Weighted Composition Operators on Some Weighted Spaces in the Unit Ball

Abstract

Let 𝔹_𝑛 be the unit ball of ℂ^𝑛, 𝐻(𝔹_𝑛) the space of all holomorphic functions in 𝔹_𝑛. Let 𝑢∈𝐻(𝔹_𝑛) and 𝛼 be a holomorphic self-map of 𝔹_𝑛. For 𝑓∈𝐻(𝔹_𝑛), the weigthed composition operator 𝑢𝐶_𝛼 is defined by (𝑢𝐶_𝛼𝑓)(𝑧)=𝑢(𝑧)𝑓(𝛼(𝑧)),𝑧∈𝔹_𝑛. The boundedness and compactness of the weighted composition operator on some weighted spaces on the unit ball are studied in this paper.

1. Introduction

Let 𝔹_𝑛 be the unit ball of ℂ^𝑛, 𝐻(𝔹_𝑛) the space of all holomorphic functions in 𝔹_𝑛, and 𝐻^∞=𝐻^∞(𝔹_𝑛) the space of all bounded holomorphic functions in the unit ball. For 𝑓∈𝐻(𝔹_𝑛), letℜ𝑓(𝑧)=^𝑛_𝑗=1𝑧_𝑗𝜕𝑓𝜕𝑧_𝑗(𝑧) (1.1)be the radial derivative of 𝑓.

A positive continuous function 𝜇 on [0,1) is called normal if there exist positive numbers 𝛼 and 𝛽,0<𝛼<𝛽, and 𝛿∈[0,1) such that (see, e.g., [1, 2])𝜇(𝑟)(1−𝑟)^𝛼isdecreasingon[𝛿,1),lim_𝑟→1𝜇(𝑟)(1−𝑟)^𝛼=0,𝜇(𝑟)(1−𝑟)^𝛽isincreasingon[𝛿,1),lim_𝑟→1𝜇(𝑟)(1−𝑟)^𝛽=∞. (1.2)

An 𝑓∈𝐻(𝔹_𝑛) is said to belong to the weighted-type space, denoted by 𝐻^∞_𝜇=𝐻^∞_𝜇(𝔹_𝑛), if‖𝑓‖_𝐻^∞_𝜇=sup_{𝑧∈𝔹𝑛}𝜇|𝑧|𝑓(𝑧)<∞, (1.3)where 𝜇 is normal on [0,1) (see [3]). 𝐻^∞_𝜇 is a Banach space with the norm ‖⋅‖_𝐻^∞_𝜇.

The little weighted-type space, denoted by 𝐻^∞_𝜇,0, is the subspace of 𝐻^∞_𝜇 consisting of those 𝑓∈𝐻^∞_𝜇 such thatlim_|𝑧|→1𝜇|𝑧|𝑓(𝑧)=0. (1.4) When 𝜇(𝑟)=(1−𝑟²)^𝛼, the induced spaces 𝐻^∞_𝜇 and 𝐻^∞_𝜇,0 become the (classical) weighted spaces 𝐻^∞_𝛼 and 𝐻^∞_𝛼,0 respectively.

An 𝑓∈𝐻(𝔹_𝑛) is said to belong to the logarithmic-type space 𝐻^∞_log if‖𝑓‖_𝐻^∞_log=sup_{𝑧∈𝔹𝑛}𝑓(𝑧)log𝑒/1−|𝑧|²<∞. (1.5)It is easy to see that 𝐻^∞_log becomes a Banach space under the norm ‖⋅‖_𝐻^∞_log, and that the following inclusions hold:𝐻^∞⊂ℬ⊂𝐻^∞_log⊂𝐻^∞_𝛼,𝛼>0, (1.6)where ℬ is the Bloch space defined byℬ=𝑓∈𝐻(𝔹_𝑛)∶sup_{𝑧∈𝔹𝑛}1−|𝑧|²ℜ𝑓(𝑧)<∞. (1.7)For some information on the Bloch and related spaces see, for example, [4–13] and the references therein. For some information on the space 𝐻^∞_log in the unit disk see [14].

Let 𝑢∈𝐻(𝔹_𝑛), and let 𝜑 be a holomorphic self-map of 𝔹_𝑛. For 𝑓∈𝐻(𝔹_𝑛), the weighted composition operator 𝑢𝐶_𝜑 is defined by𝑢𝐶_𝜑𝑓(𝑧)=𝑢(𝑧)𝑓𝜑(𝑧),𝑧∈𝔹_𝑛. (1.8)The weighted composition operator can be regarded as a generalization of a multiplication operator and a composition operator, which is defined by (𝐶_𝜑𝑓)(𝑧)=𝑓(𝜑(𝑧)). The work in [15] contains much information on this topic.

In the setting of the unit ball, Zhu studied the boundedness and compactness of the weighted composition operator between Bergman-type spaces and 𝐻^∞ in [16]. More general results can be found in [17, 18]. Some necessary and sufficient conditions for the weighted composition operator to be bounded and compact between the Bloch space and 𝐻^∞ are given in [19]. In the setting of the unit polydisk, some necessary and sufficient conditions for a weighted composition operator to be bounded and compact between the Bloch space and 𝐻^∞(𝔹^𝑛₁) are given in [20, 21] (see also [22] for the case of composition operators). Other related results can be found, for example, in [3, 23–32].

In this paper, we study the weighted composition operator from 𝐻^∞_log to the spaces 𝐻^∞_𝜇 and 𝐻^∞_log. Some necessary and sufficient conditions for the weighted composition operator 𝑢𝐶_𝜑 to be bounded and compact are given.

Throughout the paper, constants are denoted by 𝐶; they are positive and may not be the same in every occurrence.

2. Main Results and Proofs

In this section, we give our main results and their proofs. Before stating these results, we need some auxiliary results, which are incorporated in the lemmas which follow.

Lemma 2.1. Assume that 𝑢∈𝐻(𝔹_𝑛), 𝜑 is a holomorphic self-map of 𝔹_𝑛, and 𝜇 is a normal function on [0,1). Then, 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is compact if and only if 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is bounded, and for any bounded sequence (𝑓_𝑘)_𝑘∈ℕ in 𝐻^∞_log which converges to zero uniformly on compact subsets of 𝔹_𝑛 as 𝑘→∞, one has ‖𝑢𝐶_𝜑𝑓_𝑘‖_𝐻^∞_𝜇→0 as 𝑘→∞.

The proof of Lemma 2.1 follows by standard arguments (see, e.g., [15, Proposition 3.11] as well as the proofs of the corresponding results in [7, 22, 33, 34]). Hence, we omit the details.

Lemma 2.2. Assume that 𝜇 is normal. A closed set 𝐾 in 𝐻^∞_𝜇,0 is compact if and only if it is bounded and satisfies lim_|𝑧|→1sup_𝑓∈𝐾𝜇|𝑧|𝑓(𝑧)=0. (2.1)

The proof of Lemma 2.2 is similar to the proof of Lemma 1 in [35]. We omit the details.

Now, we are in a position to state and prove our main results.

Theorem 2.3. Assume that 𝑢∈𝐻(𝔹_𝑛), 𝜑 is a holomorphic self-map of 𝔹_𝑛, and 𝜇 is normal on [0,1). Then, 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is bounded if and only if 𝑀=sup_{𝑧∈𝔹𝑛}𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²<∞. (2.2)

Proof. Assume that 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is bounded. For 𝑎∈𝔹_𝑛, set𝑓_𝑎𝑒(𝑧)=log1−⟨𝑧,𝑎⟩. (2.3)It is easy to see that 𝑓_𝑎∈𝐻^∞_log and sup_{𝑎∈𝔹𝑛}‖𝑓_𝑎‖_𝐻^∞_log<∞.
For any 𝑏∈𝔹_𝑛, we have∞>𝑢𝐶_𝜑𝑓_{𝜑(𝑏)𝐻∞𝜇}=sup_{𝑧∈𝔹𝑛}𝜇|𝑧|𝑢𝐶_𝜑𝑓_𝜑(𝑏)(𝑧)=sup_{𝑧∈𝔹𝑛}𝜇|𝑧||𝑢(𝑧)||𝑓_𝜑(𝑏)|𝑒𝜑(𝑧)≥𝜇|𝑏||𝑢(𝑏)|log1−|𝜑(𝑏)|², (2.4)which implies (2.2).
Conversely, assume that (2.2) holds. Then, for any 𝑓∈𝐻^∞_log, we have𝜇|𝑧|𝑢𝐶_𝜑𝑓𝑓𝑒(𝑧)=𝜇|𝑧|𝑢(𝑧)𝜑(𝑧)≤𝜇|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²‖𝑓‖_𝐻^∞_log. (2.5)Taking the supremum in (2.5) over 𝔹_𝑛 and using condition (2.2), the boundedness of the operator 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 follows, as desired.

Theorem 2.4. Assume that 𝑢∈𝐻(𝔹_𝑛), 𝜑 is a holomorphic self-map of 𝔹_𝑛, and 𝜇 is a normal function on [0,1). Then, 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is compact if and only if 𝑢∈𝐻^∞_𝜇 and lim_{|𝜑(𝑧)|→1}𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²=0. (2.6)

Proof. Assume that 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is compact. Then, it is obvious that 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is bounded. Taking the function 𝑓(𝑧)=1∈𝐻^∞_log, we see that 𝑢∈𝐻^∞_𝜇. Let (𝜑(𝑧_𝑘))_𝑘∈ℕ be a sequence in 𝔹_𝑛 such that lim_𝑘⇀∞|𝜑(𝑧_𝑘)|=1. Set𝑓_𝑘𝑒(𝑧)=log1−𝑧,𝜑𝑧_𝑘²𝑒log1−𝜑𝑧_𝑘²⁻¹,𝑘∈ℕ. (2.7)It is easy to see that sup_𝑘∈ℕ‖𝑓_𝑘‖_𝐻^∞_log<∞. Moreover, 𝑓_𝑘→0 uniformly on compact subsets of 𝔹_𝑛 as 𝑘→∞. By Lemma 2.1,lim_𝑘→∞𝑢𝐶_𝜑𝑓_{𝑘𝐻∞𝜇}=0. (2.8)We have𝑢𝐶_𝜑𝑓_{𝑘𝐻∞𝜇}=sup_{𝑧∈𝔹𝑛}𝜇|𝑧|𝑢(𝑧)𝑓_𝑘𝑧𝜑(𝑧)≥𝜇_𝑘𝑢𝑧_𝑘𝑒log1−|𝜑𝑧_𝑘|², (2.9)which together with (2.8) implies thatlim_𝑘→∞𝜇𝑧_𝑘𝑢𝑧_𝑘𝑒log1−𝜑𝑧_𝑘²=0. (2.10)This proves that (2.6) holds.
Conversely, assume that 𝑢∈𝐻^∞_𝜇 and (2.6) holds. From this, it follows that (2.2) holds; hence 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is bounded. In order to prove that 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is compact, according to Lemma 2.1, it suffices to show that if (𝑓_𝑘)_𝑘∈ℕ is a bounded sequence in 𝐻^∞_log converging to 0 uniformly on compact subsets of 𝔹_𝑛, thenlim_𝑘→∞𝑢𝐶_𝜑𝑓_{𝑘𝐻∞𝜇}=0. (2.11)Let (𝑓_𝑘)_𝑘∈ℕ be a bounded sequence in 𝐻^∞_log such that 𝑓_𝑘→0 uniformly on compact subsets of 𝔹_𝑛 as 𝑘→∞. By (2.6), we have that for any 𝜀>0, there is a constant 𝛿∈(0,1) such that𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²<𝜀 (2.12)whenever 𝛿<|𝜑(𝑧)|<1. Let 𝐾={𝑤∈𝔹_𝑛∶|𝑤|≤𝛿}. Equation (2.12) along with the fact that 𝑢∈𝐻^∞_𝜇 implies𝑢𝐶_𝜑𝑓_{𝑘𝐻∞𝜇}=sup_{𝑧∈𝔹𝑛}𝜇|𝑧|𝑢𝐶_𝜑𝑓_𝑘(𝑧)=sup_{𝑧∈𝔹𝑛}𝜇|𝑧|𝑢(𝑧)𝑓_𝑘≤𝜑(𝑧)sup_{{𝑧∈𝔹𝑛∶|𝜑(𝑧)|≤𝛿}}+sup_{{𝑧∈𝔹𝑛∶𝛿≤|𝜑(𝑧)|<1}}𝜇|𝑧|𝑢(𝑧)𝑓_𝑘𝜑(𝑧)≤‖𝑢‖_𝐻^∞_𝜇sup_𝑤∈𝐾𝑓_𝑘(𝑤)+sup_{{𝑧∈𝔹𝑛∶𝛿≤|𝜑(𝑧)|<1}}𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²𝑓_{𝑘𝐻∞log}≤‖𝑢‖_𝐻^∞_𝜇sup_𝑤∈𝐾𝑓_𝑘(𝑤)+𝐶𝜀. (2.13)Observe that 𝐾 is a compact subset of 𝔹_𝑛 so thatlim_𝑘→∞sup_𝑤∈𝐾𝑓_𝑘(𝑤)=0. (2.14)With the aid of the above inequality, we can deduce thatlimsup_𝑘→∞𝑢𝐶_𝜑𝑓_{𝑘𝐻∞𝜇}≤𝐶𝜀 (2.15)by letting 𝑘→∞. Since 𝜀 is an arbitrary positive number, it follows that the last limit is equal to zero. Therefore, 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is compact. The proof is complete.

Theorem 2.5. Assume that 𝑢∈𝐻(𝔹_𝑛), 𝜑 is a holomorphic self-map of 𝔹_𝑛, and 𝜇 is a normal function on [0,1). Then, 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇,0 is compact if and only if 𝑢∈𝐻^∞_𝜇,0 and lim_{|𝜑(𝑧)|→1}𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²=0. (2.16)

Proof. Assume that 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇,0 is compact. Then, it is clear that 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇 is compact, and hence (2.16) holds. In addition, taking the function given by 𝑓(𝑧)=1, we get 𝑢∈𝐻^∞_𝜇,0.
Conversely, suppose that 𝑢∈𝐻^∞_𝜇,0 and (2.16) holds. In the proof of the implication we follow the lines, for example, of the proof of Lemma 4.2 in [24]. From (2.16), it follows that for every 𝜀>0, there exists a 𝛿∈(0,1) such that𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²<𝜀 (2.17)when 𝛿<|𝜑(𝑧)|<1. From the assumption 𝑢∈𝐻^∞_𝜇,0, we have that for the above 𝜀, there exists an 𝑟∈(0,1) such that when 𝑟<|𝑧|<1,𝜇<𝜀|𝑧|𝑢(𝑧)log𝑒/1−𝛿². (2.18)Therefore, if 𝑟<|𝑧|<1 and 𝛿<|𝜑(𝑧)|<1, we obtain𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²<𝜀. (2.19)If |𝜑(𝑧)|≤𝛿 and 𝑟<|𝑧|<1, we have that𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²𝑒≤𝜇|𝑧|𝑢(𝑧)log1−𝛿²<𝜀. (2.20)Combining (2.19) with (2.20), we getlim_|𝑧|→1𝜇𝑒|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²=0. (2.21) On the other hand, from (1.5) we have that𝜇|𝑧|𝑢𝐶_𝜑𝑓𝑒(𝑧)≤𝜇|𝑧|𝑢(𝑧)log1−𝜑(𝑧)²‖𝑓‖_𝐻^∞_log. (1)Taking the supremum in the above inequality over all 𝑓∈𝐻^∞_log such that ‖𝑓‖_𝐻^∞_log≤1, then letting |𝑧|→1, by (2.21) it follows thatlim_|𝑧|→1sup_{‖𝑓‖𝐻∞log≤1}𝜇|𝑧|𝑢𝐶_𝜑(𝑓)(𝑧)=0. (2.22)From this and by employing Lemma 2.2, we see that 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_𝜇,0 is compact. The proof is complete.

Similar to the proofs of Theorems 2.3 and 2.4, we easily get the following two results. We also omit their proofs.

Theorem 2.6. Assume that 𝑢∈𝐻(𝔹_𝑛) and 𝜑 is a holomorphic self-map of 𝔹_𝑛. Then, the following statements hold.
(a) 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_log is bounded if and only ifsup_{𝑧∈𝔹𝑛}𝑢(𝑧)log𝑒/1−𝜑(𝑧)²log𝑒/1−|𝑧|²<∞. (2.23)
(b) 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞_log is compact if and only if 𝑢∈𝐻^∞_log andlim_{|𝜑(𝑧)|→1}𝑢(𝑧)log𝑒/1−𝜑(𝑧)²log𝑒/1−|𝑧|²=0. (2.24)

Theorem 2.7. Assume that 𝑢∈𝐻(𝔹_𝑛) and 𝜑 is a holomorphic self-map of 𝔹_𝑛. Then, the following statements hold.
(a) 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞ is bounded if and only ifsup_{𝑧∈𝔹𝑛}𝑒𝑢(𝑧)log1−𝜑(𝑧)²<∞. (2.25)
(b) 𝑢𝐶_𝜑∶𝐻^∞_log→𝐻^∞ is compact if and only if 𝑢∈𝐻^∞ andlim_{|𝜑(𝑧)|→1}𝑒𝑢(𝑧)log1−𝜑(𝑧)²=0. (2.26)

Acknowledgment

X. Fu is supported in part by the Natural Science Foundation of Guangdong Province (no. 73006147).