Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to 𝐹(𝑝,𝑞,𝑠) Space on the Unit Ball

Abstract

This paper characterizes the boundedness and compactness of the product of extended Cesàro operator and composition operator from Lipschitz space to 𝐹(𝑝,𝑞,𝑠) space on the unit ball of ℂ^𝑛.

1. Introduction

Let 𝔹 be the unit ball in the 𝑛-dimensional complex space ℂ^𝑛, the closure of 𝔹 will be written as 𝔹. By 𝑑𝑣 we denote the Lebesgue measure on 𝔹 normalized so that 𝑣(𝔹)=1 and by 𝑑𝜎 the normalized rotation invariant measure on the boundary 𝑆=𝜕𝔹 of 𝔹. Let 𝐻(𝔹) be the class of all holomorphic functions on 𝔹 and 𝑆(𝔹) the collection of all the holomorphic self-mappings of 𝔹. Denote by 𝐴(𝔹) the unit ball algebra of all continuous functions on 𝔹 that are holomorphic on 𝔹.

For 𝑓∈𝐻(𝔹), let ℜ𝑓(𝑧)=^𝑛_𝑗=1𝑧_𝑗𝜕𝑓𝜕𝑧_𝑗(𝑧)(1.1) be the radial derivative of 𝑓.

We recall that the 𝛼-Bloch space ℬ^𝛼 for 𝛼≥0 consists of all 𝑓∈𝐻(𝔹) such that ℬ_𝛼𝑓=sup_𝑧∈𝔹1−|𝑧|^2𝛼ℜ𝑓(𝑧)<∞.(1.2) The expression ℬ_𝛼(𝑓) defines a seminorm while the natural norm is given by ‖𝑓‖_ℬ^𝛼=|𝑓(0)|+ℬ_𝛼(𝑓). This norm makes ℬ^𝛼 into a Banach space. When 𝛼=1, ℬ₁=ℬ is the well known Bloch space.

For 𝛼∈(0,1), ℒ_𝛼(𝔹) denotes the holomorphic Lipschitz space of order 𝛼 which is the set of all 𝑓∈𝐻(𝔹) such that, for some 𝐶>0,𝑓(𝑧)−𝑓(𝑤)≤𝐶|𝑧−𝑤|^𝛼(1.3) for every 𝑧,𝑤∈𝔹. It is clear that each space ℒ_𝛼(𝔹) contains the polynomials and is contained in the ball algebra 𝐴(𝔹). It is well known that ℒ_𝛼(𝔹) is endowed with a complete norm ‖⋅‖_ℒ𝛼 that is given by𝑓_ℒ𝛼=𝑓(0)+sup_{𝑧≠𝑤;𝑧,𝑤∈𝔹}𝑓(𝑧)−𝑓(𝑤)|𝑧−𝑤|^𝛼.(1.4) See [1, 2] for more information of the Lipschitz spaces on 𝔹.

For 𝑎∈𝔹, let 𝑔(𝑧,𝑎)=log|𝜑_𝑎(𝑧)|⁻¹ be Green's function on 𝔹 with logarithmic singularity at 𝑎, where 𝜑_𝑎 is the Möbius transformation of 𝔹 with 𝜑_𝑎(0)=𝑎, 𝜑_𝑎(𝑎)=0, and 𝜑_𝑎=𝜑_𝑎⁻¹.

Let 0<𝑝,𝑠<∞, −𝑛−1<𝑞<∞, a function 𝑓∈𝐻(𝔹) is said to belong to 𝐹(𝑝,𝑞,𝑠) if (see, e.g., [3–5])𝑓^𝑝_{𝐹𝑝,𝑞,𝑠}=𝑓(0)^𝑝+sup_{𝑎∈𝔹𝔹}ℜ𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜐(𝑧)<∞.(1.5)

If 𝑋 is a Banach space of holomorphic functions on a domain Ω and if 𝜑 is a (holomorphic) self-map of Ω, the composition operator of symbol 𝜑 is defined by 𝐶_𝜑(𝑓)=𝑓∘𝜑. The study of composition operators consists in the comparison of the properties of the operator 𝐶_𝜑 with that of the function 𝜑 itself, which is called the symbol of 𝐶_𝜑. One can characterize boundedness and compactness of 𝐶_𝜑 and many other properties. We refer to the books in [6, 7] and to some recent papers in [4, 5, 8] to learn much more on this subject.

Let ℎ∈𝐻(𝔹), the following integral-type operator was first introduced in [9] 𝑇_ℎ𝑓(𝑧)=¹₀𝑓(𝑡𝑧)ℜℎ(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹),𝑧∈𝔹.(1.6) This operator is called generalized Cesàro operator. It has been well studied in many papers, see, for example, [3, 9–24] as well as the related references therein.

It is natural to discuss the product of extended Cesàro operator and composition operator. For ℎ∈𝐻(𝔹) and 𝜑∈𝑆(𝔹), the product can be expressed as𝑇_ℎ𝐶_𝜑𝑓(𝑧)=¹₀𝑓𝜑(𝑡𝑧)ℜℎ(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻(𝔹),𝑧∈𝔹.(1.7) It is interesting to characterize the boundedness and compactness of the product operator on all kinds of function spaces. Even on the disk of ℂ, some properties are not easily managed; see some recent papers in [18, 25–28].

Building on those foundations, the present paper continues this line of research and discusses the operator in high dimension. The remainder is assembled as follows: in Section 2, we state a couple of lemmas. In Section 3, we characterize the boundedness and compactness of the product 𝑇_ℎ𝐶_𝜑 of extended Cesàro operator and composition operator from Lipschitz spaces to 𝐹(𝑝,𝑞,𝑠) spaces on the unit ball of ℂ^𝑛.

Throughout the remainder of this paper, 𝐶 will denote a positive constant, the exact value of which will vary from one appearance to the next. The notation 𝐴≍𝐵 means that there is a positive constant 𝐶 such that 𝐵/𝐶≤𝐴≤𝐶𝐵.

2. Some Lemmas

To begin the discussion, let us state a couple of lemmas, which are used in the proofs of the main results.

Lemma 2.1. Suppose that 𝑓,ℎ∈𝐻(𝔹). Then, ℜ𝑇_ℎ𝐶_𝜑𝑓(𝑧)=𝑓𝜑(𝑧)ℜℎ(𝑧).(2.1)

Proof. The proof of this Lemma follows by standard arguments (see, e.g., [9, 29, 30]).

Lemma 2.2 (see [2, 31]). If 0<𝛼<1, then ℬ^1−𝛼=ℒ_𝛼(𝔹); furthermore, 𝑓_𝐵1−𝛼≍𝑓_ℒ𝛼(2.2) as 𝑓 varies through ℒ_𝛼(𝔹).

The following criterion for compactness follows from standard arguments similar to the corresponding lemma in [6]. Hence, we omit the details.

Lemma 2.3. Assume that ℎ∈𝐻(𝔹) and 𝜑∈𝑆(𝔹). Suppose that 𝑋 or 𝑌  is one of the following spaces ℒ_𝛼(𝔹), 𝐹(𝑝,𝑞,𝑠). Then, 𝑇_ℎ𝐶_𝜑∶𝑋→𝑌 is compact if and only if 𝑇_ℎ𝐶_𝜑∶𝑋→𝑌 is bounded, and for any bounded sequence {𝑓_𝑘}_k∈𝑁 in 𝑋 which converges to zero uniformly on compact subsets of 𝔹 as 𝑘→∞, one has ‖𝑇_ℎ𝐶_𝜑𝑓_𝑘‖_𝑌→0 as 𝑘→∞.

Lemma 2.4 (see [4, 5]). If 𝑓∈ℬ^𝛼, then 𝑓(𝑧)≤𝐶𝑓_𝐵𝑎,0<𝛼<1,(2.3)𝑓(𝑧)≤𝐶𝑓_𝐵𝑎𝑒ln1−|𝑧|²,𝛼=1,(2.3')𝑓(𝑧𝑓)≤𝐶^𝐵𝑎1−|𝑧|^2𝛼−1,𝛼>1.(2.3'')

The next lemma was obtained in [32].

Lemma 2.5. If 𝑎>0, 𝑏>0, then the elementary inequality holds (𝑎+𝑏)^𝑝≤𝑎^𝑝+𝑏^𝑝2,0<𝑝<1,_𝑝−1𝑎^𝑝+𝑏^𝑝,𝑝≥1.(2.4)

It is obvious that Lemma 2.5 holds for the sum of finite number 𝑘, that is, 𝑎₁+⋯+𝑎_𝑘^𝑝≤𝐶𝑎^𝑝₁+⋯+𝑎^𝑝_𝑘,(2.5) where 𝑎₁,…,𝑎_𝑘>0 and 𝐶 is a positive constant.

Lemma 2.6 (see [4, 5]). For 0<𝑝, 𝑠<+∞, −𝑛−1<𝑞<+∞, 𝑞+𝑠>−1, there exists 𝐶>0 such that sup_{𝑎∈𝔹𝔹}1−|𝑤|^2𝑝|1−⟨𝑧,𝑤⟩|^{𝑛+1+𝑞+𝑝}1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤𝐶(2.6) for every 𝜔∈𝔹.

Lemma 2.7 (see [4]). There is a constant 𝐶>0 so that, for all 𝑡>−1 and 𝑧∈𝔹, one has _𝔹1ln1−⟨𝑧,𝑤⟩²1−|𝑤|^2𝑡|1−⟨𝑧,𝑤⟩|^𝑛+1+𝑡(𝑧)1𝑑𝜈≤𝐶ln1−|𝑧|²².(2.7)

Lemma 2.8 (see [4, 5]). Suppose that 0<𝑝, 𝑠<∞, −𝑛−1<𝑞<∞, and 𝑞+𝑠>−1. If 𝑓∈𝐹(𝑝,𝑞,𝑠), then 𝑓∈ℬ^{(𝑛+1+𝑞)/𝑝}, and ‖𝑓‖_ℬ^{(𝑛+1+𝑞)/𝑝}≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}.

Lemma 2.9. Let {𝑓_𝑘}_𝑘∈𝑁 be a bounded sequence in 𝐹(𝑝,𝑞,𝑠) which converges to zero uniformly on compact subsets of the unit ball 𝔹, where (𝑛+1+𝑞)/𝑝<1. Then, lim_𝑘→∞sup_𝑧∈𝔹|𝑓_𝑘(𝑧)|=0.

Proof. It follows from Lemma 2.8 that 𝐹(𝑝,𝑞,𝑠)⊆ℬ^{(𝑛+1+𝑞)/𝑝} and ‖𝑓‖_ℬ^{(𝑛+1+𝑞)/𝑝}≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)} for any 𝑓∈𝐹(𝑝,𝑞,𝑠). So, when (𝑛+1+𝑞)/𝑝<1, the proof of this lemma is similar to that of Lemma 3.6 of [33], hence the proof is omitted.

3. The Boundedness and Compactness of the Operator 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼(𝔹)→𝐹(𝑝,𝑞,𝑠)

Theorem 3.1. Assume that 𝛼∈(0,1), 0<𝑝, 𝑠<∞,  −𝑛−1<𝑞<∞, 𝑞+𝑠>−1, 𝜑∈𝑆(𝔹), and ℎ∈𝐻(𝔹). Then, 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded if and only if ℎ∈𝐹(𝑝,𝑞,𝑠).

Proof. Assume that ℎ∈𝐹(𝑝,𝑞,𝑠). Since 0<1−𝛼<1, by Lemmas 2.2 and 2.4, for any 𝑓∈ℒ_𝛼, we have 𝑓(𝑧)≤𝐶𝑓_ℬ1−𝛼≤𝐶𝑓_ℒ𝛼.(3.1) Since |𝑇_ℎ𝐶_𝜑𝑓(0)|=0, by using Lemma 2.1 and relations (2.3) and (3.1), we have 𝑇_ℎ𝐶_𝜑𝑓^𝑝_{𝐹𝑝,𝑞,𝑠}=sup_{𝑎∈𝔹𝔹}𝑓𝜑(𝑧)ℜℎ(𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤𝐶sup_{𝑎∈𝔹𝔹}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)𝑓^𝑝_ℬ1−𝛼≤𝐶‖ℎ‖^𝑝_{𝐹𝑝,𝑞,𝑠}𝑓^𝑝_ℒ𝛼<∞.(3.2) Thus 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded.
Conversely, suppose that 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded. Taking the function 𝑓(𝑧)=1∈ℒ_𝛼, then 𝑇_ℎ𝐶_𝜑𝑓^𝑝_{𝐹𝑝,𝑞,𝑠}=𝑇_ℎ𝐶_𝜑𝑓(0)^𝑝+sup_{𝑎∈𝔹𝔹}ℜ𝑇_ℎC_𝜑𝑓(𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)=sup_{𝑎∈𝔹𝔹}𝑓𝜑(𝑧)ℜℎ(𝑧)^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)=sup_{𝑎∈𝔹𝔹}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)=‖ℎ‖^𝑝_{𝐹(𝑝,𝑞,𝑠)}.(3.3) From which, the boundedness of 𝑇_ℎ𝐶_𝜑 implies that ℎ∈𝐹(𝑝,𝑞,𝑠). This completes the proof of this theorem.

Next, we characterize the compactness of 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠).

Theorem 3.2. Assume that 𝛼∈(0,1), 0<𝑝, 𝑠<∞, −𝑛−1<𝑞<∞, 𝑞+𝑠>−1, 𝜑∈𝑆(𝔹), and ℎ∈𝐻(𝔹). Then, 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is compact if and only if 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded, and lim_𝑟→1sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)=0.(3.4)

Proof. Assume that 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded and (3.4) holds. It follows from Theorem 3.1 that ℎ∈𝐹(𝑝,𝑞,𝑠).
Now, let {𝑓_𝑗}_𝑗∈𝑁 be a bounded sequence of functions in ℒ_𝛼 such that 𝑓_𝑗→0 uniformly on the compact subsets of 𝔹 as 𝑗→∞. Suppose that sup_𝑗∈𝑁‖𝑓_𝑗‖_ℒ𝛼≤𝐿. It follows from (3.4) that, for any 𝜀>0, there exists 𝑟₀∈(0,1) such that, for every 𝑟₀<𝑟<1, sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)<𝜀.(3.5) Set 𝑟₀<𝑟<1, then 𝑇_ℎ𝐶_𝜑𝑓_𝑗^𝑝_{𝐹𝑝,𝑞,𝑠}=sup_{𝑎∈𝔹𝔹}𝑓_𝑗𝜑(𝑧)^𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤sup_{𝑎∈𝔹{|𝜑(𝑧)|≤𝑟}}𝑓_𝑗𝜑(𝑧)^𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)+sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}𝑓_𝑗𝜑(𝑧)^𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)=𝐼₁+𝐼₂,(3.6) where 𝐼₁∶=sup_{𝑎∈𝔹{|𝜑(𝑧)|≤𝑟}}𝑓_𝑗𝜑(𝑧)^𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠()(𝑧),𝐼𝑧,𝑎𝑑𝜈₂∶=sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}𝑓_𝑗𝜑(𝑧)^𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧).(3.7)
Let 𝐾={𝑤∶|𝑤|≤𝑟}, then 𝐾 is a compact subset of 𝔹. Since 𝑓_𝑗→0 uniformly on compact subsets of 𝔹 as 𝑗→∞ and ℎ∈𝐹(𝑝,𝑞,𝑠), we get 𝐼₁≤sup_𝑤∈𝐾𝑓_𝑗(𝑤)^𝑝sup_{𝑎∈𝔹{|𝜑(𝑧)|≤𝑟}}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠()(𝑧)≤‖ℎ‖𝑧,𝑎𝑑𝜈_{𝑝𝐹𝑝,𝑞,𝑠}sup_𝑤∈𝐾𝑓_𝑗(𝑤)^𝑝≤𝐶sup_𝑤∈𝐾𝑓_𝑗(𝑤)^𝑝⟶0,𝑗⟶∞.(3.8)
On the other hand, by (3.5) and Lemmas 2.2 and 2.4, it follows that 𝐼₂≤𝐶𝑓_𝑗^𝑝_ℬ1−𝛼sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤𝐶‖𝑓_𝑗‖^𝑝_ℒ𝛼𝜀≤𝐶𝐿^𝑝𝜀.(3.9) Since 𝜀 is arbitrary, from the above inequalities, we get lim_𝑗→∞𝑇_ℎ𝐶_𝜑𝑓_{𝑗𝐹(𝑝,𝑞,𝑠)}=0.(3.10) Hence, by (3.10) and Lemma 2.3, we conclude that 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is compact.
For the converse direction, we suppose that 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is compact. It is obvious that 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded.
Now, we prove (3.4). Setting the test functions 𝑓_𝑙^(𝑚)(𝑧)=𝑧^𝑚_𝑙 for fixed 𝑙∈{1,…,𝑛}, where 𝑧=(𝑧₁,…,𝑧_𝑛) and 𝑚=1,2,…. It is easy to check that ‖𝑓_𝑙^(𝑚)‖_ℒ𝛼≤𝐶, and 𝑓_𝑙^(𝑚)→0 uniformly on the compact subsets of 𝔹 as 𝑚→∞. Write 𝜑=(𝜑₁,…,𝜑_𝑛), since 𝑇_ℎ𝐶_𝜑∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is compact, by Lemma 2.3, it follows that, as 𝑚→∞, 𝑇_ℎ𝐶_𝜑𝑓_𝑙^(𝑚)𝑝_{𝐹𝑝,𝑞,𝑠}=sup_{𝑎∈𝔹𝔹}𝜑_𝑙(𝑧)^𝑚𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)⟶0.(3.11) Note that |𝜑(𝑧)|²=|𝜑₁(𝑧)|²+⋯+|𝜑_𝑛(𝑧)|²≤(|𝜑₁(𝑧)|+⋯+|𝜑_𝑛(𝑧)|)²; by the relation (3.11) and Lemma 2.5, we have sup_{𝑎∈𝔹𝔹}𝜑(𝑧)^𝑚𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤sup_{𝑎∈𝔹𝔹}^𝑛_𝑙=1𝜑_𝑙(𝑧)^𝑚𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤𝐶sup_{𝑎∈𝔹𝔹}^𝑛_𝑙=1𝜑_𝑙(𝑧)^𝑚𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)⟶0,𝑚⟶∞.(3.12) This means that, for every 𝜀>0, there is 𝑚₀∈𝑁 such that, for every 𝑟∈(0,1), 𝑟^𝑚0𝑝sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)=sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}𝑟^𝑚0𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}𝜑(𝑧)^𝑚0𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)≤sup_{𝑎∈𝔹𝔹}𝜑(𝑧)^𝑚0𝑝|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)<𝜀.(3.13) Thus, when 𝑟>2^{−(1/𝑚0𝑝)}, by the above inequality, we obtain sup_{𝑎∈𝔹{|𝜑(𝑧)|>𝑟}}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)𝑑𝜈(𝑧)<2𝜀.(3.14) From which, the desired result (3.4) holds. This completes the proof of this theorem.

Remark 3.3. When 𝜑(𝑧)=𝑧, the product of extended Cesàro operator 𝑇_ℎ𝐶_𝜑 is the generalized extended Cesàro operator 𝑇_ℎ; thus, by Theorems 3.1 and 3.2, we have the following two corollaries.

Corollary 3.4. Assume that 𝛼∈(0,1), 0<𝑝, 𝑠<∞, −𝑛−1<𝑞<∞, 𝑞+𝑠>−1, and ℎ∈𝐻(𝔹). Then, 𝑇_ℎ∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded if and only if ℎ∈𝐹(𝑝,𝑞,𝑠).

Corollary 3.5. Assume that 𝛼∈(0,1), 0<𝑝, 𝑠<∞, −𝑛−1<𝑞<∞, 𝑞+𝑠>−1, and ℎ∈𝐻(𝔹). Then, 𝑇_ℎ∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is compact if and only if 𝑇_ℎ∶ℒ_𝛼→𝐹(𝑝,𝑞,𝑠) is bounded, and lim_𝑟→1sup_{𝑎∈𝔹|𝑧|>𝑟}|ℜℎ(𝑧)|^𝑝1−|𝑧|^2𝑞𝑔^𝑠(𝑧,𝑎)=0.(3.15)

Acknowledgments

The authors would like to thank the editor and referees for carefully reading the paper and providing corrections and suggestions for improvements. Z.-H. Zhou was supported in part by the National Natural Science Foundation of China (Grant nos. 10971153 and 10671141).