Weighted Iterated Radial Composition Operators between Some Spaces of Holomorphic Functions on the Unit Ball

Abstract

The boundedness and compactness of weighted iterated radial composition operators from the mixed-norm space to the weighted-type space and the little weighted-type space on the unit ball are characterized here. We also calculate the Hilbert-Schmidt norm of the operator on the weighted Bergman-Hilbert space as well as on the Hardy 𝐻² space.

1. Introduction

Let 𝑧=(𝑧₁,…,𝑧_𝑛) and 𝑤=(𝑤₁,…,𝑤_𝑛) be points in ℂ^𝑛, ⟨𝑧,𝑤⟩=∑^𝑛_𝑘=1𝑧_𝑘𝑤_𝑘, and |𝑧|=⟨𝑧,𝑧⟩. Let 𝔹={𝑧∈ℂ^𝑛∶|𝑧|<1} be the open unit ball in ℂ^𝑛, 𝜕𝔹 its boundary, and 𝐻(𝔹) the class of all holomorphic functions on 𝔹.

For an 𝑓∈𝐻(𝔹) with the Taylor expansion 𝑓(𝑧)=∑_|𝛽|≥0𝑎_𝛽𝑧^𝛽, let ℜ𝑓(𝑧)=_𝛽≥0𝛽𝑎_𝛽𝑧^𝛽 (1.1) 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 [1]. It is easy to see that ℜ𝑓(𝑧)=∇𝑓(𝑧),𝑧, (1.2) where ∇𝑓 is the complex gradient of function 𝑓.

The iterated radial derivative operator ℜ^𝑚𝑓 is defined inductively by ℜ^𝑚𝑓=ℜℜ^𝑚−1𝑓,𝑚∈ℕ⧵{1}. (1.3)

A positive continuous function 𝜈 on the interval [0,1) is called normal [2] if there are 𝛿∈[0,1) and 𝜏 and 𝑡, 0<𝜏<𝑡 such that 𝜈(𝑟)(1−𝑟)^𝜏isdecreasingon[𝛿,1),lim_𝑟→1𝜈(𝑟)(1−𝑟)^𝜏𝜈(𝑟)=0,(1−𝑟)^𝑡isincreasingon[𝛿,1),lim_𝑟→1𝜈(𝑟)(1−𝑟)^𝑡=∞. (1.4) If we say that a function 𝜈∶𝔹→[0,∞) is normal, we also assume that it is radial, that is, 𝜈(𝑧)=𝜈(|𝑧|), 𝑧∈𝔹.

Strictly positive continuous functions on 𝔹 are called weights.

The weighted-type space 𝐻^∞_𝜇(𝔹)=𝐻^∞_𝜇 consists of all 𝑓∈𝐻(𝔹) such that 𝑓_𝐻∞𝜇∶=sup_𝑧∈𝔹𝜇(𝑧)𝑓(𝑧)<∞, (1.5) where 𝜇 is a weight (see, e.g., [3, 4] as well as [5] for a related class of spaces).

The little weighted-type space 𝐻^∞_𝜇,0(𝔹)=𝐻^∞_𝜇,0 is a subspace of 𝐻^∞_𝜇 consisting of all 𝑓∈𝐻(𝔹) such that lim_|𝑧|→1𝜇(𝑧)𝑓(𝑧)=0. (1.6)

For 0<𝑝, 𝑞<∞, and 𝜙 normal, the mixed-norm space 𝐻(𝑝,𝑞,𝜙)(𝔹)=𝐻(𝑝,𝑞,𝜙) consists of all functions 𝑓∈𝐻(𝔹) such that𝑓_{𝐻(𝑝,𝑞,𝜙)}=¹₀𝑀^𝑝_𝑞𝜙(𝑓,𝑟)^𝑝(𝑟)1−𝑟𝑑𝑟^1/𝑝<∞, (1.7) where 𝑀_𝑞𝑓,𝑟=_𝜕𝔹𝑓(𝑟𝜁)^𝑞𝑑𝜎(𝜁)^1/𝑞, (1.8) and 𝑑𝜎 is the normalized surface measure on 𝜕𝔹. For 𝑝=𝑞, 𝜙(𝑟)=(1−𝑟²)^{(𝛼+1)/𝑝}, and 𝛼>−1, the space is equivalent with the weighted Bergman space 𝐴^𝑝_𝛼(𝔹)=𝐴^𝑝_𝛼, which is defined as the class of all 𝑓∈𝐻(𝔹) such that 𝑓^𝑝_𝐴𝑝𝛼∶=_𝔹𝑓(𝑧)^𝑝1−|𝑧|^2𝛼𝑑𝑉(𝑧)<∞, (1.9) where 𝑑𝑉(𝑧) is the Lebesgue volume measure on 𝔹. Some facts on mixed-norm spaces in various domains in ℂ^𝑛 can be found, for example, in [6–8] (see also the references therein).

For 0<𝑝<∞ the Hardy space 𝐻^𝑝(𝔹)=𝐻^𝑝 consists of all 𝑓∈𝐻(𝔹) such that 𝑓_𝐻𝑝∶=sup_{0<𝑟<1𝜕𝔹}𝑓(𝑟𝜁)^𝑝𝑑𝜎(𝜁)^1/𝑝<∞. (1.10) For 𝑝=2 the Hardy and the weighted Bergman space are Hilbert.

Let 𝜑 be a holomorphic self-map of 𝔹, 𝑢∈𝐻(𝔹), and 𝑚∈ℕ₀. For 𝑓∈𝐻(𝔹), the weighted iterated radial composition operator is defined by ℜ^𝑚_𝑢,𝜑𝑓(𝑧)=𝑢(𝑧)ℜ^𝑚𝑓𝜑(𝑧),𝑧∈𝔹. (1.11) Note that the operator is the composition of the multiplication, composition and the iterated radial operator, that is ℜ^𝑚_𝑢,𝜑=𝑀_𝑢∘𝐶_𝜑∘ℜ^𝑚. (1.12) This is one of the product operators suggested by this author to be investigated at numerous talks (e.g., in [9]). Note that for 𝑚=0 the operator ℜ^𝑚_𝑢,𝜑 becomes the weighted composition operator (see, e.g., [4, 8, 10]). It is of interest to provide function-theoretic characterizations for when 𝜑 and 𝑢 induce bounded or compact weighted iterated radial composition operators on spaces of holomorphic functions. Studying products of some concrete linear operators on spaces of analytic functions attracted recently some attention see, for example, [11–32] as well as the related references therein. Some operators on mixed-norm spaces have been studied, for example, in [8, 10, 11, 16, 25, 26, 29, 33] (see also the references therein).

Here we study the boundedness and compactness of weighted iterated radial composition operators from mixed-norm spaces to weighted-type spaces on the unit ball for the case 𝑚∈ℕ. We also calculate the Hilbert-Schmidt norm of the operator on the weighted Bergman-Hilbert space 𝐴²_𝛼(𝔹) as well as on the Hardy 𝐻²(𝔹) space.

In 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. If both 𝑎⪯𝑏 and 𝑏⪯𝑎 hold, then one says that 𝑎≍𝑏.

2. Auxiliary Results

In this section we quote several lemmas which are used in the proofs of the main results.

The next characterization of compactness is proved in a standard way, hence we omit its proof (see, e.g., [34]).

Lemma 2.1. Assume 𝑝,𝑞>0, 𝜑 is a holomorphic self-map of 𝔹, 𝑢∈𝐻(𝔹), 𝜙 is normal and 𝜇 is a weight. Then the operator ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is compact if and only if for every bounded sequence (𝑓_𝑘)_𝑘∈ℕ⊂𝐻(𝑝,𝑞,𝜙) converging to 0 uniformly on compacts of 𝔹 as 𝑘→∞, one has lim_𝑘→∞ℜ^𝑚_𝑢,𝜑𝑓_{𝑘𝐻∞𝜇}=0. (2.1)

The following lemma is a slight modification of Lemma 2.5 in [8] and is proved similar to Lemma 1 in [35].

Lemma 2.2. Assume 𝜇 is a normal weight. Then a closed set 𝐾 in 𝐻^∞_𝜇,0 is compact if and only if it is bounded and lim_|𝑧|→1sup_𝑓∈𝐾𝜇(𝑧)𝑓(𝑧)=0. (2.2)

The following lemma is folklore and in the next form it can be found in [36].

Lemma 2.3. Assume that 0<𝑝, 𝑞<∞, 𝜙 is normal, and 𝑚∈ℕ. Then for every 𝑓∈𝐻(𝔹) the following asymptotic relationship holds: ¹₀𝑀^𝑝_𝑞𝜙𝑓,𝑟^𝑝(𝑟)𝑓(01−𝑟𝑑𝑟≍)^𝑝+¹₀𝑀^𝑝_𝑞ℜ^𝑚𝑓,𝑟(1−𝑟)^𝑚𝑝𝜙^𝑝(𝑟)1−𝑟𝑑𝑟. (2.3)

Lemma 2.4. Assume that 𝑚∈ℕ, 0<𝑝, 𝑞<∞, 𝜙 is normal and 𝑓∈𝐻(𝑝,𝑞,𝜙). Then, there is a positive constant 𝐶 independent of 𝑓 such that ℜ^𝑚𝑓(𝑧)≤𝐶𝑓_{𝐻(𝑝,𝑞,𝜙)}|𝑧|𝜙(|𝑧|)1−|𝑧|^{2𝑛/𝑞+𝑚}. (2.4)

Proof. Let 𝑔=ℜ^𝑚−1𝑓 and 𝑧∈𝔹. By the definition of the radial derivative, the Cauchy-Schwarz inequality and the Chauchy inequality, we have that ℜ𝑔(𝑧)≤|𝑧|∇𝑔(𝑧)≤𝐶|𝑧|sup_{𝐵(𝑧,1−|𝑧|)/4)}𝑔(𝑤)|𝑧|.1− (2.5)
From (2.3) with 𝑚→𝑚−1 we easily obtain the following inequality (see, e.g., [8, Lemma 2.1]):𝑔(𝑧𝑓)≤𝐶^{𝐻(𝑝,𝑞,𝜙)}𝜙(|𝑧|)1−|𝑧|^{2𝑛/𝑞+𝑚−1}. (2.6) From (2.5) and (2.6) and the asymptotic relations |𝑤||𝑧|𝑧𝑤|𝑧|1−≍1−,𝜙(||)≍𝜙(||),for𝑤∈𝐵𝑧,1−2, (2.7) inequality (2.4) follows.

Lemma 2.5. Let 𝑓_𝑎,𝑠(𝑧)=1(1−⟨𝑧,𝑎⟩)^𝑠,𝑧∈𝔹. (2.8) Then, ℜ^𝑚𝑓_𝑎,𝑠(𝑧)𝑃=𝑠^𝑚(⟨𝑧,𝑎⟩)(1−⟨𝑧,𝑎⟩)^𝑠+𝑚, (2.9) where 𝑃_𝑚(𝑤)=𝑠^𝑚−1𝑤^𝑚+𝑝^(𝑚)_𝑚−1(𝑠)𝑤^𝑚−1+⋯+𝑝₂^(𝑚)(𝑠)𝑤²+𝑤, (2.10) and where 𝑝_𝑗^(𝑚)(𝑠), 𝑗=2,…,𝑚−1 are nonnegative polynomials for 𝑠>0.

Proof. We prove the lemma by induction. For 𝑚=1, ℜ𝑓_𝑎,𝑠(𝑧)⟨⟩=𝑠𝑧,𝑎(1−⟨𝑧,𝑎⟩)^𝑠+1,𝑧∈𝔹, (2.11) which is formula (2.9) with 𝑃₁(𝑤)=𝑤.
Assume (2.9) is true for every 𝑚∈{1,…,𝑙}. Taking the radial derivative operator on equality (2.9) with 𝑚=𝑙, we obtain ℜ^𝑙+1𝑓_𝑎,𝑠(𝑧)𝑠=𝑠ℜ^𝑙−1⟨𝑧,𝑎⟩^𝑙+𝑝^(𝑙)_𝑙−1(𝑠)⟨𝑧,𝑎⟩^𝑙−1+⋯+𝑝₂^(𝑙)(𝑠)⟨𝑧,𝑎⟩²+⟨𝑧,𝑎⟩(1−⟨𝑧,𝑎⟩)^𝑠+𝑙()𝑠=𝑠𝑠+𝑙^𝑙−1⟨𝑧,𝑎⟩^𝑙+1+𝑝^(𝑙)_𝑙−1(𝑠)⟨𝑧,𝑎⟩^𝑙+⋯+𝑝₂^(𝑙)(𝑠)⟨𝑧,𝑎⟩³+⟨𝑧,𝑎⟩²(1−⟨𝑧,𝑎⟩)^𝑠+𝑙+1+𝑠𝑙𝑠^𝑙−1⟨𝑧,𝑎⟩^𝑙+(𝑙−1)𝑝^(𝑙)_𝑙−1(𝑠)⟨𝑧,𝑎⟩^𝑙−1+⋯+2𝑝₂^(𝑙)(𝑠)⟨𝑧,𝑎⟩²+⟨𝑧,𝑎⟩(1−⟨𝑧,𝑎⟩)(1−⟨𝑧,𝑎⟩)^𝑠+𝑙+1,𝑠=𝑠^𝑙⟨𝑧,𝑎⟩^𝑙+1+⋯+(𝑠+𝑙−2)𝑝₂^(𝑙)(𝑠)+3𝑝₃^(𝑙)(𝑠)⟨𝑧,𝑎⟩³(1−⟨𝑧,𝑎⟩)^𝑠+𝑙+1+(𝑠+𝑙−1)+2𝑝₂^(𝑙)(𝑠)⟨𝑧,𝑎⟩²+⟨𝑧,𝑎⟩(1−⟨𝑧,𝑎⟩)^𝑠+𝑙+1 (2.12) from which the inductive proof easily follows.

3. Boundedness and Compactness of ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇(or 𝐻^∞_𝜇,0)

This section characterizes the boundedness and compactness of ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇(or𝐻^∞_𝜇,0).

Theorem 3.1. Assume 𝑚∈ℕ, 0<𝑝, 𝑞<∞, 𝜙 is normal, 𝜇 is a weight, 𝜑 is a holomorphic self-map of 𝔹, and 𝑢∈𝐻(𝔹). Then ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded if and only if 𝑀₁∶=sup_𝑧∈𝔹𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}<∞. (3.1) Moreover, if ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded, then the following asymptotic relationship holds ℜ^𝑚_{𝑢,𝜑𝐻(𝑝,𝑞,𝜙)→𝐻∞𝜇}≍𝑀₁. (3.2)

Proof. Assume (3.1) holds. Then by Lemma 2.4 for each 𝑓∈𝐻(𝑝,𝑞,𝜙), we have that 𝜇(𝑧)ℜ^𝑚_𝑢,𝜑𝑓(𝑧)≤𝐶𝑓_{𝐻(𝑝,𝑞,𝜙)}𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}. (3.3) Taking the supremum over the unit ball in (3.3) and using (3.1) the boundedness of operator ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 follows and ℜ^𝑚_{𝑢,𝜑𝐻(𝑝,𝑞,𝜙)→𝐻∞𝜇}≤𝐶𝑀₁. (3.4)
Now assume that operator ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded. By using the test functions𝑓_𝑗(𝑧)=𝑧_𝑗∈𝐻𝑝,𝑞,𝜙,𝑗=1,…,𝑛, (3.5) we obtain ℜ^𝑚_𝑢,𝜑𝑓_𝑗∈𝐻^∞_𝜇,𝑗=1,…,𝑛, (3.6) that is, for each 𝑗=1,…,𝑛, holds ℜ^𝑚_𝑢,𝜑𝑓_{𝑗𝐻∞𝜇}=sup_𝑧∈𝔹𝜇(𝑧)|𝑢(𝑧)|𝜑_𝑗(𝑧)≤𝑧_{𝑗𝐻(𝑝,𝑞,𝜙)}ℜ^𝑚_{𝑢,𝜑𝐻(𝑝,𝑞,𝜙)→𝐻∞𝜇}<∞, (3.7) which implies that sup_𝑧∈𝔹𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)≤^𝑛_𝑗=1sup_𝑧∈𝔹𝜇(𝑧)|𝑢(𝑧)|𝜑_𝑗(𝑧≤ℜ_{𝑚𝑢,𝜑𝐻(𝑝,𝑞,𝜙)→𝐻∞𝜇𝑛𝑗=1})𝑧_{𝑗𝐻(𝑝,𝑞,𝜙)}<∞. (3.8)
Let 𝑓_𝑤(𝑧)=1−|𝑤|^2𝑡+1𝜙(𝑤)(1−⟨𝑧,𝑤⟩)^{𝑛/𝑞+𝑡+1}. (3.9) It is known that 𝐿₁∶=sup_𝑤∈𝔹‖𝑓_𝑤‖_{𝐻(𝑝,𝑞,𝜙)}<∞ (see [8, Theorem 3.3]). From this, using the boundedness of ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 and by Lemma 2.5, we have that for each 𝑎∈𝔹𝐿₁ℜ^𝑚_{𝑢,𝜑𝐻(𝑝,𝑞,𝜙)→𝐻∞𝜇}≥ℜ^𝑚_𝑢,𝜑(𝑓_𝜑(𝑎))_𝐻∞𝜇=sup_𝑧∈𝔹𝜇(𝑧)|𝑢(𝑧)|ℜ^𝑚𝑓_𝜑(𝑎)𝜑(𝑧)≥𝑛𝑞𝜇(𝑎𝑢(𝑎𝑃+𝑡+1)|)|_𝑚𝜑(𝑎)²𝜙𝜑(𝑎)1−𝜑(𝑎)^{2𝑛/𝑞+𝑚}≥𝑛𝑞𝜇(𝑎𝑢(𝑎+𝑡+1)|)|𝜑(𝑎)²𝜙𝜑(𝑎)1−𝜑(𝑎)^{2𝑛/𝑞+𝑚}. (3.10)
From (3.10), we have that ∞>sup_{|𝜑(𝑧)|>1/2}𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)²𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}≥sup_{|𝜑(𝑧)|>1/2}𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)2𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}. (3.11)
On the other hand, from (3.8) and since 𝜙 is normal, we obtain sup_{|𝜑(𝑧)|≤1/2}𝜇𝜑(𝑧)|𝑢(𝑧)|𝜑(𝑧)²𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}≤𝐶sup_𝑧∈𝔹𝜇𝜑(𝑧)|𝑢(𝑧)|𝜑(𝑧)<∞. (3.12) From (3.8), (3.10), (3.11), and (3.12) condition (3.1) follows, and moreover 𝑀₁≤𝐶ℜ^𝑚_{𝑢,𝜑𝐻(𝑝,𝑞,𝜙)→𝐻∞𝜇}. (3.13) From (3.4) and (3.13) asymptotic relationship (3.2) follows, finishing the proof of the theorem.

Theorem 3.2. Assume 𝑚∈ℕ, 0<𝑝, 𝑞<∞, 𝜙 is normal, 𝜇 is a weight, 𝜑 is a holomorphic self-map of 𝔹 and 𝑢∈𝐻(𝔹). Then ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is compact if and only if ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded and lim_{|𝜑(𝑧)|→1}𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}=0. (3.14)

Proof. Suppose that ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is compact. Then it is clear that ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded. If ‖𝜑‖_∞<1, then (3.14) is vacuously satisfied. Hence assume that ‖𝜑‖_∞=1. Let (𝑧_𝑘)_𝑘∈ℕ be a sequence in 𝔹 such that |𝜑(𝑧_𝑘)|→1 as 𝑘→∞, and 𝑓_𝑘(𝑧)=𝑓_{𝜑(𝑧𝑘)}(𝑧),𝑘∈ℕ, where 𝑓_𝑤 is defined in (3.9). Then sup_𝑘∈ℕ‖𝑓_𝑘‖_{𝐻(𝑝,𝑞,𝜙)}<∞, 𝑓_𝑘→0 uniformly on compacts of 𝔹 as 𝑘→∞ since lim_𝑘→∞1−𝜑𝑧_𝑘^2𝑡+1𝜙𝜑𝑧_𝑘=0, (3.15) so that lim_𝑘→∞ℜ^𝑚_𝑢,𝜑𝑓_{𝑘𝐻∞𝜇}=0. (3.16) On the other hand, by (3.10), we have 𝜇𝑧_𝑘𝑢𝑧_𝑘𝜑(𝑧_𝑘)²𝜙𝜑𝑧_𝑘1−𝜑𝑧_𝑘^{2𝑛/𝑞+𝑚}≤𝐶ℜ^𝑚_𝑢,𝜑𝑓_{𝑘𝐻∞𝜇}. (3.17) From (3.16) and (3.17), equality (3.14) easily follows.
Conversely, assume that ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded and (3.14) holds. From the proof of Theorem 3.1 we know that (3.1) holds. On the other hand, from (3.14), we have that, for every 𝜀>0, there is a 𝛿∈(0,1), such that𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}<𝜀 (3.18) whenever 𝛿<|𝜑(𝑧)|<1.
Assume (𝑓_𝑘)_𝑘∈ℕ is a sequence in 𝐻(𝑝,𝑞,𝜙) such that sup_𝑘∈ℕ‖𝑓_𝑘‖_{𝐻(𝑝,𝑞,𝜙)}≤𝐿 and 𝑓_𝑘 converges to 0 uniformly on compact subsets of 𝔹 as 𝑘→∞. Let 𝐾={𝑧∈𝔹∶|𝜑(𝑧)|≤𝛿}. Then from (3.18), and by Lemma 2.4, it follows thatℜ^𝑚_𝑢,𝜑𝑓_{𝑘𝐻∞𝜇}=sup_𝑧∈𝔹𝜇(𝑧)|𝑢(𝑧)|ℜ^𝑚𝑓_𝑘𝜑(𝑧)≤sup_𝑧∈𝐾𝜇(𝑧)|𝑢(𝑧)|ℜ^𝑚𝑓_𝑘𝜑(𝑧)+sup_{𝑧∈𝔹⧵𝐾}𝜇(𝑧)|𝑢(𝑧)|ℜ^𝑚𝑓_𝑘𝜑(𝑧)≤sup_𝑧∈𝐾𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)∇ℜ^𝑚−1𝑓_𝑘𝜑(𝑧)+𝐶𝑓_{𝑘𝐻(𝑝,𝑞,𝜙)}sup_{𝑧∈𝔹⧵𝐾}𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}≤𝐾₂𝐶sup_{|𝜁|≤𝛿}∇ℜ^𝑚−1𝑓_𝑘(𝜁)+𝐶𝜀𝑓_{𝑘𝐻(𝑝,𝑞,𝜙)}, (3.19) where 𝐾₂∶=sup_𝑧∈𝔹𝜇(𝑧)|𝑢(𝑧)||𝜑(𝑧)| (see (3.8)). Therefore ℜ^𝑚_𝑢,𝜑𝑓_{𝑘𝐻∞𝜇}≤𝐾₂sup_{|𝜁|≤𝛿}∇ℜ^𝑚−1𝑓_𝑘(𝜁)+𝐶𝐿𝜀. (3.20) Since (𝑓_𝑘)_𝑘∈ℕ converges to zero on compact subsets of 𝔹 as 𝑘→∞, by Cauchy's estimates it follows that the sequence (|∇ℜ^𝑚−1𝑓_𝑘|)_𝑘∈ℕ also converges to zero on compact subsets of 𝔹 as 𝑘→∞, in particular lim_𝑘→∞sup_{|𝜁|≤𝛿}∇ℜ^𝑚−1𝑓_𝑘(𝜁)=0. (3.21) Using these facts and letting 𝑘→∞ in (3.20), we obtain that limsup_𝑘→∞ℜ^𝑚_𝑢,𝜑𝑓_{𝑘𝐻∞𝜇}≤𝐶𝐿𝜀. (3.22) Since 𝜀 is an arbitrary positive number it follows that the last limit is equal to zero. Applying Lemma 2.1, the implication follows.

Theorem 3.3. Assume 𝑚∈ℕ, 0<𝑝, 𝑞<∞, 𝜙, 𝜇 are normal, 𝜑 is a holomorphic self-map of 𝔹 and 𝑢∈𝐻(𝔹). Then ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇,0 is bounded if and only if ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded and lim_|𝑧|→1𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)=0. (3.23)

Proof. First assume that ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇,0 is bounded. Then, it is clear that ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded, and as in the proof of Theorem 3.1, by taking the test functions 𝑓_𝑗(𝑧)=𝑧_𝑗, 𝑗=1,…,𝑛, we obtain (3.23).
Conversely, assume that the operator ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇 is bounded and condition (3.23) holds. Then, for each polynomial 𝑝, we have𝜇(𝑧)ℜ^𝑚_𝑢,𝜑𝑝(𝑧)≤𝜇(𝑧)|𝑢(𝑧)|ℜ^𝑚𝑝𝜑(𝑧)≤𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)∇ℜ^𝑚−1𝑝_∞, (3.24) from which along with condition (3.23) it follows that ℜ^𝑚_𝑢,𝜑𝑝∈𝐻^∞_𝜇,0. Since the set of all polynomials is dense in 𝐻(𝑝,𝑞,𝜙), we see that for every 𝑓∈𝐻(𝑝,𝑞,𝜙) there is a sequence of polynomials (𝑝_𝑘)_𝑘∈ℕ such that lim_𝑘→∞𝑓−𝑝_{𝑘𝐻(𝑝,𝑞,𝜙)}=0. (3.25) From this and by the boundedness of the operator ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇, we have that ℜ^𝑚_𝑢,𝜑𝑓−ℜ^𝑚_𝑢,𝜑𝑝_{𝑘𝐻∞𝜇}≤ℜ^𝑚_{𝑢,𝜑𝐻(𝑝,𝑞,𝜙)→𝐻∞𝜇}𝑓−𝑝_{𝑘𝐻(𝑝,𝑞,𝜙)}⟶0 (3.26) as 𝑘→∞. Hence ℜ^𝑚_𝑢,𝜑(𝐻(𝑝,𝑞,𝜙))⊆𝐻^∞_𝜇,0, and consequently ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇,0 is bounded.

Theorem 3.4. Assume 𝑚∈ℕ, 0<𝑝, 𝑞<∞, 𝜙, 𝜇 are normal, 𝜑 is a holomorphic self-map of 𝔹 and 𝑢∈𝐻(𝔹). Then ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇,0 is compact if and only if lim_|𝑧|→1𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}=0. (3.27)

Proof. From (3.27), we see that (3.1) hold. This fact along with (3.3) implies that the set ℜ^𝑚_𝑢,𝜑({𝑓|‖𝑓‖_{𝐻(𝑝,𝑞,𝜙)}≤1}) is bounded in 𝐻^∞_𝜇, moreover in 𝐻^∞_𝜇,0. By taking the supremum in (3.3) over the unit ball in 𝐻(𝑝,𝑞,𝜙), using (3.27) and applying Lemma 2.2 we obtain that the operator ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇,0 is compact.
If ℜ^𝑚_𝑢,𝜑∶𝐻(𝑝,𝑞,𝜙)→𝐻^∞_𝜇,0 is compact, then by Theorem 3.2, we have that condition (3.14) holds, which implies that for every 𝜀>0 there is an 𝑟∈(0,1) such that 𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}<𝜀, (3.28) for 𝑟<|𝜑(𝑧)|<1.
As in Theorem 3.3, we have that (3.23) holds. Thus there is a 𝜎∈(0,1) such that 𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)<𝜀1−𝑟^{2𝑛/𝑞+𝑚}inf_{𝜑(𝑧)≤𝑟}𝜙𝜑(𝑧), (3.29) for 𝜎<|𝑧|<1.
If |𝜑(𝑧)|≤𝑟 and 𝜎<|𝑧|<1, then from (3.29), we obtain𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}𝜙𝜑(𝑧≤𝜇(𝑧𝑢(𝑧𝜑(𝑧))|)|)1−𝑟^{2𝑛/𝑞+𝑚}inf_{|𝜑(𝑧)|≤𝑟}𝜙𝜑(𝑧)<𝜀. (3.30)
Using (3.30) and the fact that from (3.28), we have𝜇(𝑧)|𝑢(𝑧)|𝜑(𝑧)𝜙𝜑(𝑧)1−𝜑(𝑧)^{2𝑛/𝑞+𝑚}<𝜀, (3.31) for 𝜎<|𝑧|<1 and 𝑟<|𝜑(𝑧)|<1, we get (3.27).

4. Hilbert-Schmidt Norm of ℜ^𝑚_𝑢,𝜑∶𝐴²_𝛼→𝐴²_𝛼 and ℜ^𝑚_𝑢,𝜑∶𝐻²→𝐻²

In this section we calculate Hilbert-Schmidt norm of the operators ℜ^𝑚_𝑢,𝜑∶𝐴²_𝛼→𝐴²_𝛼 and ℜ^𝑚_𝑢,𝜑∶𝐻²→𝐻². For some related results see [37, 38].

If ℋ is a separable Hilbert space, then the Hilbert-Schmidt norm ‖𝑇‖_HS of an operator 𝑇∶ℋ→ℋ is defined by ‖𝑇‖_HS=^∞_𝑛=1𝑇𝑒_𝑛^21/2, (4.1) where {𝑒_𝑛}_𝑛∈ℕ is an orthonormal basis on ℋ. The right-hand side in (4.1) does not depend on the choice of basis. Hence, it is larger than the operator norm ‖𝑇‖_op of 𝑇.

Let ⟨⋅,⋅⟩_𝛼, 𝛼≥−1, be the usual scalar product on 𝐴²_𝛼, where we regard that 𝐴²₋₁=𝐻₂. Since for each multi-index 𝛽=(𝛽₁,…,𝛽_𝑛)∈ℕ^𝑛_0𝔹𝑧^𝛽2𝑑𝑣_𝛼(𝑧)=𝛽!Γ(𝑛+𝛼+1)Γ𝛽,𝑛++𝛼+1 (4.2) where 𝛽!=𝛽₁!⋯𝛽_𝑛!, and _𝔹𝑧^𝛽𝑧^𝛾𝑑𝑣_𝛼(𝑧)=0,𝛽≠𝛾 (4.3) (see, e.g., [1]), we have that the vectors 𝑒_𝛽(𝑧)=Γ𝑛+𝛽+𝛼+1()𝑧𝛽!Γ𝑛+𝛼+1^𝛽,𝛽∈ℕ^𝑛₀, (4.4) form an orthonormal basis in 𝐴²_𝛼.

Theorem 4.1. Let 𝑚∈ℕ₀. Then Hilbert-Schmidt norm of the operator ℜ^𝑚_𝑢,𝜑 on 𝐴²_𝛼, 𝛼>−1 is ℜ^𝑚_𝑢,𝜑HS=_𝔹|𝑢(𝑧)|²ℜ^𝑚11−∑^𝑛_𝑗=1𝑤_𝑗^𝑛+𝛼+1_{𝑤𝑗=|𝜑𝑗(𝑧)|2}𝑑𝑣_𝛼(𝑧)^1/2. (4.5)

Proof. By using the definition of the Hilbert-Schmidt norm and the monotone convergence theorem, we have ℜ^𝑚_𝑢,𝜑²_HS=_𝛽ℜ^𝑚_𝑢,𝜑(𝑒_𝛽)²_2,𝛼=_𝛽Γ𝑛+𝛽+𝛼+1()ℜ𝛽!Γ𝑛+𝛼+1^{𝑚𝑢,𝜑}(𝑧^𝛽)²_2,𝛼=_𝛽Γ𝑛+𝛽+𝛼+1()𝛽𝛽!Γ𝑛+𝛼+1^𝑚_𝔹|𝑢(𝑧)|^2𝑛_𝑗=1𝜑_𝑗(𝑧)^2𝛽𝑗𝑑𝑣_𝛼(𝑧)=_𝔹|𝑢(𝑧)|²_𝛽𝛽^𝑚Γ𝑛+𝛽+𝛼+1𝛽!Γ(𝑛+𝛼+1)^𝑛_𝑗=1𝜑_𝑗(𝑧)^2𝛽𝑗𝑑𝑣_𝛼(𝑧). (4.6) We also have that 1−^𝑛_𝑗=1𝑤_𝑗^{−(𝑛+𝛼+1)}=^∞_𝑘=0^𝑛_𝑗=1𝑤_𝑗^𝑘Γ(𝑛+𝛼+𝑘+1)()=𝑘!Γ𝑛+𝛼+1_{∞𝑘=0|𝑙|=𝑘}𝑘!𝑙!^𝑛_𝑗=1𝑤^𝑙𝑗_𝑗Γ(𝑛+𝛼+𝑘+1)𝑘!Γ(𝑛+𝛼+1)=_𝑙Γ(𝑛+𝛼+|𝑙|+1)𝑙!Γ(𝑛+𝛼+1)^𝑛_𝑗=1𝑤^𝑙𝑗_𝑗, (4.7) from which by taking the radial derivatives it follows that ℜ^𝑚11−∑^𝑛_𝑗=1𝑤_𝑗^𝑛+𝛼+1=_𝑙|𝑙|^𝑚Γ(𝑛+𝛼+|𝑙|+1)𝑙!Γ(𝑛+𝛼+1)^𝑛_𝑗=1𝑤^𝑙𝑗_𝑗. (4.8) From (4.6) and (4.8) the result easily follows.

Similar to Theorem 4.1 the following result regarding the case of the Hardy space is proved. We omit the proof.

Theorem 4.2. Let 𝑚∈ℕ₀. Then, Hilbert-Schmidt norm of the operator ℜ^𝑚_𝑢,𝜑 on 𝐻², is ℜ^𝑚_𝑢,𝜑HS=sup_0<𝑟<1𝑆𝑢(𝑟𝜁)²ℜ^𝑚11−∑^𝑛_𝑗=1𝑤_𝑗^𝑛_{𝑤𝑗=|𝜑𝑗(𝑟𝜁)|2}𝑑𝜎(𝜁)^1/2. (4.9)