(𝑧)||𝜑(𝑧)|,𝜔(𝜑(𝑧))(4.32) and so (4.30) holds.
Since 𝐼^𝑔_𝜑∶ℬ_𝜔,0→ℬ_𝜇 is also compact, by Theorem 4.5, we see that the second limit in (4.30) is equal to zero, so that (4.26) holds. This completes the proof.

Acknowledgment

This work was supported by JSPS Grant-in-Aid for Young Scientists (Start-up: no. 20840004) and Ibaraki University.