Weighted Composition Operators and Integral-Type Operators between Weighted Hardy Spaces on the Unit Ball

Abstract

We study the boundedness and compactness of the weighted composition operators as well as integral-type operators between weighted Hardy spaces on the unit ball.

1. Introduction

Let 𝔹 denote the open unit ball of the 𝑛-dimensional complex vector space ℂ^𝑛, 𝜕𝔹 its boundary, and let 𝐻(𝔹) denote the space of all holomorphic functions on 𝔹. For 0<𝑝<∞ and 𝛼≥0 we define the weighted Hardy space  𝐻^𝑝_𝛼(𝔹) as follows: 𝐻^𝑝_𝛼(𝔹)={𝑓∈𝐻(𝔹)∶sup_0<𝑟<1(1−𝑟)^𝛼_𝜕𝔹|𝑓(𝑟𝜁)|^𝑝𝑑𝜎(𝜁)<∞}, (1.1) where 𝑑𝜎 is the normalized Lebesgue measure on 𝜕𝔹 (see, also [1], as well as [2], for an equivalent definition of the space). Note that for 𝛼=0 the weighted Hardy space becomes the Hardy space 𝐻^𝑝(𝔹). We define the norm ‖⋅‖_𝐻^𝑝_𝛼 on this space as follows: ‖𝑓‖^𝑝_𝐻𝑝𝛼=sup_0<𝑟<1(1−𝑟)^𝛼_𝜕𝔹|𝑓(𝑟𝜁)|^𝑝𝑑𝜎(𝜁). (1.2) With this norm 𝐻^𝑝_𝛼(𝔹) is a Banach space when 1≤𝑝<∞. For a related space on the unit polydisk; see [3]. In this paper, we investigate two types of operators acting between weighted Hardy spaces.

Let 𝜑 be a holomorphic self-map of 𝔹 and 𝑢∈𝐻(𝔹). Then 𝜑 and 𝑢 induce a weighted composition operator  𝑢𝐶_𝜑 on 𝐻(𝔹) which is defined by 𝑢𝐶_𝜑𝑓=𝑢(𝑓∘𝜑). This type of operators has been studied on various spaces of holomorphic functions in ℂ^𝑛, by many authors; see, for example, [4], recent papers [5–17], and the references therein.

Let 𝑔∈𝐻(𝔻) and 𝜑 be a holomorphic self-map of the open unit disk 𝔻 in the complex plane. Products of integral and composition operators on 𝐻(𝔻) were introduced by S. Li and S. Stević in a private communication (see [18–21], as well as papers [22] and [23] for closely related operators) as follows: 𝐶_𝜑𝐽_𝑔𝑓(𝑧)=₀^𝜑(𝑧)𝑓(𝜁)𝑔^′𝐽(𝜁)𝑑𝜁,_𝑔𝐶_𝜑𝑓(𝑧)=^𝑧₀𝑓(𝜑(𝜁))𝑔^′(𝜁)𝑑𝜁. (1.3)(1.4)

In [24] the first author of this paper has extended the operator in (1.4) in the unit ball settings as follows (see also [25, 26]). Assume 𝑔∈𝐻(𝔹), 𝑔(0)=0, and 𝜑 is a holomorphic self-map of 𝔹, then we define an operator on the unit ball as follows: 𝑃^𝑔_𝜑(𝑓)(𝑧)=¹₀𝑓(𝜑(𝑡𝑧))𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹),𝑧∈𝔹. (1.5) If 𝑛=1, then 𝑔∈𝐻(𝔻) and 𝑔(0)=0, so that 𝑔(𝑧)=𝑧𝑔₀(𝑧), for some 𝑔₀∈𝐻(𝔻). By the change of variable 𝜁=𝑡𝑧, it follows that 𝑃^𝑔_𝜑𝑓(𝑧)=¹₀𝑓(𝜑(𝑡𝑧))𝑡𝑧𝑔₀(𝑡𝑧)𝑑𝑡𝑡=^𝑧₀𝑓(𝜑(𝜁))𝑔₀(𝜁)𝑑𝜁. (1.6) Thus the operator (1.5) is a natural extension of operator 𝐽_𝑔𝐶_𝜑 in (1.4). For related operators see [27–33] as well as the references therein.

In this paper we study the boundedness and compactness of the weighted composition operators as well as the integral-type operator 𝑃^𝑔_𝜑, between different weighted Hardy spaces on the unit ball.

Throughout this paper, constants are denoted by 𝐶, they are positive and may differ from one occurrence to the other. The notation 𝑎⪯𝑏 means that there is a positive constant 𝐶 such that 𝑎≤𝐶𝑏. Moreover, if both 𝑎⪯𝑏 and 𝑏⪯𝑎 hold, then one says that 𝑎≍𝑏.

2. Weighted Composition Operators

This section is devoted to studying weighted composition operators between weighted Hardy spaces. Weighted composition operators between different Hardy spaces on the unit ball were previously studied in [15, 34], while the composition operators on the unit ball were studied in [35, 36]. For the case of the unit disk see also [37].

Before we formulate the main results in this section we quote several auxiliary results which will be used in the proofs of these ones.

Lemma 2.1. Let 0<𝑝<∞ and 𝛼≥0. Suppose that 𝑢∈𝐻(𝔹) and 𝜑 is a holomorphic self-map of 𝔹. Then for each 𝑓∈𝐻(𝔹)‖𝑢𝐶_𝜑𝑓‖_𝐻𝑝𝛼≤liminf_𝑅→1−‖𝑢𝐶_𝜑𝑓_𝑅‖_𝐻𝑝𝛼, (2.1) where 𝑢𝐶_𝜑𝑓_𝑅(𝑧)=𝑢(𝑧)𝑓(𝑅𝜑(𝑧)).

Proof. Fix 𝑟∈(0,1). Fatou's lemma shows that (1−𝑟)^𝛼_𝜕𝔹|𝑢(𝑟𝜁)𝑓(𝜑(𝑟𝜁))|^𝑝𝑑𝜎(𝜁)≤(1−𝑟)^𝛼liminf_{𝑅→1−𝜕𝔹}|𝑢(𝑟𝜁)𝑓(𝑅𝜑(𝑟𝜁))|^𝑝𝑑𝜎(𝜁)=liminf_𝑅→1−(1−𝑟)^𝛼_𝜕𝔹|𝑢(𝑟𝜁)𝑓(𝑅𝜑(𝑟𝜁))|^𝑝𝑑𝜎(𝜁)≤liminf_𝑅→1−‖𝑢𝐶_𝜑𝑓_𝑅‖_𝐻𝑝𝛼. (2.2) Hence we have the desired inequality.

Recall that an 𝑓∈𝐻(𝔹) has the homogeneous expansion 𝑓(𝑧)=^∞_{𝑘=0|𝛾|=𝑘}𝑐(𝛾)𝑧^𝛾, (2.3) where 𝛾=(𝛾₁,…,𝛾_𝑛) is a multi-index, |𝛾|=𝛾₁+⋯+𝛾_𝑛 and 𝑧^𝛾=𝑧₁^𝛾1⋯𝑧_𝑛^𝛾𝑛. For the homogeneous expansion of 𝑓 and any integer 𝑗≥1, let 𝑅_𝑗𝑓(𝑧)=^∞_{𝑘=𝑗|𝛾|=𝑘}𝑐(𝛾)𝑧^𝛾, (2.4) and 𝐾_𝑗=𝐼−𝑅_𝑗 where 𝐼𝑓=𝑓 is the identity operator. Note that 𝐾_𝑗 is compact operator on 𝐻^𝑝_𝛼(𝔹) for each 𝑗∈ℕ.

Lemma 2.2. If 1<𝑝<∞, then 𝑅_𝑗 converges to 0 pointwise in the Hardy space 𝐻^𝑝(𝔹)as 𝑗→∞.

Proof. See [34, Corollary 3.4] .

Lemma 2.2 and the uniform boundedness principle show that {𝑅_𝑗} is an uniformly bounded sequence in 𝐻^𝑝(𝔹).

The following lemma is proved similar to [4, Lemma 3.16]. We omit its proof.

Lemma 2.3. If 𝑢𝐶_𝜑 is bounded from 𝐻^𝑝_𝛼(𝔹) into 𝐻^𝑞_𝛽(𝔹), then ‖𝑢𝐶_𝜑‖_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}≤liminf_𝑗→∞‖𝑢𝐶_𝜑𝑅_𝑗‖_{𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}, (2.5) where ‖⋅‖_𝑒,𝐻^𝑝_{𝛼(𝔹)→𝐻}^𝑞_𝛽(𝔹) and ‖⋅‖_𝐻^𝑝_{𝛼(𝔹)→𝐻}^𝑞_𝛽(𝔹) denote the essential norm and the operator norm, respectively.

Lemma 2.4. Let 0<𝑝≤𝑞<∞. Suppose that 𝜇 is a positive Borel measure on 𝔹 which satisfies 𝜇(𝐵(𝜁,𝑡))≤𝐶₁𝑡^{𝑞𝑛/𝑝}(𝜁∈𝜕𝔹,𝑡>0), (2.6) for some positive constant 𝐶₁. Then there exists a positive constant 𝐶₂ which depends only on 𝑝,𝑞, and the dimension 𝑛 such that _𝔹|𝑓|^𝑞𝑑𝜇≤𝐶₁𝐶₂‖𝑓‖^𝑞_𝐻𝑝, (2.7) for any 𝑓∈𝐻^𝑝(𝔹). Here 𝐵(𝜁,𝑡)={𝑧∈𝔹∶|1−⟨𝑧,𝜁⟩|<𝑡}.

Proof. See [38, page 13, Theorem] or [34, Lemma 2.1] .

Let 0<𝑞<∞. For each 𝑟∈(0,1), a holomorphic self-map 𝜑 of 𝔹 and 𝑢∈𝐻(𝔹), we define a positive Borel measure 𝜇^𝑟_𝑢,𝜑 on 𝔹 by 𝜇^𝑟_𝑢,𝜑(𝐸)=_{(𝜑𝑟)−1(𝐸)}|𝑢_𝑟|^𝑞𝑑𝜎, (2.8) for all Borel sets 𝐸 of 𝔹. By the change of variables formula from measure theory, we can verify _𝔹𝑔𝑑𝜇^𝑟_𝑢,𝜑=_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞(𝑔∘𝜑)(𝑟𝜁)𝑑𝜎(𝜁), (2.9) for each nonnegative measurable function 𝑔 in 𝔹.

Theorem 2.5. Let 0<𝑝≤𝑞<∞ and 𝛼,𝛽≥0. Suppose that 𝑢∈𝐻(𝔹) and 𝜑 is a holomorphic self-map of 𝔹. Then 𝑢𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is bounded if and only if sup_𝑤∈𝔹sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)<∞. (2.10)

Proof. For 𝑤∈𝔹 we put 𝑓_𝑤(𝑧)={1−|𝑤|²(1−⟨𝑧,𝑤⟩)²}^{𝛼+𝑛/𝑝}. (2.11) Then we see that 𝑓_𝑤∈𝐻^𝑝_𝛼(𝔹) and moreover sup_𝑤∈𝔹‖𝑓_𝑤‖_𝐻^𝑝_𝛼≤𝐶. By a straightforward calculation, we have ‖𝑢𝐶_𝜑𝑓_𝑤‖^𝑞_𝐻𝑞𝛽=sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁), (2.12) for all 𝑤∈𝔹. Hence if 𝑢𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is bounded, then 𝑢 and 𝜑 satisfy the condition sup_𝑤∈𝔹sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)≤𝐶‖𝑢𝐶_𝜑‖^𝑞_{𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}<∞. (2.13)
Next we assume
𝑀∶=sup_𝑤∈𝔹sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)<∞. (2.14)
Fix 𝑟∈(0,1) and 𝑅∈(0,1), respectively. For 𝜁∈𝜕𝔹 and 𝑡,0<𝑡≤𝑡_𝑅=1−𝑅, we put 𝑤=(1−𝑡)𝜁 and 𝑤_𝑅=(1−𝑡_𝑅)𝜁. Since the function 𝑓_𝑤(𝑧), which is defined by (2.11) for this 𝑤, satisfies
|𝑓_𝑤(𝑧)|^𝑞>4^{−𝑞(𝛼+𝑛)/𝑝}𝑡^{−𝑞𝑛/𝑝}(1−𝑅)^{−𝑞𝛼/𝑝} (2.15) for all 𝑧∈𝐵(𝜁,𝑡), we have 𝜇^𝑟_𝑢,𝜑(𝐵(𝜁,𝑡))𝑡^{𝑞𝑛/𝑝}≤4^{𝑞(𝛼+𝑛)/𝑝}(1−𝑅)^{𝑞𝛼/𝑝}_{𝐵(𝜁,𝑡)}|𝑓_𝑤(𝑧)|^𝑞𝑑𝜇^𝑟_𝑢,𝜑(𝑧)≤4^{𝑞(𝛼+𝑛)/𝑝}(1−𝑅)^{𝑞𝛼/𝑝}_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)≤4^{𝑞(𝛼+𝑛)/𝑝}(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽𝑀. (2.16) By the same argument, the function 𝑓_𝑤𝑅(𝑧) gives the following estimate: 𝜇^𝑟_𝑢,𝜑(𝐵(𝜁,2𝑡_𝑅))≤9^{𝑞(𝛼+𝑛)/𝑝}(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽𝑀𝑡_𝑅^{𝑞𝑛/𝑝}. (2.17) Now we need to prove that there exists a positive constant 𝐶 such that 𝜇^𝑟_𝑢,𝜑(𝐵(𝜁,𝑡))≤𝐶(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽𝑀𝑡^{𝑞𝑛/𝑝}, (2.18) for all 𝜁∈𝜕𝔹 and 𝑡>0. By the estimate (2.16), we see that the inequality (2.18) is true for all 𝑡∈(0,𝑡_𝑅]. Thus we assume 𝑡>𝑡_𝑅. By the same argument as in [36, pages 241-242, proof of Theorem 1.1] , we see that the inequality (2.17) shows that there exists a positive constant 𝐶_𝑛 which depends only on the dimension 𝑛 such that 𝜇^𝑟_𝑢,𝜑(𝐵(𝜁,𝑡))≤𝐶_𝑛(𝑡𝑡_𝑅)^𝑛9^{𝑞(𝛼+𝑛)/𝑝}(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽𝑀𝑡_𝑅^{𝑞𝑛/𝑝}=𝐶_𝑛9^{𝑞(𝛼+𝑛)/𝑝}(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽𝑀𝑡^𝑛𝑡_𝑅^{(𝑞/𝑝−1)𝑛}≤𝐶_𝑛9^{𝑞(𝛼+𝑛)/𝑝}(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽𝑀𝑡^{𝑞𝑛/𝑝}. (2.19) Hence for 𝐶=max{4^{𝑞(𝛼+𝑛)/𝑝},𝐶_𝑛9^{𝑞(𝛼+𝑛)/𝑝}}, we have the inequality in (2.18).
For 𝑓∈𝐻^𝑝_𝛼(𝔹) the dilate function 𝑓_𝑅 belongs to the ball algebra, and so 𝑓_𝑅 is in the Hardy space 𝐻^𝑝(𝔹). Hence Lemma 2.4 gives
_𝔹|𝑓_𝑅(𝑧)|^𝑞𝑑𝜇^𝑟_𝑢,𝜑(𝑧)≤𝐶^′𝐶(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽𝑀‖𝑓_𝑅‖^𝑞_𝐻𝑝, (2.20) for some positive constant 𝐶^′ and all 𝑅∈(0,1). This implies that (1−𝑟)^𝛽_𝜕𝔹|𝑢𝐶_𝜑𝑓_𝑅(𝑟𝜁)|^𝑞𝑑𝜎(𝜁)≤𝐶^′𝐶𝑀[(1−𝑅)^𝛼_𝜕𝔹|𝑓(𝑅𝜁)|^𝑝𝑑𝜎(𝜁)]^𝑞/𝑝, (2.21) and so we have ‖𝑢𝐶_𝜑𝑓_𝑅‖^𝑞_𝐻𝑞𝛽≤𝐶^′𝐶𝑀‖𝑓‖^𝑞_𝐻𝑝𝛼, (2.22) for all 𝑅∈(0,1). By Lemma 2.1 we have ‖𝑢𝐶_𝜑𝑓‖^𝑞_𝐻𝑞𝛽≤𝐶^′𝐶‖𝑓‖^𝑞_𝐻𝑝𝛼×sup_𝑤∈𝔹sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁). (2.23) This completes the proof.

The following proposition is proved in a standard way; see, for example, the proofs of the corresponding results in [4, 32, 33, 39]. Hence we omit its proof.

Proposition 2.6. Let 0<𝑝,𝑞<∞ and 𝛼,𝛽≥0. Suppose that 𝑢∈𝐻(𝔹) and 𝜑 is a holomorphic self-map of 𝔹 which induce the bounded operator 𝑢𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹). Then 𝑢𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is compact if and only if for every bounded sequence {𝑓_𝑗}_𝑗∈ℕ in 𝐻^𝑝_𝛼(𝔹) which converges to 0 uniformly on compact subsets of 𝔹, {𝑢𝐶_𝜑𝑓_𝑗}_𝑗∈ℕ converges to 0 in 𝐻^𝑞_𝛽(𝔹).

In the proof of Theorem 2.8, we need the following lemma.

Lemma 2.7. Let 1<𝑝<∞, 𝛼≥0, and 𝑓_𝑤 be the family of test functions defined in (2.11). Then 𝑓_𝑤→0 weakly in 𝐻^𝑝_𝛼(𝔹) as |𝑤|→1⁻.

Proof. The family {𝑓_𝑤}_𝑤∈𝔹 is bounded in 𝐻^𝑝_𝛼(𝔹) and 𝑓_𝑤→0 uniformly on compact subsets of 𝔹 as |𝑤|→1⁻. By the definitions of the space 𝐻^𝑝_𝛼(𝔹) and the norm ‖⋅‖_𝐻^𝑝_𝛼, we see that 𝐻^𝑝_𝛼(𝔹) is a subspace of the weighted Bergman space 𝐴^𝑝_𝛼(𝔹) and ‖𝑓‖_𝐴^𝑝_𝛼≤𝐶(𝛼,𝑝,𝑛)‖𝑓‖_𝐻^𝑝_𝛼(𝑓∈𝐻^𝑝_𝛼(𝔹)), (2.24) for some positive constant 𝐶(𝛼,𝑝,𝑛) which depends on 𝛼,𝑝, and 𝑛. This inequality implies that the family {𝑓_𝑤}_𝑤∈𝔹 is also bounded in 𝐴^𝑝_𝛼(𝔹). Note also that the family converges to 0 uniformly on compact subsets of 𝔹 as |𝑤|→1⁻. Hence 𝑓_𝑤→0 weakly in 𝐴^𝑝_𝛼(𝔹) as |𝑤|→1⁻.
In order to prove that 𝑓_𝑤→0 weakly in 𝐻^𝑝_𝛼(𝔹) as |𝑤|→1⁻, we take an arbitrary bounded linear functional Λ on 𝐻^𝑝_𝛼(𝔹). By the Hahn-Banach theorem, Λ can be extended to a bounded linear functional Λ on 𝐴^𝑝_𝛼(𝔹) so that Λ(𝑓_𝑤)=Λ(𝑓_𝑤) for all 𝑤∈𝔹. Since 𝑓_𝑤→0 weakly in 𝐴^𝑝_𝛼(𝔹) as |𝑤|→1⁻, we have Λ(𝑓_𝑤)=Λ(𝑓_𝑤)→0 as |𝑤|→1⁻, and so 𝑓_𝑤→0 weakly in 𝐻^𝑝_𝛼(𝔹) as |𝑤|→1⁻.

Theorem 2.8. Let 1<𝑝≤𝑞<∞ and 𝛼,𝛽≥0. Suppose that 𝑢∈𝐻(𝔹) and 𝜑 is a holomorphic self-map of 𝔹 such that 𝑢𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is bounded. Then the 𝑞th power of the essential norm ‖𝑢𝐶_𝜑‖_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)} is comparable to limsup_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁). (2.25) Hence 𝑢𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is compact if and only if lim_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)=0. (2.26)

Proof. To prove a lower estimate ‖𝑢𝐶_𝜑‖^𝑞_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}≥limsup_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁), (2.27) we consider the test functions 𝑓_𝑤 defined in (2.11). The family {𝑓_𝑤}_𝑤∈𝔹 is bounded in 𝐻^𝑝_𝛼(𝔹), say by 𝐿, and 𝑓_𝑤→0 uniformly on compact subsets of 𝔹 as |𝑤|→1⁻. Thus by Lemma 2.7 we have that 𝑓_𝑤→0 weakly in 𝐻^𝑝_𝛼(𝔹) as |𝑤|→1⁻, so that ‖𝒦𝑓_𝑤‖_𝐻^𝑞_𝛽→0 as |𝑤|→1⁻ for every compact operator 𝒦∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹). Hence 𝐿‖𝑢𝐶_𝜑−𝒦‖_𝐻^𝑝_{𝛼(𝔹)→𝐻}^𝑞_𝛽(𝔹)≥limsup_{|𝑤|→1−}‖(𝑢𝐶_𝜑−𝒦)𝑓_𝑤‖_𝐻𝑞𝛽≥limsup_{|𝑤|→1−}‖𝑢𝐶_𝜑𝑓_𝑤‖_𝐻𝑞𝛽. (2.28) This inequality and (2.12) give the lower estimate for ‖𝑢𝐶_𝜑‖^𝑞_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}.
Next we prove an upper estimate. Take 𝑓∈𝐻^𝑝_𝛼(𝔹) with ‖𝑓‖_𝐻^𝑝_𝛼≤1. Fix 𝜀>0 and put
𝑀₁∶=limsup_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁). (2.29) Then we can choose 𝑅₀∈(0,1) such that sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)<𝑀₁+𝜀, (2.30) for 𝑤∈𝔹 with |𝑤|>𝑅₀. Fix 𝑟∈(0,1) and 𝑅∈(𝑅₀,1). By the same argument as in the proof of inequality (2.20) in Theorem 2.5, we obtain that _𝔹|(𝑅_𝑗𝑓)_𝑅(𝑧)|^𝑞𝑑𝜇^𝑟_𝑢,𝜑(𝑧)≤𝐶(1−𝑅)^{𝑞𝛼/𝑝}(1−𝑟)^𝛽(𝑀₁+𝜀)‖(𝑅_𝑗𝑓)_𝑅‖^𝑞_𝐻𝑝, (2.31) where the positive constant 𝐶 is independent of 𝑟, 𝑅 and a positive integer 𝑗. Since 𝑓_𝑅 is in the ball algebra, Lemma 2.2 gives ‖(𝑅_𝑗𝑓)_𝑅‖^𝑞_𝐻𝑝=‖𝑅_𝑗(𝑓_𝑅)‖^𝑞_𝐻𝑝≤sup_𝑗≥1‖𝑅_𝑗‖^𝑞_{𝐻𝑝(𝔹)→𝐻𝑝(𝔹)}‖𝑓_𝑅‖^𝑞_𝐻𝑝. (2.32) Combining this with (2.31), we have (1−𝑟)^𝛽_𝜕𝔹|𝑢𝐶_𝜑(𝑅_𝑗𝑓_𝑅)(𝑟𝜁)|^𝑞𝑑𝜎(𝜁)≤𝐶^′(𝑀₁+𝜀)[(1−𝑅)^𝛼_𝜕𝔹|𝑓_𝑅(𝜁)|^𝑝𝑑𝜎(𝜁)]^𝑞/𝑝≤𝐶^′(𝑀₁+𝜀)‖𝑓‖^𝑞_𝐻𝑝𝛼, (2.33) and so we have ‖𝑢𝐶_𝜑(𝑅_𝑗𝑓_𝑅)‖^𝑞_𝐻𝑞𝛽≤𝐶^′(𝑀₁+𝜀)‖𝑓‖^𝑞_𝐻𝑝𝛼. (2.34) Letting 𝑅→1⁻, by Lemma 2.1, we obtain ‖𝑢𝐶_𝜑𝑅_𝑗𝑓‖^𝑞_𝐻𝑞𝛽≤𝐶^′(𝑀₁+𝜀)‖𝑓‖^𝑞_𝐻𝑝𝛼. (2.35) Since 𝜀>0 is arbitrary, this estimate and Lemma 2.3 imply ‖𝑢𝐶_𝜑‖^𝑞_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}≤liminf_𝑗→∞‖𝑢𝐶_𝜑𝑅_𝑗‖^𝑞_{𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}≤𝐶^′limsup_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁), (2.36) which completes the proof.

Remark 2.9. In the above proof, we used Lemma 2.2. This lemma required the assumption 1<𝑝<∞. Hence we cannot have an upper estimate for ‖𝑢𝐶_𝜑‖_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)} in the case 0<𝑝≤1. However, Proposition 2.6 shows that the compactness of 𝑢𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹)(0<𝑝≤𝑞<∞) is equivalent to lim_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽_𝜕𝔹|𝑢(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)=0. (2.37)

3. Integral-Type Operators

Here we study the boundedness and compactness of the integral-type operators 𝑃^𝑔_𝜑 between weighted Hardy spaces on the unit ball.

For 𝑓∈𝐻(𝔹) with the Taylor expansion 𝑓(𝑧)=∑_|𝛾|≥0𝑎_𝛾𝑧^𝛾, let ℜ𝑓(𝑧)=∑_|𝛾|≥0|𝛾|𝑎_𝛾𝑧^𝛾 be the radial derivative of 𝑓.

The following lemma was proved in [24] (see also [25]).

Lemma 3.1. Assume that 𝜑 is a holomorphic self-map of 𝔹, 𝑔∈𝐻(𝔹) and 𝑔(0)=0. Then for every 𝑓∈𝐻(𝔹) one holds ℜ[𝑃^𝑔_𝜑(𝑓)](𝑧)=𝑓(𝜑(𝑧))𝑔(𝑧). (3.1)
A positive continuous function 𝜔 on the interval [0,1) is called normal [40] if there is a 𝛿∈[0,1) and 𝑎 and 𝑏, 0<𝑎<𝑏 such that
𝜔(𝑟)(1−𝑟)^𝑎isdecreasingon[𝛿,1)andlim_𝑟→1𝜔(𝑟)(1−𝑟)^𝑎=0,𝜔(𝑟)(1−𝑟)^𝑏isincreasingon[𝛿,1)andlim_𝑟→1𝜔(𝑟)(1−𝑟)^𝑏=∞. (3.2)
If one that a function 𝜔∶𝔹→[0,∞) is normal, one will also assume that it is radial.

Lemma 3.2. Assume that 0<𝑞≤∞, 𝑚 is a positive integer and 𝜔 is normal. Then for every 𝑓∈𝐻(𝔹)sup_0<𝑟<1𝜔(𝑟)𝑀_𝑞(𝑓,𝑟)≍|𝑓(0)|+sup_0<𝑟<1(1−𝑟)^𝑚𝜔(𝑟)𝑀_𝑞(ℜ^𝑚𝑓,𝑟), (3.3) where 𝑀_𝑞(𝑓,𝑟)=(_𝜕𝔹|𝑓(𝑟𝜁)|^𝑞𝑑𝜎(𝜁))^1/𝑞,𝑀_∞(𝑓,𝑟)=sup_{𝜁∈𝜕𝔹}|𝑓(𝑟𝜁)|. (3.4)

Proof. The proof of the lemma in the case 1≤𝑞≤∞ can be found in [27, Theorem 2]. However, due to an overlook, the proof for the case 𝑞∈(0,1) has a gap. Hence we will give a correct proof here in the case.
We may assume that 𝑓(0)=0, otherwise we can consider the functions ℎ(𝑧)=𝑓(𝑧)−𝑓(0). Also we may assume that 𝛿=0, to avoid some minor technical difficulties.
By [27, Lemma 1], for each fixed 𝑞∈(0,1], there is a positive constant 𝐶 depending only on 𝑞 and the dimension 𝑛 such that
𝑀_𝑞𝐶(𝑓,𝑟)≤𝑟(^𝑟₀(𝑟−𝑡)^𝑞−1𝑀^𝑞_𝑞(ℜ𝑓,𝑡)𝑑𝑡)^1/𝑞, (3.5) for every 𝑟∈(0,1) and 𝑓∈𝐻(𝔹) such that 𝑓(0)=0.
From (3.5) and the fact that 𝜔 is normal, we have
sup_0≤𝑟<1𝜔(𝑟)𝑀_𝑞(𝑓,𝑟)≤𝐶sup_0≤𝑟<11𝜔(𝑟)𝑟(^𝑟₀(𝑟−𝑡)^𝑞−1𝑀^𝑞_𝑞(ℜ𝑓,𝑡)𝑑𝑡)^1/𝑞≤𝐶sup_0≤𝑟<1(1−𝑟)^𝑎1𝑟(^𝑟₀(𝑟−𝑡)^𝑞−1𝜔^𝑞(𝑡)(1−𝑡)^𝑎𝑞𝑀^𝑞_𝑞(ℜ𝑓,𝑡)𝑑𝑡)^1/𝑞≤𝐶sup_0≤𝑟<1(1−𝑟)^𝑎1𝑟(^𝑟₀(𝑟−𝑡)^𝑞−1(1−𝑡)^{𝑎𝑞+𝑞}𝑑𝑡)^1/𝑞sup_0≤𝑡<1(1−𝑡)𝜔(𝑡)𝑀_𝑞(ℜ𝑓,𝑡)=𝐶sup_0≤𝑟<1(1−𝑟)^𝑎(¹₀(1−𝑢)^𝑞−1(1−𝑢𝑟)^{𝑎𝑞+𝑞}𝑑𝑢)^1/𝑞sup_0≤𝑡<1(1−𝑡)𝜔(𝑡)𝑀_𝑞(ℜ𝑓,𝑡). (3.6) By [40, page 291, Lemma 6] there exists a positive constant 𝐶 such that ¹₀(1−𝑢)^𝑞−1(1−𝑢𝑟)^{𝑎𝑞+𝑞}𝐶𝑑𝑢≤(1−𝑟)^𝑎𝑞, (3.7) for every 𝑟∈(0,1). Combining this with (3.6), we have sup_0≤𝑟<1𝜔(𝑟)𝑀_𝑞(𝑓,𝑟)≤𝐶sup_0≤𝑟<1(1−𝑟)^𝑎(1(1−𝑟)^𝑎𝑞)^1/𝑞sup_0≤𝑡<1(1−𝑡)𝜔(𝑡)𝑀_𝑞(ℜ𝑓,𝑡)=𝐶sup_0≤𝑡<1(1−𝑡)𝜔(𝑡)𝑀_𝑞(ℜ𝑓,𝑡). (3.8)
The reverse inequality is proved by the following inequality:
(1−𝑟)𝑀_𝑞(ℜ𝑓,𝑟)≤𝐶𝑀_𝑞(𝑓,1+𝑟2) (3.9) and the fact that 𝜔(𝑟)≍𝜔(1+𝑟/2) for 𝜔 normal (see [27]). Hence, we obtain the result for the case 𝑚=1.
For 𝑚≥2 it should be only noticed that (1−𝑟)^𝑚𝜔(𝑟) is still normal, that ℜ^𝑚𝑓(0)=0 for every integer 𝑚≥1, and use the method of induction.

Theorem 3.3. Let 0<𝑝≤𝑞<∞ and 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹. Then 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is bounded if and only if sup_𝑤∈𝔹sup_0<𝑟<1(1−𝑟)^𝛽+𝑞_𝜕𝔹|𝑔(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)<∞. (3.10)

Proof. Take 𝑓∈𝐻^𝑝_𝛼(𝔹) with ‖𝑓‖_𝐻^𝑝_𝛼≤1. Since the function (1−𝑟)^𝛽/𝑞 for 𝛽>0 and 0<𝑞<∞ is normal, Lemma 3.2 gives sup_0<𝑟<1(1−𝑟)^𝛽/𝑞𝑀_𝑞(𝑃^𝑔_𝜑𝑓,𝑟)≍|𝑃^𝑔_𝜑𝑓(0)|+sup_0<𝑟<1(1−𝑟)^{(𝛽/𝑞)+1}𝑀_𝑞(ℜ[𝑃^𝑔_𝜑𝑓],𝑟). (3.11) The assumption 𝑔(0)=0 implies 𝑃^𝑔_𝜑𝑓(0)=0, and Lemma 3.1 shows ℜ[𝑃^𝑔_𝜑𝑓]=𝑔𝐶_𝜑𝑓. Hence we obtain sup_0<𝑟<1(1−𝑟)^𝛽𝑀^𝑞_𝑞(𝑃^𝑔_𝜑𝑓,𝑟)≍sup_0<𝑟<1(1−𝑟)^𝛽+𝑞𝑀^𝑞_𝑞(𝑔𝐶_𝜑𝑓,𝑟), (3.12) and so we obtain ‖𝑃^𝑔_𝜑𝑓‖^𝑞_𝐻𝑞𝛽≍‖𝑔𝐶_𝜑𝑓‖^𝑞_{𝐻𝑞𝛽+𝑞}. This implies that the boundedness of 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is equivalent to the boundedness of 𝑔𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽+𝑞(𝔹). So Theorem 2.5 shows that the condition sup_𝑤∈𝔹sup_0<𝑟<1(1−𝑟)^𝛽+𝑞_𝜕𝔹|𝑔(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)<∞ (3.13) is a necessary and sufficient condition for the boundedness of 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹). This completes the proof.

The next proposition is proved similar to Proposition 2.6.

Proposition 3.4. Let 0<𝑝,𝑞<∞, and 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹 which induce the bounded operator 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹). Then 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is compact if and only if for every bounded sequence {𝑓_𝑗}_𝑗∈ℕ in 𝐻^𝑝_𝛼(𝔹) which converges to 0 uniformly on compact subsets of 𝔹, {𝑃^𝑔_𝜑𝑓_𝑗}_𝑗∈ℕ converges to 0 in 𝐻^𝑞_𝛽(𝔹).

Theorem 3.5. Let 0<𝑝≤𝑞<∞ and 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹 which induce the bounded operator 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹). Then 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is compact if and only if lim_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽+𝑞_𝜕𝔹|𝑔(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁)=0. (3.14)

Proof. First we assume that condition (3.14) holds. Take a bounded sequence {𝑓_𝑗}_𝑗∈ℕ⊂𝐻^𝑝_𝛼(𝔹) which converges to 0 uniformly on compact subsets of 𝔹. Theorem 2.8 and the remark in Section 2 show that 𝑔𝐶_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽+𝑞(𝔹) is compact. Thus Proposition 2.6 implies that lim_𝑗→∞‖𝑔𝐶_𝜑𝑓_𝑗‖_{𝐻𝑞𝛽+𝑞}=0. (3.15) From (3.15) and since ‖𝑃^𝑔_𝜑𝑓_𝑗‖^𝑞_𝐻𝑞𝛽≍‖𝑔𝐶_𝜑𝑓_𝑗‖^𝑞_{𝐻𝑞𝛽+𝑞}, we have that ‖𝑃^𝑔_𝜑𝑓_𝑗‖^𝑞_𝐻𝑞𝛽→0 as 𝑗→∞. By Proposition 3.4, we see that 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹) is compact.
To prove the necessity of the condition in (3.14), we consider the family of test functions 𝑓_𝑤 which is defined by (2.11). Hence we have
‖𝑃^𝑔_𝜑𝑓_𝑤‖^𝑞_𝐻𝑞𝛽≍‖𝑔𝐶_𝜑𝑓_𝑤‖^𝑞_{𝐻𝑞𝛽+𝑞}=sup_0<𝑟<1(1−𝑟)^𝛽+𝑞_𝜕𝔹|𝑔(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁), (3.16) for all 𝑤∈𝔹. Since {𝑓_𝑤}_𝑤∈𝔹 is a bounded sequence in 𝐻^𝑝_𝛼(𝔹) and 𝑓_𝑤→0 uniformly on compact subsets of 𝔹 as |𝑤|→1⁻, the compactness of 𝑃^𝑔_𝜑 and Proposition 3.4 show that ‖𝑃^𝑔_𝜑𝑓_𝑤‖^𝑞_𝐻𝑞𝛽→0 as |𝑤|→1⁻. This fact along with (3.16) implies the condition in (3.14), finishing the proof of the theorem.

Theorem 3.6. Let 1<𝑝≤𝑞<∞ and 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹 which induce the bounded operator 𝑃^𝑔_𝜑∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹). Then the 𝑞th power of the essential norm of 𝑃^𝑔_𝜑 is comparable to limsup_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽+𝑞_𝜕𝔹|𝑔(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁). (3.17)

Proof. To prove a lower estimate, we take an arbitrary compact operator 𝒦∶𝐻^𝑝_𝛼(𝔹)→𝐻^𝑞_𝛽(𝔹). Since Lemma 2.7 implies that the family of functions 𝑓_𝑤 defined by (2.11) converges to 0 weakly in 𝐻^𝑝_𝛼(𝔹) as |𝑤|→1⁻, we obtain 𝐶‖𝑃^𝑔_𝜑−𝒦‖_{𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}≥limsup_{|𝑤|→1−}(‖𝑃^𝑔_𝜑𝑓_𝑤‖_𝐻𝑞𝛽−‖𝒦𝑓_𝑤‖_𝐻^𝑞_𝛽)≥limsup_{|𝑤|→1−}‖𝑃^𝑔_𝜑𝑓_𝑤‖_𝐻𝑞𝛽. (3.18) Combining this with (3.16), we have 𝐶‖𝑃^𝑔_𝜑‖^𝑞_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}≥limsup_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽+𝑞_𝜕𝔹|𝑔(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁), (3.19) which is a lower estimate.
By some modification of Lemma 2.3 and the application of Lemmas 3.1 and 3.2, we get
‖𝑃^𝑔_𝜑‖^𝑞_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}≤liminf_𝑗→∞sup_{‖𝑓‖𝐻𝑝𝛼≤1}‖𝑃^𝑔_𝜑𝑅_𝑗𝑓‖^𝑞_𝐻𝑞𝛽≤𝐶liminf_𝑗→∞sup_{‖𝑓‖𝐻𝑝𝛼≤1}‖𝑔𝐶_𝜑𝑅_𝑗𝑓‖^𝑞_{𝐻𝑞𝛽+𝑞}. (3.20) As in the proof of Theorem 2.8, we obtain liminf_𝑗→∞sup_{‖𝑓‖𝐻𝑝𝛼≤1}‖𝑔𝐶_𝜑𝑅_𝑗𝑓‖^𝑞_{𝐻𝑞𝛽+𝑞}≤𝐶limsup_{|𝑤|→1−}sup_0<𝑟<1(1−𝑟)^𝛽+𝑞×_𝜕𝔹|𝑔(𝑟𝜁)|^𝑞{1−|𝑤|²|1−⟨𝜑(𝑟𝜁),𝑤⟩|²}^{𝑞(𝛼+𝑛)/𝑝}𝑑𝜎(𝜁), (3.21) and so we have an upper estimate for ‖𝑃^𝑔_𝜑‖^𝑞_{𝑒,𝐻𝑝𝛼(𝔹)→𝐻𝑞𝛽(𝔹)}.

4. The Case 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)→𝐻^∞_𝛽(𝔹)

When 𝑝=∞ and 𝛼>0, we define the weighted-type space 𝐻^∞_𝛼(𝔹) as follows: 𝐻^∞_𝛼(𝔹)={𝑓∈𝐻(𝔹)∶sup_0<𝑟<1(1−𝑟)^𝛼𝑀_∞(𝑓,𝑟)<∞}. (4.1) It is easy to see that 𝑓∈𝐻^∞_𝛼(𝔹) if and only if sup_𝑧∈𝔹(1−|𝑧|)^𝛼|𝑓(𝑧)|<∞, so we define the norm ‖𝑓‖_𝐻∞𝛼 on 𝐻^∞_𝛼(𝔹) by this supremum.

Furthermore we consider the subspace 𝐻^∞_𝛼,0(𝔹) defined by 𝐻^∞_𝛼,0(𝔹)={𝑓∈𝐻(𝔹)∶lim_𝑟→1−(1−𝑟)^𝛼𝑀_∞(𝑓,𝑟)=0}. (4.2)

Theorem 4.1. Let 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹. Then 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽(𝔹) is bounded if and only if sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼<∞. (4.3) In this case, the operator norm ‖𝑃^𝑔_𝜑‖_{𝐻∞𝛼(𝔹)(𝑜𝑟𝐻∞𝛼,0(𝔹))→𝐻∞𝛽(𝔹)} is comparable to the above supremum.

Proof. By the definition of the space 𝐻^∞_𝛼(𝔹), 𝑓∈𝐻^∞_𝛼(𝔹) satisfies the growth condition |𝑓(𝑤)|≤(1−|𝑤|)^−𝛼‖𝑓‖_𝐻∞𝛼(𝑤∈𝔹), (4.4) so it follows from Lemma 3.1 and Lemma 3.2 that ‖𝑃^𝑔_𝜑𝑓‖_𝐻∞𝛽≍sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔𝐶_𝜑𝑓(𝑧)|≤‖𝑓‖_𝐻∞𝛼sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼, (4.5) for every 𝑓∈𝐻^∞_𝛼(𝔹).
Hence we obtain
‖𝑃^𝑔_𝜑‖_{𝐻∞𝛼(𝔹)(o𝑟𝐻∞𝛼,0(𝔹))→𝐻∞𝛽(𝔹)}≤𝐶sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼. (4.6)
Now we prove the reverse inequality. For 𝑤∈𝔹, we put
𝑓_𝑤1(𝑧)=(1−⟨𝑧,𝑤⟩)^𝛼. (4.7) Note that 𝑓_𝑤∈𝐻^∞_𝛼,0(𝔹) for each 𝑤∈𝔹 and moreover sup_𝑤∈𝔹‖𝑓_𝑤‖_𝐻∞𝛼≤1.
When 𝜑(𝑧)≠0, we have
‖𝑃^𝑔_𝜑‖_{𝐻∞𝛼,0(𝔹)→𝐻∞𝛽(𝔹)}≥‖𝑃^𝑔_𝜑𝑓_{𝑡(𝜑(𝑧)/|𝜑(𝑧)|)}‖_𝐻∞𝛽≍sup_𝑤∈𝔹(1−|𝑤|)^𝛽+1|𝑔𝐶_𝜑𝑓_{𝑡(𝜑(𝑧)/|𝜑(𝑧)|)}(𝑤)|≥(1−|𝑧|)^𝛽+1|𝑔(𝑧)||𝑓_{𝑡(𝜑(𝑧)/|𝜑(𝑧)|)}=(𝜑(𝑧))|(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−𝑡|𝜑(𝑧)|)^𝛼, (4.8) for all 𝑡∈(0,1). Letting 𝑡→1⁻ in (4.8), we have ‖𝑃^𝑔_𝜑‖_{𝐻∞𝛼,0(𝔹)→𝐻∞𝛽(𝔹)}≥𝐶(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼. (4.9)
For the constant function 1∈𝐻^∞_𝛼,0(𝔹) we obtain
‖𝑃^𝑔_𝜑1‖_𝐻𝑞𝛽≍sup_𝑤∈𝔹(1−|𝑤|)^𝛽+1|𝑔𝐶_𝜑1(𝑤)|≥(1−|𝑧|)^𝛽+1|𝑔(𝑧)|. (4.10) Inequality (4.10) shows that the estimate in (4.9) also holds when 𝜑(𝑧)=0.
Hence, from (4.9) we obtain
‖𝑃^𝑔_𝜑‖_{𝐻∞𝛼,0(𝔹)→𝐻∞𝛽(𝔹)}≥𝐶sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼, (4.11) which along with the obvious inequality ‖𝑃^𝑔_𝜑‖_{𝐻∞𝛼(𝔹)→𝐻∞𝛽(𝔹)}≥‖𝑃^𝑔_𝜑‖_{𝐻∞𝛼,0(𝔹)→𝐻∞𝛽(𝔹)} (4.12) completes the proof of the theorem.

For the compactness of 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽(𝔹), we can also prove the following proposition which is similar to Proposition 2.6.

Proposition 4.2. Let 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹 which induce the bounded operator 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽(𝔹). Then 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽(𝔹) is compact if and only if for every bounded sequence {𝑓_𝑗}_𝑗∈ℕ in 𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹)) which converges to 0 uniformly on compact subsets of  𝔹, {𝑃^𝑔_𝜑𝑓_𝑗}_𝑗∈ℕ converges to 0 in 𝐻^∞_𝛽(𝔹).

Theorem 4.3. Let 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹 such that 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽(𝔹) is bounded. Then the essential norm ‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼(𝔹)(𝑜𝑟𝐻∞𝛼,0(𝔹))→𝐻∞𝛽(𝔹)} is comparable to limsup_{|𝜑(𝑧)|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼. (4.13) In particular, 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽(𝔹) is compact if and only if lim_{|𝜑(𝑧)|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼=0. (4.14)

Proof. First we consider the family {𝑓_𝑤}_𝑤∈𝔹 where 𝑓_𝑤(𝑧)=1−|𝑤|(1−⟨𝑧,𝑤⟩)^𝛼+1. (4.15) We can easily check that 𝑓_𝑤∈𝐻^∞_𝛼,0(𝔹), ‖𝑓_𝑤‖_𝐻∞𝛼≤1 for all 𝑤∈𝔹 and 𝑓_𝑤→0 uniformly on compact subsets of 𝔹 as |𝑤|→1⁻. Hence [40, page 296, Theorem 2] implies that 𝑓_𝑤→0 weakly in 𝐻^∞_𝛼,0(𝔹) as |𝑤|→1⁻.
If ‖𝜑‖_∞<1, then as in the proof of [26, Theorem 3] it can be seen that the operator 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(or𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽(𝔹) is compact, so that
‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼(𝔹)(𝑜𝑟𝐻∞𝛼,0(𝔹))→𝐻∞𝛽(𝔹)}=0. (4.16) On the other hand, the limit in (4.13) is vacuously equal to zero, from which the result follows in this case. If ‖𝜑‖_∞=1, then take a sequence {𝜑(𝑧_𝑗)}_𝑗∈ℕ in 𝔹 with |𝜑(𝑧_𝑗)|→1 as 𝑗→∞ and put 𝐹_𝑗(𝑧)=𝑓_{𝜑(𝑧𝑗)}(𝑧) for each 𝑗∈ℕ. Then {𝐹_𝑗}_𝑗∈ℕ is a bounded sequence in 𝐻^∞_𝛼,0(𝔹) and {𝐹_𝑗}_𝑗∈ℕ converges to 0 weakly in 𝐻^∞_𝛼,0(𝔹), as 𝑗→∞. Hence for every compact operator 𝒦∶𝐻^∞_𝛼,0(𝔹)→𝐻^∞_𝛽(𝔹) we have ‖𝒦𝐹_𝑗‖_𝐻∞𝛽→0 as 𝑗→∞. So we have ‖𝑃^𝑔_𝜑−𝒦‖_{𝐻∞𝛼,0(𝔹)→𝐻∞𝛽(𝔹)}≥limsup_𝑗→∞‖𝑃^𝑔_𝜑𝐹_𝑗−𝒦𝐹_𝑗‖_𝐻∞𝛽≥limsup_𝑗→∞‖𝑃^𝑔_𝜑𝐹_𝑗‖_𝐻∞𝛽≍limsup_𝑗→∞sup_𝑤∈𝔹(1−|𝑤|)^𝛽+1|𝑔(𝑤)||𝐹_𝑗(𝜑(𝑤))|≥limsup_𝑗→∞(1−|𝑧_𝑗|)^𝛽+1|𝑔(𝑧_𝑗)||𝐹_𝑗(𝜑(𝑧_𝑗≥1))|2^𝛼+1limsup_𝑗→∞(1−|𝑧_𝑗|)^𝛽+1|𝑔(𝑧_𝑗)|(1−|𝜑(𝑧_𝑗)|)^𝛼, (4.17) for all compact operators 𝒦∶𝐻^∞_𝛼,0(𝔹)→𝐻^∞_𝛽(𝔹). Taking the infimum over the set of all compact operators 𝒦∶𝐻^∞_𝛼,0(𝔹)→𝐻^∞_𝛽(𝔹), we obtain ‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼,0(𝔹)→𝐻∞𝛽(𝔹)}≥𝐶limsup_𝑗→∞(1−|𝑧_𝑗|)^𝛽+1|𝑔(𝑧_𝑗)|(1−|𝜑(𝑧_𝑗)|)^𝛼. (4.18) Combining this with the estimate ‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼(𝔹)→𝐻∞𝛽(𝔹)}≥‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼,0(𝔹)→𝐻∞𝛽(𝔹)}, we have ‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼(𝔹)→𝐻∞𝛽(𝔹)}≥𝐶limsup_{|𝜑(𝑧)|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼. (4.19)
Next we prove an upper estimate. Assume that {𝑟_𝑙}_𝑙∈ℕ⊂(0,1) is a sequence which increasingly converges to 1. For this {𝑟_𝑙}_𝑙∈ℕ, we define the operators defined by
𝑃^𝑔_𝑟𝑙𝜑𝑓(𝑧)=¹₀𝑔(𝑡𝑧)𝑓(𝑟_𝑙𝜑(𝑡𝑧))𝑑𝑡𝑡. (4.20) As in the proof of [26, Theorem 3], Proposition 4.2 shows that 𝑃^𝑔_𝑟𝑙𝜑∶𝐻^∞_𝛼(𝔹)→𝐻^∞_𝛽(𝔹) is compact for each 𝑙∈ℕ.
Put
𝑀₂∶=limsup_{|𝜑(𝑧)|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼, (4.21) and fix 𝜀>0. Then we can choose 𝑅∈(0,1) such that (1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼<𝑀₂+𝜀, (4.22) if 𝑅<|𝜑(𝑧)|<1. Take 𝑓∈𝐻^∞_𝛼(𝔹) with ‖𝑓‖_𝐻∞𝛼≤1 and an integer 𝑙∈ℕ. By Lemma 3.1 and Lemma 3.2, we have ‖𝑃^𝑔_𝜑𝑓−𝑃^𝑔_𝑟𝑙𝜑𝑓‖_𝐻∞𝛽≍sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|ℜ[𝑃^𝑔_𝜑𝑓](𝑧)−ℜ[𝑃^𝑔_𝑟𝑙𝜑𝑓](𝑧)|=sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|=sup_{|𝜑(𝑧)|≤𝑅}(1−|𝑧|)^𝛽+1|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|+sup_{𝑅<|𝜑(𝑧)|<1}(1−|𝑧|)^𝛽+1|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|. (4.23)
By using the mean value theorem and the asymptotic relation
sup_𝑧∈𝔹(1−|𝑧|)^𝛼+1|∇𝑓(𝑧)|≍sup_𝑧∈𝔹(1−|𝑧|)^𝛼+1|ℜ𝑓(𝑧)|, (4.24) we obtain sup_{|𝜑(𝑧)|≤𝑅}|𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|≤sup_{|𝜑(𝑧)|≤𝑅}(1−𝑟_𝑙)|𝜑(𝑧)|sup_{|𝑤|≤𝑅}≤|∇𝑓(𝑤)|(1−𝑟_𝑙)𝑅(1−𝑅)^𝛼+1sup_𝑤∈𝔹(1−|𝑤|)^𝛼+1≍|∇𝑓(𝑤)|(1−𝑟_𝑙)𝑅(1−𝑅)^𝛼+1sup_𝑤∈𝔹(1−|𝑤|)^𝛼+1≍|ℜ𝑓(𝑤)|(1−𝑟_𝑙)𝑅(1−𝑅)^𝛼+1‖𝑓‖_𝐻∞𝛼. (4.25) Since the boundedness of 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)→𝐻^∞_𝛽(𝔹) implies that 𝑃^𝑔_𝜑1∈𝐻^∞_𝛽(𝔹), we see sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)|<∞, (4.26) and so we have sup_{‖𝑓‖𝐻∞𝛼≤1}sup_{|𝜑(𝑧)|≤𝑅}(1−|𝑧|)^𝛽+1|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|≤𝐶(1−𝑟_𝑙)𝑅(1−𝑅)^𝛼+1sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)|. (4.27) On the other hand, the monotonicity of 𝑀_∞(𝑓,𝑟) shows |𝑓(𝑟_𝑙𝜑(𝑧))|≤(1−|𝜑(𝑧)|)^−𝛼‖𝑓_𝑟𝑙‖_𝐻∞𝛼≤(1−|𝜑(𝑧)|)^−𝛼‖𝑓‖_𝐻∞𝛼. (4.28) Thus we have sup_{‖𝑓‖𝐻∞𝛼≤1}sup_{𝑅<|𝜑(𝑧)|<1}(1−|𝑧|)^𝛽+1|𝑔(𝑧)||𝑓(𝜑(𝑧))−𝑓(𝑟_𝑙𝜑(𝑧))|≤2sup_{𝑅<|𝜑(𝑧)|<1}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼≤2(𝑀₂+𝜀). (4.29) From (4.23), (4.27), (4.29), and the compactness of 𝑃^𝑔_𝑟𝑙𝜑, we obtain ‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼(𝔹)→𝐻∞𝛽(𝔹)}≤‖𝑃^𝑔_𝜑−𝑃^𝑔_𝑟𝑙𝜑‖_{𝐻∞𝛼(𝔹)→𝐻∞𝛽(𝔹)}≤𝐶(1−𝑟_𝑙)𝑅(1−𝑅)^𝛼+1sup_𝑧∈𝔹(1−|𝑧|)^𝛽+1|𝑔(𝑧)|+2(𝑀₂+𝜀). (4.30) Letting 𝑙→∞ and 𝜀→0, we have ‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼(𝔹)→𝐻∞𝛽(𝔹)}≤2limsup_{|𝜑(𝑧)|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼. (4.31) This completes the proof.

When 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽,0(𝔹) is bounded, we see that 𝑔∈𝐻^∞_𝛽+1,0(𝔹). By a standard argument as in the proof of [26, Corollary 3], we have limsup_{|𝑧|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼=limsup_{|𝜑(𝑧)|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼, (4.32) and so ‖𝑃^𝑔_𝜑‖_{𝑒,𝐻∞𝛼(𝔹)(or𝐻∞𝛼,0(𝔹))→𝐻∞𝛽,0(𝔹)}≍limsup_{|𝑧|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼. (4.33) Hence we obtain the following characterization for the compactness of the operator 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(𝑜𝑟𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽,0(𝔹).

Corollary 4.4. Let 𝛼,𝛽>0. Suppose that 𝑔∈𝐻(𝔹) with 𝑔(0)=0 and 𝜑 is a holomorphic self-map of 𝔹 such that 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(or𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽,0(𝔹) is bounded. Then 𝑃^𝑔_𝜑∶𝐻^∞_𝛼(𝔹)(or𝐻^∞_𝛼,0(𝔹))→𝐻^∞_𝛽,0(𝔹) is compact if and only if lim_{|𝑧|→1−}(1−|𝑧|)^𝛽+1|𝑔(𝑧)|(1−|𝜑(𝑧)|)^𝛼=0. (4.34)

Acknowledgment

The second author of this paper is supported by Grant-in-Aid for Young Scientists (Start-up; no.20840004).