On an Integral-Type Operator from Zygmund-Type Spaces to Mixed-Norm Spaces on the Unit Ball

Abstract

The boundedness and compactness of an integral-type operator recently introduced by the author from Zygmund-type spaces to the mixed-norm space on the unit ball are characterized here.

1. Introduction

Let 𝔹={𝑧∈ℂ^𝑛∶|𝑧|<1} be the open unit ball in ℂ^𝑛, 𝜕𝔹 its boundary,𝑑𝑉_𝑁 the normalized volume measure on 𝔹, and 𝐻(𝔹) the class of all holomorphic functions on 𝔹. Strictly positive, bounded, continuous functions on 𝔹 are called weights.

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.

A positive, continuous function 𝜈 on the interval [0,1) is called normal [1] if there are 𝛿∈[0,1) and 𝑎 and 𝑏, 0<𝑎<𝑏 such that𝜈(𝑟)(1−𝑟)^𝑎isdecreasingon[𝛿,1),lim_𝑟→1𝜈(𝑟)(1−𝑟)^𝑎𝜈(𝑟)=0,(1−𝑟)^𝑏isincreasingon[𝛿,1),lim_𝑟→1𝜈(𝑟)(1−𝑟)^𝑏=∞. (1.2) 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.

Let 𝜇 be a weight. By 𝒵_𝜇(𝔹)=𝒵_𝜇, we denote the class of all 𝑓∈𝐻(𝔹) such that 𝑧𝑓∶=sup_𝑧∈𝔹𝜇(𝑧)ℜ²𝑓(𝑧)<∞, (1.3) and call it the Zygmund-type class. The quantity 𝑧(𝑓) is a seminorm. A norm on 𝒵_𝜇 can be introduced by ‖𝑓‖_𝒵=|𝑓(0)|+𝑧(𝑓). Zygmund-type class with this norm will be called the Zygmund-type space.

The little Zygmund-type space on 𝔹, denoted by 𝒵_𝜇,0(𝔹)=𝒵_𝜇,0, is the closed subspace of 𝒵_𝜇 consisting of functions 𝑓 satisfying the following condition lim_|𝑧|→1𝜇(𝑧)ℜ²𝑓(𝑧)=0. (1.4)

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 that 𝑓_{𝐻(𝑝,𝑞,𝜙)}=¹₀𝑀^𝑝_𝑞𝜙𝑓,𝑟^𝑝(𝑟)1−𝑟𝑑𝑟^1/𝑝<∞, (1.5) where 𝑀_𝑞𝑓,𝑟=_𝜕𝔹𝑓(𝑟𝜁)^𝑞𝑑𝜎(𝜁)^1/𝑞, (1.6) 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).

In [2], the present author has introduced products of integral and composition operators on 𝐻(𝔹) as follows (see also [3–5]). Assume 𝑔∈𝐻(𝔹),𝑔(0)=0, and 𝜑 is a holomorphic self-map of 𝔹, then we define an operator on 𝐻(𝔹) by 𝑃^𝑔_𝜑𝑓(𝑧)=¹₀𝑓𝜑(𝑡𝑧)𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹),𝑧∈𝔹. (1.7) The operator is an extension of the operator introduced in [6]. Here, we continue to study operator 𝑃^𝑔_𝜑 by characterizing the boundedness and compactness of the operator between Zygmund-type spaces and the mixed-norm space. For some results on related integral-type operators mostly in ℂ^𝑛, see, for example, [3, 6–27] and the references therein.

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 first lemma was proved in [2].

Lemma 2.1. Assume that 𝜑 is a holomorphic self-map of 𝔹,𝑔∈𝐻(𝔹), and 𝑔(0)=0. Then, for every 𝑓∈𝐻(𝔹) it holdsℜ𝑃^𝑔_𝜑𝑓(𝑧)=𝑓𝜑(𝑧)𝑔(𝑧). (2.1)

The next Schwartz-type characterization of compactness [28] is proved in a standard way (see, e.g., the proof of the corresponding lemma in [11]), hence we omit its proof.

Lemma 2.2. Assume 𝑝,𝑞>0,𝜑 is a holomorphic self-map of 𝔹,𝑔∈𝐻(𝔹),𝑔(0)=0,𝜙 is normal, and 𝜇 is a weight. Then, the operator 𝑃^𝑔_𝜑∶𝒵_𝜇(or𝒵_𝜇,0)→𝐻(𝑝,𝑞,𝜙) is compact if and only if for every bounded sequence (𝑓_𝑘)_𝑘∈ℕ⊂𝒵_𝜇(or𝒵_𝜇,0) converging to 0 uniformly on compacts of 𝔹 we have lim_𝑘→∞‖𝑃^𝑔_𝜑𝑓_𝑘‖_{𝐻(𝑝,𝑞,𝜙)}=0.

The next lemma is folklore and can be found, for example, in [6] (one-dimensional case for standard power weights is due to Flett [29, Theorems 6 and 7]).

Lemma 2.3. Assume that 0<𝑝, 𝑞<∞,𝜙 is normal, and 𝑚∈ℕ. Then, the following asymptotic relationship holds for every 𝑓∈𝐻(𝔹), ¹₀𝑀^𝑝_𝑞𝜙𝑓,𝑟^𝑝(𝑟)𝑓(01−𝑟𝑑𝑟≍)^𝑝+¹₀𝑀^𝑝_𝑞ℜ^𝑚𝑓,𝑟(1−𝑟)^𝑚𝑝𝜙^𝑝(𝑟)1−𝑟𝑑𝑟. (2.2)

Lemma 2.4. Assume that 𝜇 is normal and 𝑓∈𝒵_𝜇. Then, 𝑓(𝑧)≤𝐶𝑓_𝒵𝜇1+₀^|𝑧|𝑡₀𝑑𝑠𝜇(𝑠)𝑑𝑡,𝑧∈𝔹. (2.3) Moreover, if ¹₀^𝑡₀𝑑𝑠𝜇(𝑠)𝑑𝑡<∞, (2.4) then 𝑓(𝑧)≤𝐶𝑓_𝒵𝜇, (2.5) for any 𝑧∈𝔹.

Proof. By Lemma 3.1 in [21] applied to ℜ𝑓 we have that ℜ𝑓(𝑧)≤𝐶𝑓_𝒵𝜇1+₀^|𝑧|𝑑𝑠𝜇(𝑠). (2.6) Hence, for |𝑧|≥1/2, we have that 𝑓(𝑧)𝑧−𝑓2≤¹_1/2ℜ𝑓(𝑡𝑧)𝑑𝑡𝑡≤𝐶𝑓_𝒵𝜇¹_1/21+₀^𝑡|𝑧|𝑑𝑠𝜇(𝑠)𝑑(𝑡|𝑧|)|𝑧|, (2.7) so that 𝑓(𝑧)≤𝑀_∞1𝑓,2+𝐶𝑓_𝒵𝜇1+₀^|𝑧|𝑡₀𝑑𝑠𝜇(𝑠)𝑑𝑡, (2.8) where 𝑀_∞(𝑓,1/2)=max_|𝑧|≤1/2|𝑓(𝑧)|.
If |𝑧|≤1/2, then by the mean value property of the function 𝑓(𝑧)−𝑓(0) (see [30]), Jensen’s inequality, and Parseval’s formula, we obtain max_|𝑧|≤1/2𝑓(𝑧)−𝑓(0)²≤4^𝑛_|𝑧|≤3/4𝑓(𝑤)−𝑓(0)²𝑑𝑉_𝑁(𝑤)≤4^𝑛_|𝑧|≤3/4ℜ𝑓(𝑤)²𝑑𝑉_𝑁(𝑤)≤3^𝑛max_|𝑧|≤3/4ℜ𝑓(𝑧)². (2.9)
From (2.9) and (2.6), we obtain 𝑀_∞𝑓,1/2≤𝑓(0)+√3^𝑛max_|𝑧|≤3/4(𝑧≤𝑓(0+√ℜ𝑓))3^𝑛𝐶𝑓_𝒵𝜇1+₀^3/4𝑑𝑠𝜇(𝑠)≤𝐶𝑓_𝒵𝜇. (2.10) From (2.8) and (2.10), (2.3) follows, from which by (2.4) the second statement follows.

Lemma 2.5. Assume 𝜇 is normal and (2.4) holds. Then, for every bounded sequence (𝑓_𝑘)_𝑘∈ℕ⊂𝒵_𝜇 converging to 0 uniformly on compacts of 𝔹, we have that lim_𝑘→∞sup_𝑧∈𝔹𝑓_𝑘(𝑧)=0. (2.11)

Proof. From (2.4), we have that for every 𝜀>0, there is a 𝛿∈(0,min{𝜀,1/2}) such that _(𝑧|^|𝑧|_1−𝛿)|^𝑡₀𝑑𝑠𝜇(𝑠)𝑑𝑡<𝜀, (2.12) for |𝑧|>1−𝛿.
Hence, from (2.12) it follows that for each 𝑘∈ℕ and |𝑧|≥1−𝛿𝑓_𝑘(𝑧)−𝑓_𝑘((1−𝛿)𝑧)≤¹_1−𝛿ℜ𝑓_𝑘(𝑡𝑧)𝑑𝑡𝑡≤𝐶𝑓_𝑘𝒵𝜇¹_1−𝛿1+₀^𝑡|𝑧|𝑑𝑠𝜇(𝑠)𝑓𝑑𝑡≤𝐶_𝑘𝒵𝜇𝜀+_(𝑧|^|𝑧|_1−𝛿)|^𝑡₀𝑑𝑠𝜇(𝑠)𝑑𝑡. (2.13)
From (2.12) and (2.13), we obtain 𝑓_𝑘(𝑧)≤sup_{|𝑤|≤1−𝛿}𝑓_𝑘(𝑤)+2𝐶𝜀sup_𝑘∈ℕ𝑓_𝑘𝒵𝜇. (2.14) Letting 𝑘→∞ in this inequality, using the assumption that 𝑓_𝑘 converges to 0 on the compact |𝑤|≤1−𝛿, and using the fact that 𝜀 is an arbitrary positive number, the lemma follows.

3. The Boundedness and Compactness of 𝑃^𝑔_𝜑∶𝒵_𝜇(or𝒵_𝜇,0)→𝐻(𝑝,𝑞,𝜙)

The boundedness and compactness of the operator 𝑃^𝑔_𝜑∶𝒵_𝜇(or𝒵_𝜇,0)→𝐻(𝑝,𝑞,𝜙) are characterized in this section.

Theorem 3.1. Assume that 𝑝,𝑞>0,𝜑 is a holomorphic self-map of 𝔹,𝑔∈𝐻(𝔹),𝑔(0)=0,𝜙 and 𝜇 are normal, and  𝜇 satisfies condition (2.4). Let 𝐺(𝑧)=¹₀𝑔(𝑡𝑧)𝑑𝑡𝑡. (3.1) Then, the following statements are equivalent:
(a) 𝑃^𝑔_𝜑∶𝒵_𝜇,0→𝐻(𝑝,𝑞,𝜙) is bounded;
(b) 𝑃^𝑔_𝜑∶𝒵_𝜇→𝐻(𝑝,𝑞,𝜙) is bounded;
(c) 𝑃^𝑔_𝜑∶𝒵_𝜇,0→𝐻(𝑝,𝑞,𝜙) is compact;
(d) 𝑃^𝑔_𝜑∶𝒵_𝜇→𝐻(𝑝,𝑞,𝜙) is compact;
(e) 𝐺∈𝐻(𝑝,𝑞,𝜙).
Moreover, if 𝑃^𝑔_𝜑∶𝒵_𝜇→𝐻(𝑝,𝑞,𝜙) is bounded, then the following asymptotic relations hold: 𝑃^𝑔_{𝜑𝒵𝜇→𝐻(𝑝,𝑞,𝜙)}≍𝑃^𝑔_{𝜑𝒵𝜇,0→𝐻(𝑝,𝑞,𝜙)}≍‖𝐺‖_{𝐻(𝑝,𝑞,𝜙)}. (3.2)

Proof. The implications (d)⇒(b),(b)⇒(a),(d)⇒(c), and (c)⇒(a) are obvious.
(a)⇒(e) Since 𝑃^𝑔_𝜑∶𝒵_𝜇,0→𝐻(𝑝,𝑞,𝜙) is bounded and 𝑓(𝑧)≡1∈𝒵_𝜇,0, by Lemma 2.1 we have that 𝐺(𝑧)=𝑃^𝑔_𝜑(1)(𝑧)∈𝐻(𝑝,𝑞,𝜙). Moreover, ‖𝐺‖_{𝐻(𝑝,𝑞,𝜙)}=𝑃^𝑔_𝜑(1)_{𝐻(𝑝,𝑞,𝜙)}≤𝑃^𝑔_{𝜑𝒵𝜇,0→𝐻(𝑝,𝑞,𝜙)}. (3.3)
(e)⇒(d) Assume that (𝑓_𝑘)_𝑘∈ℕ⊂𝒵_𝜇 is a bounded sequence converging to 0 uniformly on compacts of 𝔹. Then, by Lemmas 2.1, 2.3, and 2.5, we have 𝑃^𝑔_𝜑𝑓_{𝑘𝐻(𝑝,𝑞,𝜙)}≍𝑃^𝑔_𝜑𝑓_𝑘(0)+¹₀𝑀^𝑝_𝑞𝑔𝑓_𝑘𝜙∘𝜑,𝑟^𝑝(𝑟)(1−𝑟)^1−𝑝𝑑𝑟^1/𝑝≤𝐶‖𝐺‖_{𝐻(𝑝,𝑞,𝜙)}sup_𝑧∈𝔹𝑓_𝑘(𝑧)⟶0,as𝑘⟶∞, (3.4) which along with Lemma 2.2 implies the compactness of 𝑃^𝑔_𝜑∶𝒵_𝜇→𝐻(𝑝,𝑞,𝜙).
From (2.4) and by Lemmas 2.3 and 2.4, we have 𝑃^𝑔_𝜑𝑓_{𝐻(𝑝,𝑞,𝜙)}≤𝐶¹₀𝑀^𝑝_𝑞𝜙𝑔𝑓∘𝜑,𝑟^𝑝(𝑟)(1−𝑟)^1−𝑝𝑑𝑟^1/𝑝≤𝐶𝑓_𝒵𝜇¹₀𝑀^𝑝_𝑞𝜙𝑔,𝑟^𝑝(𝑟)(1−𝑟)^1−𝑝𝑑𝑟^1/𝑝≤𝐶𝑓_𝒵𝜇‖𝐺‖_{𝐻(𝑝,𝑞,𝜙)}. (3.5) This, together with (3.3) and the inequality 𝑃^𝑔_{𝜑𝒵𝜇,0→𝐻(𝑝,𝑞,𝜙)}≤𝑃^𝑔_{𝜑𝒵𝜇→𝐻(𝑝,𝑞,𝜙)}, (3.6) implies the asymptotic relations in (3.2), as desired.