On a New Integral-Type Operator from the Weighted Bergman Space to the Bloch-Type Space on the Unit Ball

Abstract

We introduce an integral-type operator, denoted by 𝑃^𝑔𝜑, on the space of holomorphic functions on the unit ball 𝔹⊂ℂ^𝑛, which is an extension of the product of composition and integral operators on the unit disk. The operator norm of 𝑃^𝑔𝜑 from the weighted Bergman space 𝐴^𝑝𝛼(𝔹) to the Bloch-type space ℬ_𝜇(𝔹) or the little Bloch-type space ℬ_𝜇,0(𝔹) is calculated. The compactness of the operator is characterized in terms of inducing functions 𝑔 and 𝜑. Upper and lower bounds for the essential norm of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼(𝔹)→ℬ_𝜇(𝔹), when 𝑝>1, are also given.

1. Introduction

Let 𝔹 be the open unit ball in the complex vector space ℂ^𝑛,𝑆=𝜕𝔹 its boundary, 𝔻 the open unit disk in the complex plane ℂ,𝑑𝑉(𝑧) the Lebesgue measure on 𝔹,𝑑𝑉_𝛼(𝑧)=𝑐_𝛼(1−|𝑧|²)^𝛼𝑑𝑉(𝑧), where 𝛼>−1 and where the constant 𝑐_𝛼 is chosen such that 𝑉_𝛼(𝔹)=1,𝑑𝜎 the normalized rotation invariant measure on 𝑆 (that is, 𝜎(𝑆)=1), 𝐻(𝔹) the class of all holomorphic functions on the unit ball and 𝐻^∞=𝐻^∞(𝔹) the space 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,⟨𝑧,𝑤⟩=^𝑛_𝑘=1𝑧_𝑘𝑤_𝑘(1.2)and |𝑧|=√⟨𝑧,𝑧⟩.

For 𝑓∈𝐻(𝔹) with the Taylor expansion 𝑓(𝑧)=∑_|𝛽|≥0𝑎_𝛽𝑧^𝛽, letℜ𝑓(𝑧)=_|𝛽|≥0|𝛽|𝑎_𝛽𝑧^𝛽(1.3)be the radial derivative of 𝑓, where 𝛽=(𝛽₁,𝛽₂,…,𝛽_𝑛) is a multi-index, |𝛽|=𝛽₁+⋯+𝛽_𝑛 and 𝑧^𝛽=𝑧^𝛽1₁⋯𝑧^𝛽𝑛_𝑛. It is well known (see, e.g., [1]) thatℜ𝑓(𝑧)=^𝑛_𝑗=1𝑧_𝑗𝜕𝑓𝜕𝑧_𝑗(𝑧).(1.4)

For 𝑝>0 the Hardy space 𝐻^𝑝=𝐻^𝑝(𝔹) consists of all 𝑓∈𝐻(𝔹) such that‖𝑓‖^𝑝𝑝=sup_0<𝑟<1_𝑆||𝑓(𝑟𝜁)||^𝑝𝑑𝜎(𝜁)<∞.(1.5)It is well known that for every 𝑓∈𝐻^𝑝 the radial limit𝑓^∗(𝜁)∶=lim_𝑟→1𝑓(𝑟𝜁)(1.6)exists almost everywhere on 𝜁∈𝑆.

The weighted Bergman space 𝐴^𝑝𝛼=𝐴^𝑝𝛼(𝔹),𝑝>0,𝛼>−1, consists of all 𝑓∈𝐻(𝔹) such that‖𝑓‖^𝑝𝐴_𝑝𝛼=_𝔹||𝑓(𝑧)||^𝑝𝑑𝑉_𝛼(𝑧)<∞.(1.7)When 𝑝≥1, the weighted Bergman space with the norm ‖⋅‖_𝐴^𝑝𝛼 becomes a Banach space. If 𝑝∈(0,1), it is a Frechet space with the translation invariant metric𝑑(𝑓,𝑔)=‖𝑓−𝑔‖^𝑝𝐴_𝑝𝛼.(1.8)

Since for every 𝑓∈𝐻^𝑝lim_{𝛼→−1+0}_𝔹||𝑓(𝑧)||^𝑝𝑑𝑉_𝛼(𝑧)=_𝑆||𝑓^∗(𝜁)||^𝑝𝑑𝜎(𝜁),(1.9)we will also use the notation 𝐴^𝑝−1 for the Hardy space 𝐻^𝑝.

A positive continuous function 𝜙 on [0,1) is called normal (see [2]) if there is 𝛿∈[0,1) and 𝑎 and 𝑏,0<𝑎<𝑏 such that𝜙(𝑟)(1−𝑟)^𝑎isdecreasingon[𝛿,1),lim_𝑟→1𝜙(𝑟)(1−𝑟)^𝑎𝜙(𝑟)=0;(1−𝑟)^𝑏isincreasingon[𝛿,1),lim_𝑟→1𝜙(𝑟)(1−𝑟)^𝑏=∞.(1.10)From now on if we say that a function 𝜇∶𝔹→[0,∞) is normal, we will also assume that it is radial, that is, 𝜇(𝑧)=𝜇(|𝑧|),𝑧∈𝔹.

The weighted space 𝐻^∞𝜇=𝐻^∞𝜇(𝔹) consists of all 𝑓∈𝐻(𝔹) such that‖𝑓‖_𝐻^∞𝜇=sup_𝑧∈𝔹𝜇(𝑧)||𝑓(𝑧)||<∞,(1.11)where 𝜇 is normal. For 𝜇(𝑧)=(1−|𝑧|²)^𝛽, 𝛽>0, we obtain the weighted space 𝐻^∞𝛽=𝐻^∞𝛽(𝔹) (see, e.g., [3–5]).

The little weighted space 𝐻^∞𝜇,0=𝐻^∞𝜇,0(𝔹) is a subspace of 𝐻^∞𝜇 consisting of all 𝑓∈𝐻(𝔹) such thatlim_|𝑧|→1𝜇(𝑧)||𝑓(𝑧)||=0.(1.12)

The class of all 𝑓∈𝐻(𝔹) such that𝐵_𝜇(𝑓)=sup_𝑧∈𝔹𝜇(𝑧)||ℜ𝑓(𝑧)||<∞,(1.13)where 𝜇 is normal, is called the Bloch-type space, and is denoted by ℬ_𝜇=ℬ_𝜇(𝔹). With the norm‖𝑓‖_ℬ𝜇=||𝑓(0)||+𝐵_𝜇(𝑓),(1.14)the Bloch-type space becomes a Banach space.

The little Bloch-type space ℬ_𝜇,0 is a subspace of ℬ_𝜇 consisting of those 𝑓∈ℬ_𝜇 such thatlim_|𝑧|→1𝜇(𝑧)||ℜ𝑓(𝑧)||=0.(1.15)The 𝛼-Bloch space ℬ^𝛼 is obtained for 𝜇(𝑧)=(1−|𝑧|²)^𝛼,𝛼∈(0,∞) (see, e.g., [6–11]). For 𝛼=1 the space ℬ¹=ℬ becomes the classical Bloch space.

Let 𝜑 be a holomorphic self-map of 𝔹. For any 𝑓∈𝐻(𝔹), the composition operator is defined by𝐶_𝜑𝑓(𝑧)=𝑓𝜑(𝑧),𝑧∈𝔹.(1.16)It is of interest to provide function theoretic characterizations when 𝜑 induces bounded or compact composition operators on spaces of holomorphic functions. For some classical results in the topic (see, e.g., [12]). For some recent results see, for example, [3–5, 7, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23] and the references therein.

Let 𝑔∈𝐻(𝔻) and 𝜑 be a holomorphic self-map of 𝔻. For 𝑓∈𝐻(𝔻), products of integral-type and composition operator are defined as follows:𝐶_𝜑𝐽_𝑔𝑓(𝑧)=^0𝜑(𝑧)𝑓(𝜁)𝑔^(𝜁)𝑑𝜁,𝐽_𝑔𝐶_𝜑𝑓(𝑧)=^𝑧0𝑓𝜑(𝜁)𝑔^(𝜁)𝑑𝜁.(1.17)

When 𝜑(𝑧)=𝑧, operators in (1.17) are reduced to the integral operator introduced in [24]. For some other results on the operator; see, for example, [25, 26], and related references therein. Some results on related integral-type operators on spaces of holomorphic functions in ℂ^𝑛 can be found, for example, in [27–41] (see also the references therein).

In [42], among other results, we proved the following theorem regarding the boundedness of the operator 𝐽_𝑔𝐶_𝜑∶𝐴^𝑝𝛼(𝔻)→ℬ_𝜇(𝔻).

Theorem 1.1. Assume that 𝑝>0,𝛼>−1,𝑔∈𝐻(𝔻),𝜇 is normal, and 𝜑 is a holomorphic self-map of 𝔻. Then 𝐽_𝑔𝐶_𝜑∶𝐴^𝑝𝛼(𝔻)→ℬ_𝜇(𝔻) is bounded if and only ifsup_𝑧∈𝔻𝜇(𝑧)||𝑔^(𝑧)||1−||𝜑(𝑧)||²^{(𝛼+2)/𝑝}<∞.(1.18)

One of the interesting questions is to extend operators in (1.17) in the unit ball settings and to study their function theoretic properties on spaces of holomorphic functions on the unit ball in terms of inducing functions.

Assume that 𝑔∈𝐻(𝔹),𝑔(0)=0, and 𝜑 is a holomorphic self-map of 𝔹. We introduce the following important integral-type operator on the space of holomorphic functions on 𝔹: 𝑃^𝑔𝜑(𝑓)(𝑧)=¹⁰𝑓𝜑(𝑡𝑧)𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹),𝑧∈𝔹.(1.19)

First note that when 𝑛=1, the operator is reduced to an operator of the form as the second operator in (1.17). Indeed, since 𝑔∈𝐻(𝔻) and 𝑔(0)=0, it follows that 𝑔(𝑧)=𝑧𝑔₀(𝑧),𝑧∈𝔻 for some 𝑔₀∈𝐻(𝔻). By using this fact and the change of variables 𝜁=𝑡𝑧, we obtain𝑃^𝑔𝜑𝑓(𝑧)=¹⁰𝑓𝜑(𝑡𝑧)𝑡𝑧𝑔₀(𝑡𝑧)𝑑𝑡𝑡=^𝑧0𝑓𝜑(𝜁)𝑔₀(𝜁)𝑑𝜁.(1.20)Hence operator (1.19) is a natural extension of the second operator in (1.17).

Now we formulate the following big research project related to the operator 𝑃^𝑔𝜑.

Research Project 1. Let 𝑋 and 𝑌 be two Banach spaces of holomorphic functions on the unit ball in ℂ^𝑛 (e.g., the weighted Bergman space 𝐴^𝑝𝛼, the Bloch-type space ℬ_𝜇, the Hardy space 𝐻^𝑝 space, the weighted space 𝐻^∞𝜇, the Besov space 𝐵^𝑝, BMOA etc.) Characterize the boundedness, compactness, essential norms, and other operator theoretic properties of the operator 𝑃^𝑔𝜑∶𝑋→𝑌 in terms of function theoretic properties of inducing functions 𝜑 and 𝑔.

Another interesting question is to find the exact value of the norm of operators on spaces of holomorphic functions. Majority of papers in the area only find asymptotics of the operator norm of certain linear operators on some spaces of holomorphic functions. There are a few papers which calculate the operator norm of these operators. Recently in [4] we calculated operator norm of the weighted composition operator 𝑢𝐶_𝜑 mapping the Bloch space ℬ to the weighted space 𝐻^∞𝜇, which motivates us to find the norms of weighted composition and other closely related operators between various spaces of holomorphic functions.

Research Project 2. Let 𝑋 and 𝑌 be two Banach spaces of holomorphic functions as in Research project 1. Calculate the operator norm of 𝑃^𝑔𝜑∶𝑋→𝑌 in terms of inducing functions 𝜑 and 𝑔.

In this paper, among other results, we will calculate the operator norm of 𝑃^𝑔𝜑∶𝐴^𝑝𝛼(𝔹)→ℬ_𝜇(𝔹). We will also characterize the boundedness, compactness, and the essential norm of the operator. These results partially solve problems posed in the above research projects.

Throughout the paper, 𝐶 denotes a positive constant not necessarily the same at each occurrence. The notation 𝐴≍𝐵 means that there is a positive constant 𝐶 such that 𝐴/𝐶≤𝐵≤𝐶𝐴.

2. Auxiliary Results

In this section, we give several auxiliary results, which are used in the proofs of the main results.

Lemma 2.1 (see [43, Corollary 3.5]). Suppose that 𝑝∈(0,∞) and 𝛼≥−1. Then for all 𝑓∈𝐴^𝑝𝛼(𝔹) and 𝑧∈𝔹, the following inequality holds:||𝑓(𝑧)||≤‖𝑓‖^𝐴𝑝𝛼1−|𝑧|²^{(𝑛+1+𝛼)/𝑝}.(2.1)

The following criterion for the compactness follows by standard arguments (see, e.g., [12, 20, 34–36]). Hence, we omit its proof.

Lemma 2.2. Suppose that 0<𝑝<∞,𝛼≥−1,𝑔∈𝐻(𝔹),𝜇 is normal, and 𝜑 is a holomorphic self-map of 𝔹. Then the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is compact if and only if 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded and for every bounded sequence (𝑓_𝑘)_𝑘∈ℕ in 𝐴^𝑝𝛼 converging to zero uniformly on compacts of 𝔹, one has ‖𝑃^𝑔𝜑𝑓_𝑘‖_ℬ𝜇→0 as 𝑘→∞.

The following result can be found in [44]. For closely related results, see also [11, 45–52] and the references therein.

Lemma 2.3. Suppose that 0<𝑝<∞,𝛼>−1, then‖𝑓‖^𝑝𝐴_𝑝𝛼≍||𝑓(0)||^𝑝+_𝔹||∇𝑓(𝑧)||^𝑝1−|𝑧|²^𝑝+𝛼𝑑𝑉(𝑧),(2.2)for every 𝑓∈𝐴^𝑝𝛼 (here ∇𝑓=((𝜕𝑓/𝜕_𝑧1),…,(𝜕𝑓/𝜕_𝑧𝑛))).

The following lemma can be proved similar to [53, Lemma 1].

Lemma 2.4. Suppose that 𝜇 is normal. A closed set 𝐾 in ℬ_𝜇,0 is compact if and only if it is bounded andlim_|𝑧|→1sup_𝑓∈𝐾𝜇(𝑧)||ℜ𝑓(𝑧)||=0.(2.3)

The following lemma is related to [32, Lemma 1] and [34, Lemma 2].

Lemma 2.5. Assume that 𝑓,𝑔∈𝐻(𝔹) and 𝑔(0)=0. Thenℜ𝑃^𝑔𝜑(𝑓)(𝑧)=𝑓(𝜑(𝑧))𝑔(𝑧).(2.4)

Proof. Since the function 𝑓(𝜑(𝑧))𝑔(𝑧) is holomorphic and 𝑔(0)=0, it has the Taylor expansion in the following form ∑_𝛼≠0𝑎_𝛼𝑧^𝛼. Thenℜ𝑃^𝑔𝜑(𝑓)(𝑧)=ℜ¹⁰_𝛼_≠0𝑎_𝛼(𝑡𝑧)^𝛼𝑑𝑡𝑡=ℜ_𝛼≠0𝑎_𝛼|𝛼|𝑧^𝛼=_𝛼≠0𝑎_𝛼𝑧^𝛼,(2.5)as claimed.

3. The Norm of the Operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇

In this section, we calculate the norm ‖𝑃^𝑔𝜑‖_𝐴^𝑝𝛼_→ℬ𝜇.

Theorem 3.1. Assume that 𝑝>0,𝛼≥−1,𝑔∈𝐻(𝔹),𝜇 is normal, 𝜑 is a holomorphic self-map of 𝔹, and 𝑃^𝑔𝜑∶𝐴^𝑝𝛼(𝔹)→ℬ_𝜇(𝔹) is bounded. Then‖𝑃^𝑔𝜑‖_𝐴^𝑝𝛼_→ℬ𝜇=‖𝑃^𝑔𝜑‖_𝐴^𝑝𝛼_→ℬ𝜇,0=sup_𝑧∈𝔹𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}=∶𝑀.(3.1)

Proof. If 𝑓∈𝐴^𝑝𝛼, then by Lemmas 2.5 and 2.1 we obtain‖𝑃^𝑔𝜑𝑓‖_ℬ𝜇=sup_𝑧∈𝔹𝜇(𝑧)||𝑔(𝑧)𝑓𝜑(𝑧)||≤‖𝑓‖_𝐴^𝑝𝛼sup_𝑧∈𝔹𝜇(𝑧)||𝑔(𝑧)||(1−||𝜑(𝑧)||²)^{(𝑛+1+𝛼)/𝑝},(3.2)from which it follows that‖𝑃^𝑔𝜑‖_𝐴^𝑝𝛼_→ℬ𝜇≤𝑀.(3.3)
Now we prove the reverse inequality. For 𝑤∈𝔹 fixed, set𝑓_𝑤(𝑧)=1−|𝑤|²^{(𝑛+1+𝛼)/𝑝}1−⟨𝑧,𝑤⟩^{2(𝑛+1+𝛼)/𝑝},𝑧∈𝔹.(3.4)We have that ‖𝑓_𝑤‖_𝐴^𝑝𝛼=1, for each 𝑤∈𝔹. For 𝛼>−1 this fact is well known. The proof for the case 𝛼=−1 could be less known, and we give a proof of it for the lack of a specific reference and for the benefit of the reader. Let 𝑧=𝑟𝜁, 𝜁∈𝑆, then we have ‖𝑓_𝑤‖^𝑝𝑝=sup_0<𝑟<1_𝑆(1−|𝑤|²^𝑛||1−⟨𝑧,𝑤⟩||^2𝑛=1−|𝑤|₂𝑑𝜎(𝜁)=1−|𝑤|₂^𝑛sup_0<𝑟<1_𝑆||1−⟨𝑧,𝑤⟩^−𝑛||²𝑑𝜎(𝜁)^𝑛sup_0<𝑟<1_𝑆||||^∞_𝑘=0Γ(𝑛+𝑘)Γ(𝑘+1)Γ(𝑛)𝑟^𝑘⟨𝜁,𝑤⟩^𝑘||||²=1−|𝑤|₂𝑑𝜎(𝜁)=1−|𝑤|₂^𝑛sup_0<𝑟<1_{𝑆∞𝑘=0}Γ(Γ𝑘(+𝑛+1)𝑘Γ)(𝑛)²𝑟^2𝑘||⟨𝜁,𝑤⟩||^2𝑘𝑑𝜎(𝜁)^𝑛sup_∞0<𝑟<1_𝑘=0Γ(𝑛+𝑘)Γ(𝑘+1)Γ(𝑛)𝑟^2𝑘|𝑤|^2𝑘=1−|𝑤|²^𝑛sup_0<𝑟<111−𝑟²|𝑤|²^𝑛=1,(3.5)where we have used the following formula (see, e.g., [1])_𝑆||⟨𝜁,𝑤⟩||^2𝑘𝑑𝜎(𝜁)=Γ(𝑘+1)Γ(𝑛)Γ(𝑛+𝑘)|𝑤|^2𝑘.(3.6)
From this and the boundedness of 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇, we have‖‖𝑃^𝑔𝜑‖‖_𝐴^𝑝𝛼_→ℬ𝜇=‖‖𝑓_𝜑(𝑤)‖‖_𝐴^𝑝𝛼‖‖𝑃^𝑔𝜑‖‖_𝐴^𝑝𝛼_→ℬ𝜇≥‖‖𝑃^𝑔𝜑𝑓_𝜑(𝑤)‖‖_ℬ𝜇≥=𝜇su(p𝑤_𝑧∈𝔹)𝜇||𝑔(𝑧(𝑤)||𝑔)||(||𝑧𝑓)||||𝑓_{𝜑(𝑤)𝜑(𝑤)}𝜑(𝑧)||=𝜇(𝑤)||𝑔(𝑤)||𝜑(𝑤)||1−||𝜑(𝑤)||²^{(𝑛+1+𝛼)/𝑝}.(3.7)Taking the supremum in (3.7) over 𝑤∈𝔹, we obtain‖‖𝑃^𝑔𝜑‖‖_𝐴^𝑝𝛼_→ℬ𝜇≥𝑀.(3.8)From (3.3) and (3.8), it follows that ‖𝑃^𝑔𝜑‖_𝐴^𝑝𝛼_→ℬ𝜇=𝑀.
Since‖‖𝑃^𝑔𝜑‖‖_𝐴^𝑝𝛼_→ℬ𝜇,0≤‖‖𝑃^𝑔𝜑‖‖_𝐴^𝑝𝛼_→ℬ𝜇(3.9)and the proof of (3.8) does not depend on the space ℬ_𝜇 (we may replace it by ℬ_𝜇,0) the second equality in (3.1) also holds.

Corollary 3.2. Assume that 𝑝>0,𝛼≥−1,𝑔∈𝐻(𝔹),𝜇 is normal, and 𝜑 is a holomorphic self-map of 𝔹. Then 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded if and only ifsup_𝑧∈𝔹𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}<∞.(3.10)

Proof. If 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded, then (3.10) follows from Theorem 3.1. If (3.10) holds, then the boundedness of 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 follows from (3.3).

4. The Boundedness of the Operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0

Here we characterize the boundedness of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0.

Theorem 4.1. Assume that 𝑝>0,𝛼≥−1,𝑔∈𝐻(𝔹),𝜇 is normal, and 𝜑 is a holomorphic self-map of 𝔹. Then 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0 is bounded if and only if 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded and 𝑔∈𝐻^∞𝜇,0.

Proof. Assume that 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded and 𝑔∈𝐻^∞𝜇,0. Then, for each polynomial 𝑝, we have𝜇(𝑧)||ℜ𝑃^𝑔𝜑𝑝(𝑧)||=𝜇(𝑧)||𝑔(𝑧)𝑝𝜑(𝑧)||≤𝜇(𝑧)||𝑔(𝑧)||‖𝑝‖_∞⟶0,as|𝑧|⟶1(4.1)from which it follows that 𝑃^𝑔𝜑(𝑝)∈ℬ_𝜇,0.
Since the set of all polynomials is dense in 𝐴^𝑝𝛼, we have that for every 𝑓∈𝐴^𝑝𝛼 there is a sequence of polynomials (𝑝_𝑘)_𝑘∈ℕ such thatlim_𝑘→∞‖‖𝑓−𝑝_𝑘‖‖_𝐴^𝑝𝛼=0.(4.2)From this and since the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded, it follows that‖‖𝑃^𝑔𝜑𝑓−𝑃^𝑔𝜑𝑝_𝑘‖‖_ℬ𝜇≤‖‖𝑃^𝑔𝜑‖‖_𝐴^𝑝𝛼_→ℬ𝜇‖‖𝑓−𝑝_𝑘‖‖_𝐴^𝑝𝛼⟶0,(4.3)as 𝑘→∞. Hence 𝑃^𝑔𝜑(𝐴^𝑝𝛼)⊂ℬ_𝜇,0. Since ℬ_𝜇,0 is a closed subset of ℬ_𝜇, the boundedness of 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0 follows.
Now assume that 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0 is bounded. Then clearly 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded. Taking the test function 𝑓(𝑧)=1∈𝐴^𝑝𝛼, we obtain 𝑔∈𝐻^∞𝜇,0.

5. Compactness of the Operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇

This section is devoted to studying of the compactness of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇. We prove the following result.

Theorem 5.1. Assume that 𝑝>0,𝛼≥−1,𝑔∈𝐻(𝔹),𝜇 is normal, 𝜑 is a holomorphic self-map of 𝔹, and the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded. Then the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is compact if and only iflim_{|𝜑(𝑧)|→1}𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}=0.(5.1)

Proof. First assume that the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is compact. If ‖𝜑‖_∞<1, then condition (5.1) is vacuously satisfied. Hence, assume that ‖𝜑‖_∞=1 and assume to the contrary that (5.1) does not hold. Then there is a sequence (𝑧_𝑘)_𝑘∈ℕ satisfying the condition |𝜑(𝑧_𝑘)|→1 as 𝑘→∞ and 𝛿>0 such that𝜇(𝑧_𝑘)||𝑔(𝑧_𝑘)||1−||𝜑(𝑧_𝑘)||²^{(𝑛+1+𝛼)/𝑝}≥𝛿,𝑘∈ℕ.(5.2)For 𝑤∈𝔹 fixed, set𝐹_𝑘(𝑧)=𝑓_{𝜑(𝑧𝑘)}(𝑧),𝑘∈ℕ,(5.3)where 𝑓_𝑤 is defined in (3.4). Recall that ‖𝑓_𝑤‖_𝐴^𝑝𝛼=1, for each 𝑤∈𝔹. Then ‖𝐹_𝑘‖_𝐴^𝑝𝛼=1,𝑘∈ℕ and it is easy to see that 𝐹_𝑘→0 uniformly on compacts of 𝔹 as 𝑘→∞. Hence, by Lemma 2.2, it follows thatlim_𝑘→∞‖‖𝑃^𝑔𝜑𝐹_𝑘‖‖_ℬ𝜇=0.(5.4)
On the other hand, by Lemma 2.5 and (5.2), we obtain‖‖𝑃^𝑔𝜑𝐹_𝑘‖‖_ℬ𝜇=sup_𝑧∈𝔹𝜇(𝑧)||𝑔(𝑧)||||𝐹_𝑘𝜑(𝑧)||≥𝜇(𝑧^𝑘)||𝑔𝑧_𝑘||1−||𝜑𝑧_𝑘||²^{(𝑛+1+𝛼)/𝑝}≥𝛿>0,(5.5)for every 𝑘∈ℕ, which contradicts with (5.4).
Now assume that (5.1) holds. Then for every 𝜀>0 there is an 𝑟∈(0,1) such that when 𝑟<|𝜑(𝑧)|<1,𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}<𝜀.(5.6)
On the other hand, since the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded, for 𝑓≡1∈𝐴^𝑝𝛼, we obtain ‖𝑔‖_𝐻^∞𝜇<∞.
Assume that (ℎ_𝑘)_𝑘∈ℕ is a bounded sequence in 𝐴^𝑝𝛼 converging to zero uniformly on compacts of 𝔹 as 𝑘→∞. Let sup_𝑘∈ℕ‖ℎ_𝑘‖_𝐴^𝑝𝛼=𝑀₁. Then by Lemma 2.1 and (5.6), for 𝑟<|𝜑(𝑧)|<1, we obtain𝜇(𝑧)||𝑔(𝑧)||||ℎ_𝑘𝜑(𝑧)||≤sup_𝑘∈ℕ‖ℎ_𝑘‖_𝐴^𝑝𝛼𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}<𝑀₁𝜀.(5.7)If |𝜑(𝑧)|≤𝑟, we have𝜇(𝑧)||𝑔(𝑧)||||ℎ_𝑘𝜑(𝑧)||≤‖𝑔‖_𝐻^∞𝜇sup_{|𝑤|≤𝑟}||ℎ_𝑘(𝑤)||⟶0,as𝑘⟶∞.(5.8)From (5.7) and (5.8), it follows that ‖𝑃^𝑔𝜑ℎ_𝑘‖_ℬ𝜇→0 as 𝑘→∞, from which the compactness of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 follows.

6. Compactness of the Operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0

This section characterizes the compactness of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0.

Theorem 6.1. Assume 𝑝>0,𝛼≥−1,𝑔∈𝐻(𝔹),𝜇 is normal, 𝜑 is a holomorphic self-map of 𝔹, and the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0 is bounded. Then the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0 is compact if and only iflim_|𝑧|→1𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}=0.(6.1)

Proof. Assume 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0 is compact. Then clearly 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0 is bounded and as in Theorem 4.1 we have that 𝑔∈𝐻^∞𝜇,0.
Hence if ‖𝜑‖_∞<1, thenlim_|𝑧|→1𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}≤lim_|𝑧|→1𝜇(𝑧)||𝑔(𝑧)||1−‖𝜑‖^2∞^{(𝑛+1+𝛼)/𝑝}=0,(6.2)from which the result follows in this case.
Now assume ‖𝜑‖_∞=1. By using the test functions 𝐹_𝑘(𝑧)=𝑓_{𝜑(𝑧𝑘)}(𝑧), 𝑘∈ℕ, defined in (5.3) we obtain that condition (5.1) holds, which implies that for every 𝜀>0, there is an 𝑟∈(0,1) such that for 𝑟<|𝜑(𝑧)|<1, condition (5.6) holds.
Since 𝑔∈𝐻^∞𝜇,0, there is 𝜎∈(0,1) such that for 𝜎<|𝑧|<1𝜇(𝑧)||𝑔(𝑧)||<𝜀1−𝑟²^{(𝑛+1+𝛼)/𝑝}.(6.3)
Hence, if |𝜑(𝑧)|≤𝑟 and 𝜎<|𝑧|<1, we have𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}≤𝜇(𝑧)||𝑔(𝑧)||1−𝑟²^{(𝑛+1+𝛼)/𝑝}<𝜀.(6.4)From (5.6) and (6.4), condition (6.1) follows.
Now assume that condition (6.1) holds. Then the quantity 𝑀 in Theorem 3.1 is finite. Using this fact and the following inequality𝜇(𝑧)||ℜ𝑃^𝑔𝜑𝑓(𝑧)||≤𝜇(𝑧)||𝑔(𝑧)𝑓𝜑(𝑧)||≤‖𝑓‖_𝐴^𝑝𝛼𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝},(6.5)it follows that the set 𝑃^𝑔𝜑({𝑓∶‖𝑓‖_𝐴^𝑝𝛼≤1}) is bounded in ℬ_𝜇, moreover, in view of (6.1), it is bounded in ℬ_𝜇,0. Taking the supremum in the last inequality over the unit ball in 𝐴^𝑝𝛼, then letting |𝑧|→1, using condition (6.1) and employing Lemma 2.4, we obtain the compactness of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇,0, as desired.

7. Essential Norm of 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇

Let 𝑋 and 𝑌 be Banach spaces, and let 𝐿∶𝑋→𝑌 be a bounded linear operator. The essential norm of the operator 𝐿∶𝑋→𝑌, denoted by ‖𝐿‖_{𝑒,𝑋→𝑌}, is defined as follows:‖𝐿‖_{𝑒,𝑋→𝑌}=inf‖𝐿+𝐾‖_𝑋→𝑌∶𝐾iscompactfrom𝑋𝑡𝑜𝑌,(7.1)where ‖⋅‖_𝑋→𝑌 denote the operator norm.

From this definition and since the set of all compact operators is a closed subset of the set of bounded operators, it follows that operator 𝐿 is compact if and only if ‖𝐿‖_{𝑒,𝑋→𝑌}=0.

In this section, we study the essential norm of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 for the case 𝑝>1.

Theorem 7.1. Assume that 𝑝∈(1,∞),𝛼≥−1,𝑔∈𝐻(𝔹),𝑔(0)=0,𝜑 is a holomorphic self-map of 𝔹, and 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇 is bounded. Then the following inequalities hold:limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}≤‖‖𝑃^𝑔𝜑‖‖_𝑒,𝐴^𝑝𝛼_→ℬ𝜇≤2limsup_{|𝜑(𝑧)|→1}𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}.(7.2)

Proof. Assume that (𝜑(𝑧_𝑘))_𝑘∈ℕ is a sequence in 𝔹 such that |𝜑(𝑧_𝑘)|→1 as 𝑘→∞. Note that the sequence (𝑓_{𝜑(𝑧𝑘)})_𝑘∈ℕ (where 𝑓_𝑤 is defined in (3.4)) is such that ‖𝑓_{𝜑(𝑧𝑘)}‖_𝐴^𝑝𝛼=1 for each 𝑘∈ℕ and it converges to zero uniformly on compacts of 𝔹. From this and by [11, Theorems 2.12 and 4.50], it follows that 𝑓_{𝜑(𝑧𝑘)}→0 weakly in 𝐴^𝑝𝛼, as 𝑘→∞ (here we use the condition 𝑝>1). Hence, for every compact operator 𝐾∶𝐴^𝑝𝛼→ℬ_𝜇, we have that ‖𝐾𝑓_{𝜑(𝑧𝑘)}‖_ℬ𝜇→0 as 𝑘→∞. Thus, for every such sequence and for every compact operator 𝐾∶𝐴^𝑝𝛼→ℬ_𝜇, we have that‖‖𝑃^𝑔𝜑+𝐾‖‖_𝐴^𝑝𝛼_→ℬ𝜇≥limsup_𝑘→∞‖‖𝑃^𝑔𝜑𝑓_{𝜑(𝑧𝑘)}‖‖_ℬ𝜇−‖‖𝐾𝑓_{𝜑(𝑧𝑘)}‖‖_ℬ𝜇‖‖𝑓_{𝜑(𝑧𝑘)}‖‖_𝐴^𝑝𝛼=limsup_𝑘→∞‖‖𝑃^𝑔𝜑𝑓_{𝜑(𝑧𝑘)}‖‖_ℬ𝜇=≥lliimmssuup_𝑘→∞p_𝑛→∞𝜇(𝑧_𝑘)||𝑔𝑧_𝑘𝑓_{𝜑(𝑧𝑘)}𝜑𝑧_𝑘||𝜇𝑧_𝑘||𝑔𝑧_𝑘||1−||𝜑𝑧_𝑘||²^{(𝑛+1+𝛼)/𝑝}.(7.3)
Taking the infimum in (7.3) over the set of all compact operators 𝐾∶𝐴^𝑝𝛼→ℬ_𝜇, we obtain‖‖𝑃^𝑔𝜑‖‖_𝑒,𝐴^𝑝𝛼_→ℬ𝜇≥limsup_𝑛→∞𝜇𝑧_𝑘||𝑔𝑧_𝑘||1−||𝜑𝑧_𝑘||²^{(𝑛+1+𝛼)/𝑝},(7.4)from which the first inequality follows.
In the sequel we prove the second inequality. Assume that (𝑟_𝑙)_𝑙∈ℕ is a sequence which increasingly converges to 1. Consider the operators defined by𝑃^𝑔𝑟_𝑙𝜑𝑓(𝑧)=¹⁰𝑔(𝑡𝑧)𝑓𝑟_𝑙𝜑(𝑡𝑧)𝑑𝑡𝑡,𝑙∈ℕ.(7.5)We prove that these operators are compact. Indeed, since |𝑟_𝑙𝜑(𝑧)|≤𝑟_𝑙<1, it follows that condition (5.1) in Theorem 5.1 is vacuously satisfied, from which the claim follows.
Recall that 𝑔∈𝐻^∞𝜇. Let 𝜌∈(0,1) be fixed for a moment. Employing Lemma 2.1, and using the fact‖‖𝑓−𝑓_𝑟𝑙‖‖_𝐴^𝑝𝛼≤2‖𝑓‖_𝐴^𝑝𝛼,𝑙∈ℕ,(7.6)which follows by using the triangle inequality for the norm, the monotonicity of the integral means𝑀^𝑝𝑝(𝑓,𝑟)=_𝑆||𝑓(𝑟𝜁)||^𝑝𝑑𝜎(𝜁)(7.7)and the polar coordinates, we have ‖‖𝑃^𝑔𝜑−𝑃^𝑔𝑟_𝑙𝜑‖‖_𝐴^𝑝𝛼_→ℬ𝜇=≤ssuupp_{‖𝑓‖𝐴𝑝𝛼≤1‖𝑓‖𝐴𝑝𝛼≤1}sup_𝑧∈𝔹𝜇(𝑧)||𝑔(𝑧)||||𝑓𝜑(𝑧)−𝑓𝑟_𝑙𝜑(𝑧)||+sup_{‖𝑓‖𝐴𝑝𝛼≤1}sup_{|𝜑(𝑧)|≤𝜌}𝜇(𝑧)||𝑔(𝑧)||||𝑓𝜑(𝑧)−𝑓𝑟_𝑙𝜑(𝑧)||≤‖𝑔‖_𝐻∞𝜇sup_{|𝜑(𝑧)|>𝜌}𝜇(𝑧)||𝑔(𝑧)||||𝑓𝜑(𝑧)−𝑓𝑟_𝑙𝜑(𝑧)||+2sups_{|𝜑(𝑧)|>𝜌}up_{‖𝑓‖𝐴𝑝𝛼≤1}sup_{|𝜑(𝑧)|≤𝜌}||𝑓𝜑(𝑧)−𝑓𝑟_𝑙𝜑(𝑧)||𝜇(𝑧)||𝑔(𝑧)||1−||𝜑(𝑧)||²^{(𝑛+1+𝛼)/𝑝}.(7.8)
Let𝐼_𝑙∶=sup_{‖𝑓‖𝐴𝑝𝛼≤1}sup_{|𝜑(𝑧)|≤𝜌}||𝑓(𝜑(𝑧))−𝑓𝑟_𝑙𝜑(𝑧)||.(7.9)If 𝛼>−1, then by using the mean value theorem, the subharmonicity of the partial derivatives of 𝑓 and Lemma 2.3, we have 𝐼_𝑙≤≤𝐶s_𝜌up_{‖𝑓‖𝐴𝑝𝛼≤1}sup_{|𝜑(𝑧)|≤𝜌}1−𝑟_𝑙||𝜑(𝑧)||sup_{|𝑤|≤𝜌}||∇𝑓(𝑤)||(7.10)1−𝑟_𝑙sup_{‖𝑓‖𝐴𝑝𝛼≤1}_{|𝑤|≤(1+𝜌)/2}||∇𝑓(𝑤)||^𝑝1−|𝑤|²^𝑝+𝛼𝑑𝑉(𝑤)^1/𝑝≤𝐶_𝜌1−𝑟_𝑙sup_{‖𝑓‖𝐴𝑝𝛼≤1}_𝔹||𝑓(𝑤)||^𝑝1−|𝑤|²^𝛼𝑑𝑉(𝑤)^1/𝑝≤𝐶_𝜌(1)(1−𝑟_𝑙)⟶0as𝑙⟶∞.(7.11)
If 𝛼=−1, then applying in (7.10) the known fact that for each compact 𝐾⊂𝔹,sup_𝑤∈𝐾||∇𝑓(𝑤)||≤𝐶‖𝑓‖_𝑝,(7.12) for some 𝐶 independent of 𝑓 (see [11]), we obtain that (7.11) also holds in this case.
Letting 𝑙→∞ in (7.8), using (7.11), and then letting 𝜌→1, the second inequality in (7.2) follows, finishing the proof of the theorem.

Motivated by Theorem 7.1, we leave the following open problem.

Open Problem 1. Find the exact value of the essential norm of the operator 𝑃^𝑔𝜑∶𝐴^𝑝𝛼→ℬ_𝜇.