The Multiplication Operator from 𝐹(𝑝,𝑞,𝑠) Spaces to 𝑛th Weighted-Type Spaces on the Unit Disk

Abstract

Let 𝐻(𝔻) be the space of analytic functions on 𝔻 and 𝑢∈𝐻(𝔻). The boundedness and compactness of the multiplication operator 𝑀_𝑢 from 𝐹(𝑝,𝑞,𝑠),(or𝐹₀(𝑝,𝑞,𝑠)) spaces to 𝑛th weighted-type spaces on the unit disk are investigated in this paper.

1. Introduction

Let 𝐻(𝔻) denote the space of all analytic functions in the open unit disc 𝔻 of the finite-complex plane ℂ, 𝜕𝔻 the boundary of 𝔻, ℕ₀ the set of all nonnegative integers and ℕ the set of all positive integers. Let 𝜇(𝑧) be a positive continuous function on 𝔻 (weight) such that 𝜇(𝑧)=𝜇(|𝑧|) and 𝑛∈ℕ₀. The 𝑛th weighted-type spaces on the unit disk 𝔻, denoted by 𝒲^𝜇(𝑛)(𝔻) which were introduced in [1], consist of all 𝑓∈𝐻(𝔻) such that𝑏_𝒲^𝜇(𝑛)_(𝔻)(𝑓)=sup_𝑧∈𝔻𝜇(𝑧)||𝑓^(𝑛)(𝑧)||<∞.(1.1) For 𝑛=0, the space becomes the weighted-type space 𝐻^∞𝜇(𝔻) (see, e.g., [2–4]), for 𝑛=1 the Bloch-type space ℬ_𝜇(𝔻) and for 𝑛=2 the Zygmund-type space 𝒵_𝜇(𝔻). For 𝜇(𝑧)=1−|𝑧|², we obtain correspondingly the classical weighted-type space, the Bloch space ℬ(𝔻)=ℬ, and the Zygmund space 𝒵(𝔻)=𝒵. Some information on Zygmund-type spaces on the unit disk and some operators on them, for example, in [5–9] and on the unit ball, can be found, for example, in [10, 11]. From now on, we will assume that 𝑛∈ℕ. Set‖𝑓‖_𝒲^𝜇(𝑛)_(𝔻)=^𝑛−1_𝑗=0||𝑓^(𝑗)(0)||+𝑏_𝒲^𝜇(𝑛)_(𝔻)(𝑓).(1.2) With this norm, the 𝑛th weighted-type space becomes a Banach space.

The little 𝑛th weighted-type space, denoted by 𝒲^(𝑛)𝜇,0(𝔻), is a closed subspace of 𝒲^𝜇(𝑛)(𝔻) consisting of those 𝑓 for whichlim_|𝑧|→1𝜇(𝑧)||𝑓^(𝑛)(𝑧)||=0.(1.3)

A positive continuous function 𝜙 on [0,1) is called a normal if there exist positive numbers 𝑎,𝑏,0<𝑎<𝑏, and 𝑡₀∈[0,1), such that𝜙(𝑡)1−𝑡²^𝑎decreasesfor𝑡₀≤𝑡<1andlim_𝑡→1−𝜙(𝑡)1−𝑡²^𝑎𝜙(𝑡)=0,1−𝑡²^𝑏increasesfor𝑡₀≤𝑡<1andlim_𝑡→1−𝜙(𝑡)1−𝑡²^𝑏=∞(1.4) (see [12]).

For 0<𝑝,𝑠<∞,−2<𝑞<∞, a function 𝑓∈𝐻(𝔻) is said to belong to the general function space 𝐹(𝑝,𝑞,𝑠)=𝐹(𝑝,𝑞,𝑠)(𝔻) if‖𝑓‖^{𝑝𝐹(𝑝,𝑞,𝑠)}=||𝑓(0)||^𝑝+sup_𝑧∈𝔻_𝔻||𝑓^(𝑧)||^𝑝1−|𝑧|²^𝑞1−||𝜑^𝑎(𝑧)||²^𝑠𝑑𝐴(𝑧)<∞,(1.5) where 𝜑_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧),𝑎∈𝔻. An 𝑓∈𝐻(𝔻) is said to belong to 𝐹₀(𝑝,𝑞,𝑠)=𝐹₀(𝑝,𝑞,𝑠)(𝔻) iflim_|𝑎|→1_𝔻||𝑓^(𝑧)||^𝑝1−|𝑧|²^𝑞1−||𝜑^𝑎(𝑧)||²^𝑠𝑑𝐴(𝑧)=0.(1.6) The space 𝐹(𝑝,𝑞,𝑠) was introduced by Zhao in [13]. We can get many function spaces if we take some specific parameters of 𝑝,𝑞,𝑠; for example (see [13]), 𝐹(𝑝,𝑞,𝑠)=ℬ^{(2+𝑞)/𝑝} and 𝐹₀(𝑝,𝑞,𝑠)=ℬ^{0(2+𝑞)/𝑝} for 𝑠>1; 𝐹(𝑝,𝑞,𝑠)⊂ℬ^{(2+𝑞)/𝑝} and 𝐹₀(𝑝,𝑞,𝑠)⊂ℬ^{0(2+𝑞)/𝑝} for 0<𝑠≤1; 𝐹(2,0,𝑠)=𝑄_𝑠 and 𝐹₀(2,0,𝑠)=𝑄_𝑠,0; 𝐹(2,0,1)=BMOA and 𝐹₀(2,0,1)=VMOA. Since for 𝑞+𝑠≤−1, 𝐹(𝑝,𝑞,𝑠) is the space of constant functions, we assume that 𝑞+𝑠>−1.

The multiplication operator 𝑀_𝑢 is defined by 𝑀_𝑢𝑓=𝑢𝑓. It is interesting to provide a function theoretic characterization of when 𝑢 induces a bounded or compact composition operator on various spaces (see, e.g., [14–20]). Yu and Liu in [21] studied the boundedness and compactness of the operator 𝐷𝑀_𝑢 from mixed-norm spaces to the Bloch-type space. Stević in [22] studied the boundedness and compactness of the product of the differentiation and composition operator from the space of bounded analytic functions, the Bloch space and the little Bloch space to 𝑛th weighted-type spaces on the unit disk. Zhang and Xiao in [23] studied the bounded and compact-weighted composition operator 𝑢𝐶_𝜑 from the 𝐹(𝑝,𝑞,𝑠) space to the Bloch-type space in the unit disc. Zhang and Zeng in [24] studied the boundedness and compactness of weighted differentiation composition operators from weighted bergman space to 𝑛th weighted-type spaces on the unit disk. Ye in [25] studied the boundedness and compactness of the weighted composition operator 𝑢𝐶_𝜑 from 𝐹(𝑝,𝑞,𝑠) into the logarithmic Bloch space ℬ_log on the unit disk. Yang in [26] studied the boundedness and compactness of weighted differentiation composition operators from the 𝐹(𝑝,𝑞,𝑠) space to the Bloch-type space. Yang in [27] studied the boundedness and compactness of the composition operator from the 𝐹(𝑝,𝑞,𝑠) space to 𝑛th weighted-type spaces on the unit disk. Zhou and Chen in [28] studied the weighted composition operator from the 𝐹(𝑝,𝑞,𝑠) space to the Bloch-type space in the unit ball. Zhu in [29] studied the weighted composition operator from the 𝐹(𝑝,𝑞,𝑠) space to 𝐹^∞𝜇 space in the unit ball. Stević in [30, 31] studied the boundedness and compactness of the integral operators between 𝐹(𝑝,𝑞,𝑠) spaces and Bloch-type spaces in the unit ball. This paper focuses on the boundedness and compactness of the operators 𝑀_𝑢 from 𝐹(𝑝,𝑞,𝑠)(or𝐹₀(𝑝,𝑞,𝑠)) to 𝑛th weighted-type spaces on the unit disk.

From now on, we will always assume that 0<𝑝, 𝑠<∞, −2<𝑞<∞, 𝑞+𝑠>−1, 𝜇(𝑧)=𝜇(|𝑧|) is normal and 𝑛∈ℕ. Further, for the sake of simplicity, 𝐶 will always denote an independent constant, which can be different from one display to another.

2. Auxiliary Results

In this section we formulate some auxiliary results which will be used in the proofs of the main results.

Lemma 2.1 (see [13, 27]). Assume that 𝑓∈𝐹(𝑝,𝑞,𝑠). Then, for each 𝑛∈ℕ, there is a positive constant 𝐶, independent of 𝑓 such that ‖𝑓‖_ℬ^{(2+𝑞)/𝑝}≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)} and ||𝑓^(𝑛)(𝑧)||≤𝐶‖𝑓‖^{𝐹(𝑝,𝑞,𝑠)}1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛},𝑧∈𝔻.(2.1) Moreover, if 𝑓∈𝐹₀(𝑝,𝑞,𝑠), then 𝑓∈ℬ^{0(2+𝑞)/𝑝}.

Lemma 2.2 (see [32, 33]). Let 𝛼>0 and 𝑓∈ℬ^𝛼. Then, ||𝑓(𝑧)||≤⎧⎪⎪⎨⎪⎪⎩𝐶‖𝑓‖^ℬ𝛼𝐶log,2if0<𝛼<1,1−|𝑧|²𝐶‖𝑓‖_ℬ^𝛼,if𝛼=1,(𝛼−1)1−|𝑧|²^𝛼−1‖𝑓‖_ℬ^𝛼,if𝛼>1.(2.2)

Lemma 2.3 (see [1]). Assume 𝑎>0 and 𝐷_𝑛(𝑎)=||||||||||||||||𝑎(𝑎⋮𝑎1+1)(𝑎+(𝑎1)⋮+1(𝑎1)+2)⋱⋯⋯⋯(𝑎+(𝑎𝑛+−⋮_𝑛−21𝑛1)−(𝑎1)+𝑛)_𝑗=0(𝑎+𝑗)^𝑛−2_𝑗=0(𝑎+𝑗+1)⋯^𝑛−2_𝑗=0(𝑎+𝑗+𝑛−1)||||||||||||||||.(2.3) Then, 𝐷_𝑛(𝑎)=∏^{𝑛−1𝑗=0}𝑗!.

By standard arguments (see, e.g., [34] or Lemma  3 in [35]) the following lemma follows.

Lemma 2.4. Assume that 𝑢∈𝐻(𝔻). Then, 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)(𝑜𝑟𝐹₀(𝑝,𝑞,𝑠))→𝒲^𝜇(𝑛)(𝔻) is compact if and only if 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)(𝑜𝑟𝐹₀(𝑝,𝑞,𝑠))→𝒲^𝜇(𝑛)(𝔻) is bounded and for any bounded sequence {𝑓_𝑘} in 𝐹(𝑝,𝑞,𝑠)(𝑜𝑟𝐹₀(𝑝,𝑞,𝑠)) which converges to zero uniformly on compact subsets of 𝔻 as 𝑘→∞, one has ‖𝑀_𝑢𝑓_𝑘‖_𝒲^𝜇(𝑛)_(𝔻)→0 as 𝑘→∞.

3. The Boundedness and Compactness of 𝑀_𝑢 from 𝐹(𝑝,𝑞,𝑠) to 𝒲^𝜇(𝑛)(𝔻)

In this section, we characterize the boundedness and compactness of 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻).

Theorem 3.1. Assume that 𝑢∈𝐻(𝔻) and 𝜇 is normal.(a)If 2+𝑞>𝑝, then 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded if and only if sup_𝑧∈𝔻𝜇(|𝑧|)||𝑢(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛}<∞.(3.1)(b)If 2+𝑞<𝑝, then 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded if and only if (3.1) holds and 𝑢∈𝒲^𝜇(𝑛)(𝔻).(c)If 2+𝑞=𝑝,𝑠>1, then 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded if and only if (3.1) holds and sup_𝑧∈𝔻𝜇(|𝑧|)||𝑢^(𝑛)(𝑧)||log21−|𝑧|²<∞.(3.2)

Proof. Let 0<𝑝, 𝑠<∞,−2<𝑞<∞, 𝑞+𝑠>−1. Assume that conditions (3.1) holds. Then for all 𝑧∈𝔻||𝑢(𝑧)||≤𝐶1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛}𝜇(|𝑧|).(3.3) Since 12𝜋^02𝜋1||𝑒^𝑖𝜃−𝑧||²𝑑𝜃=11−|𝑧|²,∀𝑧∈𝔻,(3.4) let 𝛿_𝑧=(1+|𝑧|)/2, then we have |𝑧/𝛿_𝑧|=2|𝑧|/(1+|𝑧|)<1, so 12𝜋^02𝜋1||𝛿_𝑧𝑒^𝑖𝜃−𝑧||²𝑑𝜃=1𝛿^2𝑧−|𝑧|².(3.5) By the Cauchy integral formula and (3.3), we obtain ||𝑢^(𝑧)||=12𝜋|||||^02𝜋𝑢𝛿_𝑧𝑒^𝑖𝜃𝛿_𝑧𝑒^𝑖𝜃−𝑧²≤𝐶1−𝛿_2𝑧𝛿_𝑧𝑒^𝑖𝜃𝑑𝜃|||||^{((2+𝑞−𝑝)/𝑝)+𝑛}𝜇𝛿_𝑧21𝜋^02𝜋𝛿_𝑧||𝛿_𝑧𝑒^𝑖𝜃−𝑧||²=𝐶1−𝛿_2𝑧𝑑𝜃^{((2+𝑞−𝑝)/𝑝)+𝑛}𝜇𝛿_𝑧𝛿^2𝑧−𝛿^𝑧|𝑧|²≤𝐶1−𝛿^2𝑧^{((2+𝑞−𝑝)/𝑝)+𝑛}𝜇𝛿_𝑧1−1𝛿_𝑧.(3.6) Note that 12(1−|𝑧|)≤1−𝛿^2𝑧=1+𝛿_𝑧1−𝛿_𝑧≤(1−|𝑧|),(3.7) and 𝜇 are normal, we have ||𝑢^(𝑧)||≤𝐶1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−1}𝜇(|𝑧|),(3.8) hence sup_𝑧∈𝔻𝜇(|𝑧|)||𝑢′(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−1}<∞.(3.9) Similarly, for 𝑗∈{2,3,…,𝑛}, we have that sup_𝑧∈𝔻𝜇(|𝑧|)||𝑢^(𝑗)(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}<∞.(3.10) Hence, if 2+𝑞>𝑝, by Lemmas 2.1 and 2.2, the Leibnitz formula, (3.1), (3.9), and (3.10), we have that |||𝜇(|𝑧|)𝑀_𝑢=𝜇(|𝑧|𝑓)||(𝑢(^(𝑛)𝑧()𝑧𝑓)|||(𝑧))_(𝑛)=𝜇(|𝑧|)|||||_𝑛||_𝑗=0≤𝜇(|𝑧|)||𝑢_(𝑛)𝐶^𝑗𝑛𝑢^(𝑗)(𝑧)𝑓^{(𝑛−𝑗)}(𝑧)|||||(𝑧)𝑓(𝑧)||+𝐶^𝑛−1_𝑗=0||𝐶^𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶𝜇(|𝑧|)||𝑢^(𝑛)(𝑧)||1−|𝑧|²^{(2+𝑞−𝑝)/𝑝}‖𝑓‖_ℬ^{(2+𝑞)/𝑝}+𝐶^𝑛−1_𝑗=0||𝐶^𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)},(3.11) for every 𝑧∈𝔻 and 𝑓∈𝐹(𝑝,𝑞,𝑠), if 2+𝑞<𝑝, using 𝑢∈𝒲^𝜇(𝑛)(𝔻), we have that |||𝜇(|𝑧|)𝑀_𝑢≤𝐶𝜇(|𝑓𝑧|)||𝑢^(𝑛)(𝑧_(𝑛))|||(𝑧)||‖𝑓‖_ℬ^{(2+𝑞)/𝑝}+𝐶^𝑛−1_𝑗=0||𝐶^𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶‖𝑢‖_𝒲^𝜇(𝑛)_(𝔻)‖𝑓‖_ℬ^{(2+𝑞)/𝑝}+𝐶^𝑛−1_𝑗=0||𝐶^𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)},(3.12) for every 𝑧∈𝔻 and 𝑓∈𝐹(𝑝,𝑞,𝑠), if 2+𝑞=𝑝, by (3.2), we have that |||𝜇(|𝑧|)𝑀_𝑢≤𝐶𝜇(|𝑓𝑧|)||𝑢^(𝑛)(𝑧_(𝑛))|||(𝑧)||log21−|𝑧|²‖𝑓‖_ℬ+𝐶^𝑛−1_𝑗=0||𝐶^𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶‖𝑓‖_ℬ+𝐶^𝑛−1_𝑗=0||𝐶^𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)||1−|𝑧|²^𝑛−𝑗‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)},(3.13) for every 𝑧∈𝔻 and 𝑓∈𝐹(𝑝,𝑞,𝑠). We also have that ||𝑀_𝑢𝑓(0)||=||𝑢(0)𝑓(0)||≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)},(3.14) and for each 𝑠∈{1,2,…,𝑛−1}, |||𝑀_𝑢𝑓^(𝑠)(0)|||=|||||^𝑠_𝑗=0𝐶^𝑗𝑠𝑢^(𝑗)(0)𝑓^{(𝑠−𝑗)}(0)|||||≤𝐶^𝑠_𝑗=0||𝐶^𝑗𝑠𝑢^(𝑗)(0)||‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}.(3.15) Using (3.11), (3.12), (3.13), (3.14), and (3.15) it follows that the operator 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded.
On the other hand, suppose that 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded, that is, there exists a constant 𝐶 such that‖‖𝑀_𝑢𝑓‖‖_𝒲^𝜇(𝑛)_(𝔻)≤𝐶‖𝑓‖_{𝐹(𝑝,𝑞,𝑠)}(3.16) for all 𝑓∈𝐹(𝑝,𝑞,𝑠). Then, we can easily obtain 𝑢∈𝒲^𝜇(𝑛)(𝔻) by taking 𝑓(𝑧)=1. For a fixed 𝑤∈𝔻, and constants 𝑐₁,𝑐₂,…,𝑐_𝑛, set 𝑓_𝑤(𝑧)=^𝑛_𝑗=1𝑐_𝑗1−|𝑤|²^𝑗+11−𝑤𝑧^𝛼+𝑗,(3.17) where 𝛼=(2+𝑞)/𝑝. A straightforward calculation shows that, for 𝑙∈{1,…,𝑛}, 𝑓^𝑤(𝑙)(𝑧)=^𝑛_𝑗=1𝑐_𝑗𝑙−1_𝑘=0(𝛼+𝑗+𝑘)𝑤^𝑙1−|𝑤|²^𝑗+11−𝑤𝑧^{𝛼+𝑗+𝑙},(3.18) so 𝑓^𝑤(𝑙)(𝑤)=𝑤^𝑙1−|𝑤|²^{𝑛𝛼−1+𝑙}_𝑗=1𝑐_𝑗𝑙−1_𝑘=0(𝛼+𝑗+𝑘).(3.19) It is easy to see that 𝑓_𝑤∈𝐹(𝑝,𝑞,𝑠) for each 𝑤∈𝔻 and ‖𝑓_𝑤‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶 by using the same methods in [23]. By Lemma 2.3, using the same method in [22, 36], we can choose 𝑐₁,𝑐₂,…,𝑐_𝑛, the corresponding function is denoted by 𝑓_𝑤, such that 𝑓^𝑤(𝑛)(𝑤)=^𝑛_𝑗=1𝑐_𝑗𝑙−1_𝑘=0(𝛼+𝑗+𝑘)𝑤^𝑛1−|𝑤|²^{𝛼−1+𝑛},𝑓^𝑤(𝑙)(𝑤)=0,𝑙∈{0,1,…,𝑛−1}.(3.20) Therefore, 𝐶≥=‖‖𝑀_𝑢𝑓_𝑤‖‖_{𝒲𝜇(𝑛)(𝔻)}≥𝜇(|𝑤|)||𝑢(𝑤)𝑓^𝑤(𝑛)(𝑤)_𝑛||_𝑗=1𝑐_𝑗𝑙−1_𝑘=0(𝛼+𝑗+𝑘)𝜇(|𝑤|)|𝑤|^𝑛||𝑢(𝑤)||1−|𝑤|²^{((2+𝑞−𝑝)/𝑝)+𝑛}.(3.21) From this, we obtain sup_|𝑤|>1/2𝜇(|𝑤|)||𝑢(𝑤)||1−|𝑤|²^{((2+𝑞−𝑝)/𝑝)+𝑛}≤𝐶.(3.22) Since 𝜇 is normal and 𝑢∈𝐻(𝔻), we get sup_|𝑤|≤1/2𝜇(|𝑤|)||𝑢(𝑤)||1−|𝑤|²^{((2+𝑞−𝑝)/𝑝)+𝑛}≤𝐶sup_|𝑤|≤1/2𝜇(|𝑤|)||𝑢(𝑤)||≤𝐶,(3.23) which along with (3.22) implies that (3.1) is necessary for all case. Let 2+𝑞=𝑝,𝑠>1, and 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) be bounded. To prove (3.2), we set 𝑔_𝑤(𝑧)=𝐴log21−𝑤𝑧+𝐵log(2/(1−𝑤𝑧))²log2/1−|𝑤|²,𝑧,𝑤∈𝔻.(3.24) By a direct calculation, we get ||𝑔^𝑤(𝑧)||≤||𝐴||||1−𝑤𝑧||+2||𝐵log2/1−𝑤𝑧||||log2/1−|𝑤|²||||1−𝑤𝑧||≤||1−𝐶𝑤𝑧||,(3.25) thus we have 𝑔_𝑤∈𝐹(𝑝,𝑞,𝑠) and sup_𝑤∈𝔻‖𝑔_𝑤‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶<∞ (see [19, 30]). On the other hand, we have that 𝑔^𝑤(𝑧)=𝐴𝑤1−𝑤𝑧+2𝐵𝑤log2/1−𝑤𝑧𝑔log2/1−|𝑤|²1−𝑤𝑧_𝑤,(𝑧)=𝐴𝑤²1−𝑤𝑧²+2𝐵𝑤²log2/1−|𝑤|²1−𝑤𝑧²+2𝐵𝑤²log2/1−𝑤𝑧log2/1−|𝑤|²1−𝑤𝑧²𝑔_{𝑤},(𝑧)=2𝐴𝑤³1−𝑤𝑧³+6𝐵𝑤³log2/1−|𝑤|²1−𝑤𝑧³+2⋅2𝐵𝑤³log2/1−𝑤𝑧log2/1−|𝑤|²1−𝑤𝑧³𝑔_𝑤(4),(𝑧)=3⋅2𝐴𝑤⁴1−𝑤𝑧⁴+22𝐵𝑤⁴log2/1−|𝑤|²1−𝑤𝑧⁴+3!2𝐵𝑤⁴log2/1−𝑤𝑧log2/1−|𝑤|²1−𝑤𝑧⁴𝑔_𝑤(5),(𝑧)=4!𝐴𝑤⁵1−𝑤𝑧⁵+100𝐵𝑤⁵log2/1−|𝑤|²1−𝑤𝑧⁵+4!2𝐵𝑤⁵log2/1−𝑤𝑧log2/1−|𝑤|²1−𝑤𝑧⁵𝑔_𝑤(𝑚)⋮,(𝑧)=(𝑚−1)!𝐴𝑤^𝑚1−𝑤𝑧^𝑚+𝐴_𝑚𝐵𝑤^𝑚log2/1−|𝑤|²1−𝑤𝑧^𝑚+2(𝑚−1)!𝐵𝑤^𝑚log2/1−𝑤𝑧log2/1−|𝑤|²1−𝑤𝑧^𝑚.(3.26) Moreover, we have that 𝑔_𝑤(𝑤)=(𝐴+𝐵)log21−|𝑤|²𝑔_𝑤,(𝑤)=(𝐴+2𝐵)𝑤1−|𝑤|²𝑔_𝑤,(𝑤)=(𝐴+2𝐵)𝑤²1−|𝑤|²²+2𝐵𝑤²log2/1−|𝑤|²1−|𝑤|²²𝑔_{𝑤},(𝑤)=2(𝐴+2𝐵)𝑤³1−|𝑤|²³+6𝐵𝑤³log2/1−|𝑤|²1−|𝑤|²³𝑔_𝑤(4),(𝑤)=6(𝐴+2𝐵)𝑤⁴1−|𝑤|²⁴+22𝐵𝑤⁴log2/1−|𝑤|²1−|𝑤|²⁴𝑔_𝑤(5),(𝑤)=24(𝐴+2𝐵)𝑤⁵1−|𝑤|²⁵+100𝐵𝑤⁵log2/1−|𝑤|²1−|𝑤|²⁵𝑔_𝑤(𝑚)⋮,(𝑤)=(𝑚−1)!(𝐴+2𝐵)𝑤^𝑚1−|𝑤|²^𝑚+𝐴_𝑚𝐵𝑤^𝑚log2/1−|𝑤|²1−|𝑤|²^𝑚.(3.27) Taking 𝐴=2, 𝐵=−1, if 𝑛=1, we have 𝜇(|𝑤|)||||𝑢^(𝑤)log21−|𝑤|²||||≤‖‖𝑀_𝑢𝑔_𝑤‖‖_{𝒲𝜇(𝑛)(𝔻)}≤𝐶‖‖𝑀_𝑢‖‖<∞,(3.28) if 𝑛≠1, we have ‖‖𝑀_𝑢𝑔_𝑤‖‖_{𝒲𝜇(𝑛)(𝔻)}≥𝜇(|𝑤|)|||||^𝑛_𝑗=0≥𝜇(|𝑤|)||𝑢_(𝑛)𝐶^𝑗𝑛𝑢^(𝑗)(𝑤)𝑔^{𝑤(𝑛−𝑗)}(𝑤)|||||(𝑤)𝑔_𝑤(𝑤)||−𝜇(|𝑤|)|||||^𝑛−1_𝑗=0≥𝜇(|𝑤|)||𝑢_(𝑛)𝐶^𝑗𝑛𝑢^(𝑗)(𝑤)𝑔^{𝑤(𝑛−𝑗)}(𝑤)|||||×|||||(𝑤)𝑔_𝑤(𝑤)||−𝜇(|𝑤|)_𝑛−1_𝑗=0𝐶^𝑗𝑛𝑢^(𝑗)(𝑤)𝐴_𝑛−𝑗𝐵𝑤^𝑛−𝑗log2/1−|𝑤|²1−|𝑤|²^𝑛−𝑗|||||.(3.29) From (3.29), (3.1), (3.9), and (3.10) we have 𝜇(|𝑤|)||||𝑢^(𝑛)(𝑤)log21−|𝑤|²≤‖‖𝑀_𝑢||||𝑔_𝑤‖‖_{𝒲𝜇(𝑛)(𝔻)}+𝐶|||||^𝑛−1_𝑗=0𝜇(|𝑤|)𝑢^(𝑗)(𝑤)1−|𝑤|²^𝑛−𝑗≤𝐶‖‖𝑀_𝑢|||||‖‖+𝐶<∞.(3.30) Using (3.28) and (3.30), it is easy to get that (3.2) holds, finishing the proof of the theorem.

Theorem 3.2. Assume that 𝑢∈𝐻(𝔻) and 𝜇 is normal.(a)If 2+𝑞>𝑝, then 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is compact if and only if lim_|𝑧|→1𝜇(|𝑧|)||𝑢(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛}=0.(3.31)(b)If 2+𝑞<𝑝, then 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is compact if and only if (3.31) holds and 𝑢∈𝒲^𝜇(𝑛)(𝔻).(c)If 2+𝑞=𝑝,𝑠>1, then 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is compact if and only if (3.31) holds and lim_|𝑧|→1𝜇(|𝑧|)||𝑢^(𝑛)(𝑧)||log21−|𝑧|²=0.(3.32)

Proof. Assume that conditions (3.31) hold. By Theorem 3.1, 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded. For any bounded sequence {𝑓_𝑘} in 𝐹(𝑝,𝑞,𝑠) with 𝑓_𝑘→0 uniformly on compact subsets of 𝔻 establish the assertion, it suffices, in view of Lemma 2.4, to show that ‖‖𝑀_𝑢𝑓_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}⟶0as𝑘⟶∞.(3.33) We assume that ‖𝑓_𝑘‖_{𝐹(𝑝,𝑞,𝑠)}≤1. From (3.31) and (3.32), given 𝜖>0, there exists a 𝛿∈(0,1), when 𝛿<|𝑧|<1, we have 𝜇(|𝑧|)||𝑢(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛}<𝜖,(3.34)𝜇(|𝑧|)||𝑢^(𝑛)(𝑧)||log21−|𝑧|²<𝜖.(3.35) By (3.34) and the Cauchy integral formula, we obtain ||𝑢^(𝑧)||=12𝜋|||||^02𝜋𝑢𝛿_𝑧𝑒^𝑖𝜃𝛿_𝑧𝑒^𝑖𝜃−𝑧²<𝜖1−𝛿_2𝑧𝛿_𝑧𝑒^𝑖𝜃𝑑𝜃|||||^{((2+𝑞−𝑝)/𝑝)+𝑛}𝜇𝛿_𝑧21𝜋^02𝜋𝛿_𝑧||𝛿_𝑧𝑒^𝑖𝜃−𝑧||²=𝜖1−𝛿_2𝑧𝑑𝜃^{((2+𝑞−𝑝)/𝑝)+𝑛}𝜇𝛿_𝑧𝛿^2𝑧−𝛿^𝑧|𝑧|²≤𝜖𝐶1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−1}𝜇(|𝑧|),(3.36) when 𝛿<|𝑧|<1. Hence, 𝜇(|𝑧|)||𝑢^(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−1}<𝐶𝜖,(3.37) when 𝜏<|𝑧|<1. Similarly, for 𝑗∈{2,…,𝑛}, we have that 𝜇(|𝑧|)||𝑢^(𝑗)(𝑧)||1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}<𝐶𝜖,(3.38) when 𝛿<|𝑧|<1. Since 𝑓_𝑘→0 uniformly on compact subsets of 𝔻, Cauchy’s estimate gives that 𝑓^𝑘(𝑗) converges to 0 uniformly on compact subsets of 𝔻 for each 𝑗∈{1,…,𝑛}, there exists a 𝐾₀∈ℕ such that 𝑘>𝐾₀ implies that ^𝑛−1_{𝑗=0𝑗−1𝑚=0}𝐶^𝑚𝑗||𝑢^(𝑚)(0)𝑓^{𝑘(𝑗−𝑚)}(0)||+sup_{|𝑧|≤𝛿}𝜇(|𝑧|)|||||^𝑛_𝑗=0𝐶^𝑗𝑛𝑢^(𝑗)(𝑧)𝑓^{𝑘(𝑛−𝑗)}(𝑧)|||||<𝐶𝜖.(3.39) If 2+𝑞>𝑝, from (3.34), (3.38), (3.39), and Lemmas 2.1 and 2.2, we have ‖‖𝑀_𝑢𝑓_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}=^𝑛−1_𝑗=0|||𝑢𝑓_𝑘≤^(𝑗)(0)|||+sup_𝑧∈𝔻𝜇(|𝑧|)|||𝑢𝑓_𝑘^(𝑛)(𝑧)_𝑛−1|||_{𝑗=0𝑗−1𝑚=0}𝐶^𝑚𝑗||𝑢^(𝑚)(0)𝑓^{𝑘(𝑗−𝑚)}(0)||+sup_{|𝑧|≤𝛿}𝜇(|𝑧|)|||||^𝑛_𝑗=0+sup_|𝑧|>𝛿𝐶^𝑗𝑛𝑢^(𝑗)(𝑧)𝑓^{𝑘(𝑛−𝑗)}(𝑧)|||||𝜇(|𝑧|)|||||^𝑛_𝑗=0<𝐶𝜖+𝐶sup_|𝑧|>𝛿𝐶^𝑗𝑛𝑢^(𝑗)(𝑧)𝑓^{𝑘(𝑛−𝑗)}(𝑧)|||||||𝜇(|𝑧|)𝑢^(𝑛)(𝑧)𝑓_𝑘(𝑧)||+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}|||||‖‖𝑓_𝑘‖‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶𝜖+𝐶sup_|𝑧|>𝛿|||||𝜇(|𝑧|)𝑢_(𝑛)(𝑧)1−|𝑧|²^{(2+𝑞−𝑝)/𝑝}|||||‖‖𝑓_𝑘‖‖_{ℬ(2+𝑞)/𝑝}+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}≤𝐶𝜖+𝐶sup_|𝑧|>𝛿||||||||||𝜇(|𝑧|)𝑢_(𝑛)(𝑧)1−|𝑧|²^{(2+𝑞−𝑝)/𝑝}|||||+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}<2+𝐶_𝑛|||||_𝑗=0𝐶^𝑗𝑛𝐶𝜖,(3.40) when 𝑘>𝐾₀. If 2+𝑞<𝑝, then 𝑓_𝑘∈𝐹(𝑝,𝑞,𝑠)⊂ℬ^{(2+𝑞)/𝑝},𝑘∈ℕ. By [37, Lemma 3.2], we have lim_𝑘→∞sup_𝑧∈𝔻||𝑓_𝑘(𝑧)||=0.(3.41) By (3.38), (3.39), Lemma 2.1, and 𝑢∈𝒲^𝜇(𝑛)(𝔻), we get ‖‖𝑀_𝑢𝑓_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}=^𝑛−1_𝑗=0|||𝑢𝑓_𝑘<𝐶𝜖+𝐶sup^(𝑗)_|𝑧|>𝛿(0)|||+sup_𝑧∈𝔻𝜇(|𝑧|)|||𝑢𝑓_𝑘^(𝑛)(𝑧)|||||𝜇(|𝑧|)𝑢^(𝑛)(𝑧)𝑓_𝑘(𝑧)||+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}|||||‖‖𝑓_𝑘‖‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶𝜖+𝐶sup_𝑧∈𝔻||𝑓_𝑘(𝑧)||+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}≤𝐶𝜖+𝐶𝜖+𝐶sup_{|𝑧|>𝛿𝑛−1}|||||_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}<2+𝐶_𝑛|||||_𝑗=0𝐶^𝑗𝑛𝐶𝜖,(3.42) when 𝑘>𝐾₀. If 2+𝑞=𝑝, from (3.35), (3.38), (3.39), and Lemmas 2.1 and 2.2, we get ‖‖𝑀_𝑢𝑓_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}=^𝑛−1_𝑗=0|||𝑢𝑓_𝑘<𝐶𝜖+𝐶sup^(𝑗)_|𝑧|>𝛿(0)|||+sup_𝑧∈𝔻𝜇(|𝑧|)|||𝑢𝑓_𝑘^(𝑛)(𝑧)|||||𝜇(|𝑧|)𝑢^(𝑛)(𝑧)𝑓_𝑘(𝑧)||+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}|||||‖‖𝑓_𝑘‖‖_{𝐹(𝑝,𝑞,𝑠)}<𝐶𝜖+𝐶sup_|𝑧|>𝛿||𝜇(|𝑧|)𝑢^(𝑛)(𝑧)||log21−|𝑧|²‖‖𝑓_𝑘‖‖_ℬ+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^{((2+𝑞−𝑝)/𝑝)+𝑛−𝑗}≤𝐶𝜖+𝐶sup_|𝑧|>𝛿|||||||𝜇(|𝑧|)𝑢^(𝑛)(𝑧)||log21−|𝑧|²‖‖𝑓_𝑘‖‖_{𝐹(𝑝,𝑞,𝑠)}+𝐶sup_{|𝑧|>𝛿𝑛−1}_𝑗=0|||||𝐶_𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)1−|𝑧|²^𝑛−𝑗<2+𝐶_𝑛|||||_𝑗=0𝐶^𝑗𝑛𝐶𝜖,(3.43) when 𝑘>𝐾₀. It follows that the operator 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is compact.
Conversely, assume that 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is compact. Taking 𝑓=1, we get 𝑢∈𝒲^𝜇(𝑛)(𝔻). Let {𝑧_𝑘} be a sequence in 𝔻 such that |𝑧_𝑘|→1 as 𝑘→∞. Taking the test functions ℎ_𝑧𝑘, where ℎ_𝑧𝑘 is defined by (3.17), we will write ℎ_𝑘(𝑧)=𝑓_𝑧𝑘(𝑧).(3.44) We obtain that ℎ_𝑘∈𝐹(𝑝,𝑞,𝑠), sup_𝑘∈ℕ‖‖ℎ_𝑘‖‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶,(3.45) and for 𝑡∈{1,…,𝑛−1}, ℎ^𝑘(𝑛)𝑧_𝑘=^𝑛_𝑗=1𝑐_𝑗𝑙−1_𝑘=0(𝛼+𝑗+𝑘)𝑧_𝑘^𝑛1−||𝑧^𝑘||²^{((2+𝑞−𝑝)/𝑝)+𝑛},ℎ^𝑘(𝑡)𝑧_𝑘=0.(3.46) For |𝑧|=𝑟<1, we have ||ℎ_𝑘(𝑧)||=|||||||^𝑛_𝑗=1𝑐_𝑗1−||𝑧^𝑘||²^𝑗+11−𝑧_𝑘𝑧^𝛼+𝑗|||||||≤_𝑛_𝑗=1𝑐_𝑗1−||𝑧^𝑘||²^𝑗+1(1−𝑟)^𝛼+𝑗⟶0as𝑘⟶∞,(3.47) that is, ℎ_𝑘 converges to 0 uniformly on compact subsets of 𝔻, using the compactness of 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻), we obtain 𝜇𝑧_𝑘||𝑢𝑧_𝑘||||𝑧_𝑘||^𝑛1−||𝑧^𝑘||²^{((2+𝑞−𝑝)/𝑝)+𝑛}≤𝐶‖‖𝑀_𝑢ℎ_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}⟶0as𝑘⟶∞,(3.48) and consequently (3.31) holds.
Assume that 2+𝑞=𝑝,𝑠>1 and 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is compact, we only need to prove (3.32) holds. To do this, let {𝑧_𝑘} be a sequence in 𝔻 such that |𝑧_𝑘|→1 as 𝑘→∞. Set 𝑎_𝑘=log21−||𝑧_𝑘||²𝐽_𝑘,(𝑧)=𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧²+𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧³.(3.49) Since ||𝐽^𝑘(𝑧)||≤||||2𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘1−𝑧_𝑘𝑧||||+||||3𝑎𝐹^2𝑘log1−2𝑧_𝑘𝑧²𝑧_𝑘1−𝑧_𝑘𝑧||||≤||1−𝐶𝑧_𝑘𝑧||,(3.50) we have 𝐽_𝑘∈𝐹(𝑝,𝑞,𝑠) and sup_𝑘∈ℕ‖𝐽_𝑘‖_{𝐹(𝑝,𝑞,𝑠)}≤𝐶 (see [19, 30]), and 𝐽_𝑘→0 uniformly on compact subsets of 𝔻. Since 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is compact, we have ‖‖𝑀_𝑢𝐽_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}⟶0as𝑘⟶∞.(3.51) We also have that 𝐽^𝑘(𝑧)=2𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘1−𝑧_𝑘𝑧+3𝑎𝐹^2𝑘log1−2𝑧_𝑘𝑧²𝑧_𝑘𝐽_𝑘1−𝑧_𝑘𝑧,(𝑧)=2𝐸𝑎_𝑘𝑧_𝑘²1−𝑧_𝑘𝑧²+2𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘²1−𝑧_𝑘𝑧²+3⋅2𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘²1−𝑧_𝑘𝑧²+3𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧²𝑧_𝑘²1−𝑧_𝑘𝑧²𝐽_{𝑘},(𝑧)=2⋅2𝐸𝑎_𝑘𝑧_𝑘³1−𝑧_𝑘𝑧³+2𝐸𝑎_𝑘𝑧_𝑘³1−𝑧_𝑘𝑧³+4𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘³1−𝑧_𝑘𝑧³+3⋅2𝐹𝑎^2𝑘𝑧_𝑘³1−𝑧_𝑘𝑧³+3⋅2⋅2𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘³1−𝑧_𝑘𝑧³+3⋅2𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘³1−𝑧_𝑘𝑧³+6𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧²𝑧_𝑘³1−𝑧_𝑘𝑧³=6𝐸,𝑎_𝑘𝑧_𝑘³1−𝑧_𝑘𝑧³+4𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘³1−𝑧_𝑘𝑧³+6𝐹𝑎^2𝑘𝑧_𝑘³1−𝑧_𝑘𝑧³+18𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘³1−𝑧_𝑘𝑧³+6𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧²𝑧_𝑘³1−𝑧_𝑘𝑧³𝐽_𝑘(4),(𝑧)=22𝐸𝑎_𝑘𝑧_𝑘⁴1−𝑧_𝑘𝑧⁴+3!2𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘⁴1−𝑧_𝑘𝑧⁴+36𝐹𝑎^2𝑘𝑧_𝑘⁴1−𝑧_𝑘𝑧⁴+66𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘⁴1−𝑧_𝑘𝑧⁴+3!3𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧²𝑧_𝑘⁴1−𝑧_𝑘𝑧⁴𝐽_𝑘(𝑚)⋮,(𝑧)=𝑏^𝑚𝐸𝑎_𝑘𝑧_𝑘^𝑚1−𝑧_𝑘𝑧^𝑚+(𝑚−1)!2𝐸𝑎_𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘^𝑚1−𝑧_𝑘𝑧^𝑚+𝑑^𝑚𝐹𝑎^2𝑘𝑧_𝑘^𝑚1−𝑧_𝑘𝑧^𝑚+𝑐^𝑚𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧𝑧_𝑘^𝑚1−𝑧_𝑘𝑧^𝑚+(𝑚−1)!3𝐹𝑎^2𝑘log1−2𝑧_𝑘𝑧²𝑧_𝑘^𝑚1−𝑧_𝑘𝑧^𝑚.(3.52) Therefore, 𝐽𝐽_𝑘𝑘𝑧_𝑘=(𝐸+𝐹)𝑎_𝑘,𝑧_𝑘=(2𝐸+3𝐹)𝑧_𝑘1−||𝑧_𝑘||²𝐽_𝑘,𝑧_𝑘=(2𝐸+3𝐹)𝑧_𝑘²1−||𝑧^𝑘||²²+2(𝐸+3𝐹)𝑎_𝑘𝑧_𝑘²1−||𝑧^𝑘||²²𝐽_{𝑘},𝑧_𝑘=2(2𝐸+3𝐹)𝑧_𝑘³1−||𝑧^𝑘||²³+6(𝐸+3𝐹)𝑎_𝑘𝑧_𝑘³1−||𝑧^𝑘||²³+6𝐹𝑎^2𝑘𝑧_𝑘³1−||𝑧^𝑘||²³𝐽_𝑘(4),𝑧_𝑘=3!(2𝐸+3𝐹)𝑧_𝑘⁴1−||𝑧^𝑘||²⁴+22(𝐸+3𝐹)𝑎_𝑘𝑧_𝑘⁴1−||𝑧^𝑘||²⁴+36𝐹𝑎^2𝑘𝑧_𝑘⁴1−||𝑧^𝑘||²⁴𝐽_𝑘(𝑚)⋮,𝑧_𝑘=(𝑚−1)!(2𝐸+3𝐹)𝑧_𝑘^𝑚1−||𝑧^𝑘||²^𝑚+𝑏^𝑚𝐸+𝑐_𝑚𝐹𝑎_𝑘𝑧_𝑘^𝑚1−||𝑧^𝑘||²^𝑚+𝑑^𝑚𝐹𝑎^2𝑘𝑧_𝑘^𝑚1−||𝑧^𝑘||²^𝑚.(3.53) Taking 𝐸=3 and 𝐹=−2, if 𝑛=1, we have 𝜇||𝑧_𝑘|||||||𝑢^𝑧_𝑘log21−||𝑧_𝑘||²|||||≤‖‖𝑀_𝑢𝐽_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}⟶0as𝑘⟶∞,(3.54) if 𝑛≠1, we have ‖‖𝑀_𝑢𝐽_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}≥𝜇||𝑧_𝑘|||||||^𝑛_𝑗=0≥𝜇||𝑧_𝑘𝐶^𝑗𝑛𝑢^(𝑗)𝑧_𝑘𝐽^{𝑘(𝑛−𝑗)}𝑧_𝑘|||||||||𝑢^(𝑛)𝑧_𝑘𝐽_𝑘𝑧_𝑘||−𝜇||𝑧_𝑘|||||||^𝑛−1_𝑗=0≥𝜇||𝑧_𝑘𝐶^𝑗𝑛𝑢^(𝑗)𝑧_𝑘𝐽^{𝑘(𝑛−𝑗)}𝑧_𝑘||||||||||||𝑢^(𝑛)𝑧_𝑘log21−||𝑧_𝑘||²|||||−𝜇||𝑧_𝑘|||||||^𝑛−1_𝑗=0𝐶^𝑗𝑛𝑢^(𝑗)𝑧_𝑘𝐽^{𝑘(𝑛−𝑗)}𝑧_𝑘|||||.(3.55) From (3.31), (3.35), (3.37), and (3.38), we have that for sufficiently large 𝑘: 𝜇||𝑧_𝑘|||||||𝑢^(𝑛)𝑧_𝑘log21−||𝑧_𝑘||²≤‖‖𝑀_𝑢|||||𝐽_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}+𝜇||𝑧_𝑘|||||||^𝑛−1_𝑗=0≤‖‖𝑀_𝑢𝐶^𝑗𝑛𝑢^(𝑗)𝑧_𝑘𝐽^{𝑘(𝑛−𝑗)}𝑧_𝑘|||||𝐽_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}+𝐶^𝑛−1_𝑗=0⎛⎜⎜⎜⎝𝜇||𝑧_𝑘||||𝑢^(𝑗)𝑧_𝑘||𝑎_𝑘1−||𝑧^𝑘||²^𝑛−𝑗+𝜇||𝑧^𝑘||||𝑢^(𝑗)𝑧_𝑘||𝑎^2𝑘1−||𝑧^𝑘||²^𝑛−𝑗≤‖‖𝑀_𝑢⎞⎟⎟⎟⎠𝐽_𝑘‖‖_{𝒲𝜇(𝑛)(𝔻)}+𝐶^𝑛−1_𝑗=0𝜇||𝑧_𝑘||||𝑢^(𝑗)𝑧_𝑘||1−||𝑧^𝑘||²^𝑛−𝑗<𝐶𝜖.(3.56) Using (3.54) and (3.56), it is easy to get that (3.32) holds. From which we obtain the desired result.

4. The Boundedness of 𝑀_𝑢 from 𝐹(𝑝,𝑞,𝑠) (or𝐹₀(𝑝,𝑞,𝑠)) to 𝒲^(𝑛)𝜇,0(𝔻)

In this section, we characterize the boundedness of 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)(or𝐹₀(𝑝,𝑞,𝑠))→𝒲^(𝑛)𝜇,0(𝔻).

Theorem 4.1. Assume that 𝑢∈𝐻(𝔻) and 𝜇 is normal. Then,(1)𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^(𝑛)𝜇,0(𝔻) is bounded if and only if 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded and lim_|𝑧|→1𝜇(𝑧)||𝑢^(𝑛)(𝑧)||=0.(4.1)(2)𝑀_𝑢∶𝐹₀(𝑝,𝑞,𝑠)→𝒲^(𝑛)𝜇,0(𝔻) is bounded if and only if 𝑀_𝑢∶𝐹₀(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded and (4.1) holds.

Proof. (1) Assume that 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded and condition (4.1) holds. Since 𝑢(𝑧)−𝑢(0)=¹⁰𝑥𝑢^(𝑧𝑥)𝑑𝑥,(4.2) it follows that 𝜇(𝑧)||𝑢(𝑧)||≤≤𝜇𝜇((𝑧𝑧))||||𝑢𝑢((00))||||++𝜇𝜇((𝑧𝑧))||||||||¹⁰_01/2𝑥𝑢′(𝑧𝑥)𝑑𝑥||||≤𝜇(𝑧)||𝑢(0)||+𝜇(𝑧)max_|𝑧|≤1/2𝑥𝑢′(𝑧𝑥)𝑑𝑥||||+𝜇(𝑧)||||^11/2𝑥𝑢′(𝑧𝑥)𝑑𝑥||||||𝑢^(𝑧)||+𝜇(𝑧)||||^11/2𝑥𝑢′(𝑧𝑥)𝑑𝑥||||.(4.3) Since 𝜇 is normal, by the monotonicity of 𝜇(𝑡)/(1−𝑡²)^𝑎, for 𝑡₀≤𝑡₁<𝑡<1, we have 𝜇(𝑡)=1−𝑡²^𝑎𝜇(𝑡)1−𝑡²^𝑎≤1−𝑡²^𝑎𝜇𝑡₁1−𝑡²¹^𝑎<𝜇𝑡₁,(4.4) that is, 𝜇 is decreasing on [𝑡₀,1), and for any 𝜖>0, there is a 𝜏>0 such that 0<𝜇(|𝑧|)<𝜖1−|𝑧|²^𝑎,(𝜏<|𝑧|<1),(4.5) which implies lim_|𝑧|→1𝜇(|𝑧|)=0. Since for 1/2<𝑥<1 and 2𝑡₀<|𝑧|<1, we have 𝜇(|𝑧|)≤𝜇(𝑥|𝑧|). From (4.3), it follows that 𝜇(|𝑧|)||𝑢(|𝑧|)||≤𝜇(|𝑧|)||𝑢(0)||+𝜇(|𝑧|)max_|𝑧|≤1/2||𝑢′(𝑧)||+^11/2𝜇(𝑥|𝑧|)||𝑢′(𝑧𝑥)||𝑑𝑥.(4.6) For 𝑗∈{2,3,…,𝑛}, by applying formula (4.6) to the function 𝑢^(𝑗−1), we get 𝜇(|𝑧|)||𝑢^(𝑗−1)(𝑧)||≤𝜇(|𝑧|)||𝑢^(𝑗−1)(0)||+𝜇(|𝑧|)max_|𝑧|≤1/2||𝑢^(𝑗)(𝑧)||+^11/2𝜇(|𝑧|𝑥)||𝑢^(𝑗)(𝑧𝑥)||𝑑𝑥,(4.7) when 2𝑡₀<|𝑧|<1. It follows from (4.1), (4.6), and (4.7) that lim_|𝑧|→1𝜇(|𝑧|)|𝑢^(𝑗−1)(𝑧)|=0 for 𝑗∈{1,3,…,𝑛}. Since, for each polynomial 𝑝, we have |||𝜇(|𝑧|)𝑀_𝑢𝑝^(𝑛)(𝑧)|||==𝜇𝜇((||𝑧𝑧||))|||||||(𝑢_𝑛(𝑧)𝑝(𝑧))^(𝑛)||_𝑗=0≤𝐶_𝑛𝐶^𝑗𝑛𝑢^(𝑗)(𝑧)𝑝^{(𝑛−𝑗)}(𝑧)|||||_𝑗=0||𝐶^𝑗𝑛𝜇(|𝑧|)𝑢^(𝑗)(𝑧)||‖‖𝑝^{(𝑛−𝑗)}‖‖_∞,(4.8) hence, 𝑀_𝑢𝑝∈𝒲^(𝑛)𝜇,0(𝔻). Since the set of all polynomials is dense in 𝐹(𝑝,𝑞,𝑠), we have that for every 𝑓∈𝐹(𝑝,𝑞,𝑠) there is a sequence of polynomials {𝑝_𝑘} such that lim_𝑘→∞‖‖𝑝_𝑘−𝑓‖‖_{𝐹(𝑝,𝑞,𝑠)}=0.(4.9) From this and since the operator 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded, we have that ‖‖𝑀_𝑢𝑝_𝑘−𝑀_𝑢𝑓‖‖_𝒲^𝜇(𝑛)_(𝔻)≤‖‖𝑀_𝑢‖‖‖‖𝑝_𝑘−𝑓‖‖_{𝐹(𝑝,𝑞,𝑠)}⟶0a𝑠𝑘⟶∞.(4.10) Since 𝒲^(𝑛)𝜇,0(𝔻) is a closed subspace of 𝒲^𝜇(𝑛)(𝔻), therefore, we have 𝑀_𝑢(𝐹(𝑝,𝑞,𝑠)⊂𝒲^(𝑛)𝜇,0(𝔻), from which the boundedness of 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^(𝑛)𝜇,0(𝔻) follows.
On the other hand, assume that 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^(𝑛)𝜇,0(𝔻) is bounded, then 𝑀_𝑢∶𝐹(𝑝,𝑞,𝑠)→𝒲^𝜇(𝑛)(𝔻) is bounded. By taking the function given by 𝑓(𝑧)=1, we obtain 𝜇(|𝑧|)||𝑢^(𝑛)(𝑧)||=|||𝜇(|𝑧|)𝑀_𝑢𝑓^(𝑛)(𝑧)|||⟶0(as|𝑧|⟶1),(4.11) as desired.
(2) The proof is similar to that of the case (1). We leave the details to the interested reader.

Acknowledgments

The authors acknowledge gratefully the support in part by the National Natural Science Foundation of China (no. 11171285) and the Grant of Natural Science Basic Research of Jiangsu Province of China for Colleges and Universities (nos. 06KJD110175; 07KJB110115). The authors also thank the referees for their thoughtful comments and helpful suggestions which greatly improved the final version of this paper.