Weighted Composition Operators from Hardy to Zygmund Type Spaces

Abstract

This paper aims at studying the boundedness and compactness of weighted composition operator between spaces of analytic functions. We characterize boundedness and compactness of the weighted composition operator 𝑢𝐶_𝜙 from the Hardy spaces 𝐻^𝑝 to the Zygmund type spaces 𝒵_𝛼={𝑓∈𝐻(𝐷)∶sup_𝑧∈𝐷(1−|𝑧|²)^𝛼|𝑓^″(𝑧)|<∞} and the little Zygmund type spaces 𝒵_𝛼,0 in terms of function theoretic properties of the symbols 𝑢 and 𝜙.

1. Introduction

Let 𝐷={𝑧∶|𝑧|<1} be the open unit disk in the complex plane 𝐂 and 𝑇={𝑧∶|𝑧|=1} its boundary, and 𝐻(𝐷) denotes the set of all analytic functions on 𝐷. An analytic self-map 𝜑∶𝐷→𝐷 induces the composition operator 𝐶_𝜑 on 𝐻(𝐷), defined by 𝐶_𝜑(𝑓)=𝑓(𝜑(𝑧)) for 𝑓 analytic on 𝐷. It is a well-known consequence of Littlewood’s subordination principle that the composition operator 𝐶_𝜑 is bounded on the classical Hardy 𝐻^𝑝  (0<𝑝≤∞) spaces, Bergman 𝐴^𝑝  (0<𝑝≤∞) spaces, and Bloch spaces (see, e.g., [1–4]).

Let 𝑢 be a fixed analytic function on the open unit disk. Define a linear operator 𝑢𝐶_𝜑 on the space of analytic functions on 𝐷, called a weighted composition operator, by 𝑢𝐶_𝜑𝑓=𝑢⋅(𝑓∘𝜑), where 𝑓 is an analytic function on 𝐷. We can regard this operator as a generalization of a multiplication operator and a composition operator. In recent years the weighted composition operator has received much attention and appears in various settings in the literature. For example, it is known that isometries of many analytic function spaces are weighted composition operators (see [5], for instance). Their boundedness and compactness have been studied on various Banach spaces of analytic functions, such as Hardy, Bergman, BMOA, Bloch-type, and Zygmund spaces; see, for example, [6–11]. Also, it has been studied from one Banach space of analytic functions to another; one may see [12–23].

The purpose of this paper is to consider the weighted composition operators from the Hardy space 𝐻^𝑝  (0<𝑝<∞) to the Zygmund type spaces 𝒵_𝛼. Our main goal is to characterize boundedness and compactness of the operators 𝑢𝐶_𝜑 from 𝐻^𝑝 to 𝒵_𝛼 in terms of function theoretic properties of the symbols 𝑢 and 𝜑.

Now we give a detailed definition of these spaces. For 0≤𝑟<1, 𝑓(𝑧)∈𝐻(𝐷), we set 𝑀_𝑝=1𝑟,𝑓2𝜋₀^2𝜋𝑓𝑟𝑒^𝑖𝜃𝑝𝑑𝜃^1/𝑝𝑀,0<𝑝<∞,_∞𝑟,𝑓=max_{0≤𝜃≤2𝜋}𝑓𝑟𝑒^𝑖𝜃. (20) For 0<𝑝≤∞, the Hardy space 𝐻^𝑝 consists of those functions 𝑓∈𝐻(𝐷), for which 𝑓_𝑝=sup_0≤𝑟<1𝑀_𝑝𝑟,𝑓<∞. (2) It is well known that with norm (2) the 𝐻^𝑝 space is a Banach space if 1≤𝑝≤∞, for 0<𝑝<1, 𝐻^𝑝 space is a nonlocally convex topological vector space, and 𝑑(𝑓,𝑔)=‖𝑓−𝑔‖^𝑝_𝑝 is a complete metric for it. For more information about the 𝐻^𝑝 space, one may see these books, for example, [24, 25].

For 𝛼>0 the 𝛼-Bloch space 𝛽_𝛼 consists of all analytic functions 𝑓 defined on 𝐷 such that 𝑓_𝛽𝛼=sup1−|𝑧|^2𝛼𝑓^′(𝑧)∶𝑧∈𝐷,0<𝛼<+∞. (3) The space 𝒵_𝛼 consists of all analytic functions 𝑓 defined on 𝐷 such that 𝑓_𝒵𝛼=sup1−|𝑧|^2𝛼𝑓^′′(𝑧)∶𝑧∈𝐷,0<𝛼<+∞. (4) When 𝛼=1, it is called the Zygmund space. From a theorem by Zygmund (see [26, vol. I, p. 263] or [24, Theorem 5.3]), we see that 𝑓∈𝒵₁ if and only if 𝑓 is continuous in the close unit disk 𝐷={𝑧∶|𝑧|≤1} and the boundary function 𝑓(𝑒^𝑖𝜃) such that 𝑓𝑒^{𝑖(𝜃+ℎ)}+𝑓𝑒^{𝑖(𝜃−ℎ)}−2𝑓𝑒^𝑖𝜃ℎ<∞. (5) When 𝛼>1, from Proposition 8 of [27], we know that 𝒵_𝛼=𝛽_𝛼−1. Then the space 𝒵_𝛼 is called a Zygmund type space if 0<𝛼≤1. However, all results in this paper are valid for all 𝒵_𝛼 spaces (𝛼>0). An analytic function 𝑓∈𝐻(𝐷) is said to belong to the little Zymund type space 𝒵_𝛼,0 which consists of all 𝑓∈𝒵_𝛼 satisfying lim_|𝑧|→1⁻(1−|𝑧|²)^𝛼|𝑓^′′(𝑧)|=0. It can be easily proved that 𝒵_𝛼 is a Banach space under the norm 𝑓_∗=𝑓(0)+𝑓^′(0)+𝑓_𝒵𝛼. (6) And the polynomials are norm-dense in closed subspace 𝒵_𝛼,0. For some other information on this space and some operators on it, see, for example, [28–31].

Throughout this paper, constants are denoted by 𝐶, 𝐶(𝑝), they are positive, and 𝐶(𝑝) are only depending on 𝑝 and may differ from one occurrence to the another.

2. Auxiliary Results

In order to prove the main results of this paper. We need some auxiliary results. The first lemma is well known.

Lemma 1 (see [24, p. 65]). For 𝑝>1, there exists a constant 𝐶(𝑝) such that ₀^2𝜋𝑑𝜃|1−𝑧|^𝑝≤𝐶𝑝1−|𝑧|^2𝑝−1forevery𝑧∈𝐷. (7)

Lemma 2. Suppose that 0<𝑝<∞, 𝑓∈𝐻^𝑝; then 𝑓^(𝑛)(𝑧≤𝑓)^𝑝1−|𝑧|^21/𝑝+𝑛 (8) for every 𝑧∈𝐷 and all nonnegative integer 𝑛=0,1,2,….

Proof. We use induction on 𝑛. The case 𝑛=0 holds because it is Exercise  5 in [25, p. 85]. Assume the case 𝑛=𝑘 holds. Fix 0<𝑟<1 and let 𝑔(𝑧)=𝑓^(𝑘)(𝑟𝑧). Then 𝑔(𝑧) is in 𝐻^∞⊂𝛽₁, and ‖𝑔‖_𝛽1≤‖𝑔‖_∞. It follows that 1−|𝑧|²𝑔^′(𝑧)=1−|𝑧|²𝑟𝑓^(𝑘+1)(≤𝑓𝑟𝑧)_𝑝1−|𝑟𝑧|^21/𝑝+𝑘≤𝑓_𝑝1−|𝑧|^21/𝑝+𝑘. (9) Let 𝑟→1⁻; we have 𝑓^(𝑘+1)(𝑧≤𝑓)^𝑝1−|𝑧|^{21/𝑝+𝑘+1}. (10) Then the case 𝑛=𝑘+1 holds. Hence (8) holds.

Lemma 3. For 0<𝑝<∞, suppose 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is a bounded operator. Then 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼 is a bounded operator.

This is obvious.

3. Boundedness of 𝑢𝐶_𝜑 from 𝐻^𝑝(0<𝑝<∞) to 𝒵_𝛼 and 𝒵_𝛼,0

In this section we characterize bounded weighted composition operators from the Hardy space 𝐻^𝑝  (0<𝑝<∞) to the Zygmund spaces 𝒵_𝛼.

Theorem 4. Let 𝛼>0,  0<𝑝<∞,  and 𝑢 be an analytic function on the unit disc 𝐷 and 𝜑 an analytic self-map of 𝐷. Then 𝑢𝐶_𝜑 is a bounded operator from 𝐻^𝑝 to the Zygmund spaces 𝒵_𝛼 if and only if the following are satisfied: sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝<∞,sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2<∞,sup_𝑧∈𝐷1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1<∞. (11)(12)(13)

Proof . Suppose 𝑢𝐶_𝜑 is bounded from 𝐻^𝑝 to the Zygmund spaces 𝒵_𝛼. Then we can easily obtain the following results by taking 𝑓(𝑧)=1 and 𝑓(𝑧)=𝑧 in 𝐻^𝑝, respectively: 𝑢∈𝒵_𝛼,sup_𝑧∈𝐷1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)+𝜑(𝑧)𝑢^′′(𝑧)<+∞. (14) By (14) and the boundedness of the function 𝜑(𝑧), we get 𝐾₁=sup_𝑧∈𝐷1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)<+∞. (15) Let 𝑓(𝑧)=𝑧² in 𝐻^𝑝 again; in the same way we have sup_𝑧∈𝐷1−|𝑧|^2𝛼4𝜑(𝑧)𝜑^′(𝑧)𝑢^′(𝑧)+𝜑²(𝑧)𝑢^′′(𝑧)+2𝑢(𝑧)𝜑(𝑧)𝜑^′′(𝑧)+𝜑^′(𝑧)²<∞. (16) Using these facts and the boundedness of the function 𝜑(𝑧) again, we get 𝐾₂=sup_𝑧∈𝐷1−|𝑧|^2𝛼𝜑^′(𝑧)²𝑢(𝑧)<+∞. (17) Fix 𝑎∈𝐷; we take the test functions 𝑓_𝑎(𝑧)|𝑎|=−1−²(1−𝑎𝑧)^1+1/𝑝|𝑎|+21−²²(1−𝑎𝑧)^2+1/𝑝−1−|𝑎|²³(1−𝑎𝑧)^3+1/𝑝 (18) for 𝑧∈𝐷. From Lemma 1 we obtain that 𝑓_𝑎∈𝐻^𝑝 and sup_𝑎‖𝑓_𝑎‖_𝑝≤𝐶(𝑝)<∞ with a direct calculation. Since 𝑓_𝑎(𝑎)=0,  𝑓^′_𝑎(𝑎)=0,  and 𝑓^′′_𝑎(𝑎)=−2𝑎²/(1−|𝑎|²)^2+1/𝑝, it follows that, for all 𝜆∈𝐷 with |𝜑(𝜆)|>1/2, we have 𝐶𝑓_𝑎∗≥𝑢𝐶_𝜑𝑓_𝑎∗≥sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓_𝑎^′′(𝑧)=sup_𝑧∈𝐷1−|𝑧|^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′_𝑎𝜑(𝑧)+𝑓^′′_𝑎𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+𝑢^′′(𝑧)𝑓_𝑎𝜑(𝑧). (19) Let 𝑎=𝜑(𝜆); it follows that 𝐶𝑓_𝑎∗≥1−|𝜆|^2𝛼×2𝜑^′(𝜆)𝑢^′(𝜆)+𝜑^′′(𝜆)𝑢(𝜆)𝑓^′_𝜑(𝜆)𝜑(𝜆)+𝑓^′′_𝜑(𝜆)𝜑(𝜆)𝜑^′(𝜆)²𝑢(𝜆)+𝑢^′′(𝜆)𝑓_𝜑(𝜆)𝜑(𝜆)=1−|𝜆|^2𝛼𝜑^′(𝜆)²𝑢(𝜆)2𝜑(𝜆)²1−𝜑(𝜆)^22+1/𝑝≥121−|𝜆|^2𝛼𝑢(𝜆)𝜑^′(𝜆)²1−𝜑(𝜆)^22+1/𝑝. (1) For all 𝜆∈𝐷 with |𝜑(𝜆)|≤1/2, by (17), we have sup_𝜆∈𝐷1−|𝜆|^2𝛼𝑢(𝜆)𝜑^′(𝜆)²1−𝜑(𝜆)^1/𝑝+2≤43^1/𝑝+2sup_𝜆∈𝐷1−|𝜆|^2𝛼𝑢(𝜆)𝜑^′(𝜆)²<+∞. (21) Hence (12) holds.
Next, fix 𝑎∈𝐷; we take another test functions 𝑔_𝑎(𝑧)|𝑎|=−31−²(1−𝑎𝑧)^1/𝑝+1|𝑎|+51−²²(1−𝑎𝑧)^1/𝑝+2|𝑎|−21−²³(1−𝑎𝑧)^1/𝑝+3 (22) for 𝑧∈𝐷. From Lemma 1 we obtain that 𝑔_𝑎∈𝐻^𝑝 and sup_𝑎‖𝑔_𝑎‖_𝑝≤𝐶(𝑝)<∞ with a direct calculation. Since 𝑔_𝑎(𝑎)=0,  𝑔^′′_𝑎(𝑎)=0,  and 𝑔^′_𝑎(𝑎)=𝑎/(1−|𝑎|²)^1/𝑝+1, it follows that, for all 𝜆∈𝐷 with |𝜑(𝜆)|>1/2, we obtain that 𝐶𝑔_𝑎∗≥𝑢𝐶_𝜑𝑔_𝑎(𝑧)_∗≥sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑔_𝑎^′′(𝑧)=sup_𝑧∈𝐷1−|𝑧|^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑔^′_𝑎𝜑(𝑧)+𝑔^′′_𝑎𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+𝑢^′′(𝑧)𝑔_𝑎𝜑(𝑧)≥1−|𝜆|^2𝛼×2𝜑^′(𝜆)𝑢^′(𝜆)+𝜑^′′(𝜆)𝑢(𝜆)𝑔^′_𝜑(𝜆)𝜑(𝜆)+𝑔^′′_𝜑(𝜆)𝜑(𝜆)𝜑^′(𝜆)²𝑢(𝜆)+𝑢^′′(𝜆)𝑔_𝜑(𝜆)𝜑(𝜆)=1−|𝜆|^2𝛼×2𝜑^′(𝜆)𝑢^′(𝜆)+𝜑^′′(𝜆)𝑢(𝜆×)𝜑(𝜆)1−𝜑(𝜆)^21/𝑝+1≥121−|𝜆|^2𝛼2𝜑^′(𝜆)𝑢^′(𝜆)+𝜑^′′(𝜆)𝑢(𝜆)1−𝜑(𝜆)^21/𝑝+1. (23) For all 𝜆∈𝐷 with |𝜑(𝜆)|≤1/2, by (15), we have sup_{𝜑(𝜆)≤1/2}1−|𝜆|^2𝛼2𝜑^′(𝜆)𝑢^′(𝜆)+𝜑^′′(𝜆)𝑢(𝜆)1−𝜑(𝜆)^21/𝑝+1≤43^1/𝑝+1sup_{𝜑(𝜆)≤1/2}1−|𝜆|^2𝛼×2𝜑^′(𝜆)𝑢^′(𝜆)+𝜑^′′(𝜆)𝑢(𝜆)<∞. (24) Hence (13) holds.
Finally, fix 𝑎∈𝐷, and, for all 𝑧∈𝐷, let ℎ_𝑎(𝑧)=−𝑝+3|𝑎|𝑝+11−²(1−𝑎𝑧)^1/𝑝+1+2𝑝+3|𝑎|𝑝+21−²²(1−𝑎𝑧)^1/𝑝+2−1−|𝑎|²³(1−𝑎𝑧)^1/𝑝+3. (25) From Lemma 1 we obtain that ℎ_𝑎∈𝐻^𝑝 and sup_𝑎‖ℎ_𝑎‖_𝑝≤𝐶(𝑝)<∞ with a direct calculation. Since ℎ^′_𝑎(𝑎)=0,  ℎ^′′_𝑎(𝑎)=0,  and ℎ_𝑎(𝑎)=−2/(𝑝+1)(𝑝+2)(1−|𝑎|²)^1/𝑝, it follows that, for all 𝜆∈𝐷, we obtain that 𝐶ℎ_𝑎∗≥𝑢𝐶_𝜑ℎ_𝑎(𝑧)_∗≥sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢𝐶_𝜑ℎ_𝑎^′′(𝑧)=sup_𝑧∈𝐷1−|𝑧|^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)ℎ^′_𝑎𝜑(𝑧)+ℎ^′′_𝑎𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+𝑢^′′(𝑧)ℎ_𝑎𝜑(𝑧)≥1−|𝜆|^2𝛼×2𝜑^′(𝜆)𝑢^′(𝜆)+𝜑^′′(𝜆)𝑢(𝜆)ℎ^′_𝜑(𝜆)𝜑(𝜆)+ℎ^′′_𝜑(𝜆)𝜑(𝜆)𝜑^′(𝜆)²𝑢(𝜆)+𝑢^′′(𝜆)ℎ_𝜑(𝜆)𝜑(𝜆)=1−|𝜆|^2𝛼𝑢^′′(𝜆)ℎ_𝜑(𝜆)𝜑(𝜆)=2|𝜆|𝑝+1𝑝+21−^2𝛼𝑢^′′(𝜆)1−𝜑(𝜆)^21/𝑝. (26) Then (11) holds.
Conversely, suppose that (11), (12), and (13) hold. For 𝑓∈𝐻^𝑝, by Lemma 2, we have the following inequality: 1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓^′′(𝑧=|𝑧|)1−^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′𝜑(𝑧)+𝑓^′′𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+𝑢^′′(𝑧)𝑓𝜑(𝑧)≤1−|𝑧|^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′𝜑(𝑧)+1−|𝑧|^2𝛼𝑓^′′𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧+|𝑧|)1−²𝑢^′′(𝑧)𝑓𝜑(𝑧)≤1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1𝑓_𝑝+1−|𝑧|^2𝛼𝜑^′(𝑧)²𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+2𝑓_𝑝+1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝𝑓_𝑝≤𝐶𝑓_𝑝,𝑢(0)𝑓𝜑(0)+𝑢^′(0)𝑓𝜑(0)+𝑢(0)𝑓^′𝜑(0)≤|𝑢(0)|1−𝜑(0)^21/𝑝+𝑢^′(0)1−𝜑(0)^21/𝑝+|𝑢(0)|1−𝜑(0)^21/𝑝+1𝑓_𝑝. (27) This shows that 𝑢𝐶_𝜑 is bounded. This completes the proof of Theorem 4.

Theorem 5. Let 𝛼>0,  0<𝑝<∞, and 𝑢 be an analytic function on the unit disc 𝐷 and 𝜑 an analytic self-map of 𝐷. Then 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is a bounded operator provided that the following are satisfied: lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝=0,lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2=0,lim_{|𝑧|→1−}1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1=0. (28)(29)(30) Conversely, if 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is a bounded operator, then 𝑢∈𝒵_𝛼,0, (11), (12), and (13) hold, and the following are satisfied: lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²=0,lim_{|𝑧|→1−}1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)=0. (31)(32)

Proof. Assume that (28), (29), and (30) hold. Then for any 𝜖>0, there is a constant 𝛿,  0<𝛿<1, such that 𝛿<|𝑧|<1 implies 1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2|𝑧|<𝜖,1−^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝|𝑧|<𝜖,1−^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1<𝜖. (33) Then, for any 𝑓∈𝐻^𝑝, from Lemma 2 we obtain that 1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓^′′(𝑧=|𝑧|)1−^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′𝜑(𝑧)+𝑓^′′𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+𝑢^′′(𝑧)𝑓𝜑(𝑧)≤1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧×1)1−𝜑(𝑧)^21/𝑝+1𝑓_𝑝+1−|𝑧|^2𝛼𝜑^′(𝑧)²𝑢(𝑧1)1−𝜑(𝑧)^21/𝑝+2𝑓_𝑝+1−|𝑧|²𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝𝑓_𝑝≤3𝑓_𝑝𝜖. (34) Hence 𝑢𝐶_𝜑𝑓∈𝒵_𝛼,0 for all 𝑓∈𝒵_𝛼,0. On the other hand, (25), (28), and (29) imply that (11), (12), and (13) hold; then 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼 is bounded by Theorem 4. So 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is bounded.
Conversely, assume that 𝑢𝐶_𝜑 is bounded from 𝐻^𝑝 to the little Zygmund type space 𝒵_𝛼,0. Then 𝑢=𝑢𝐶_𝜑1∈𝒵_𝛼,0. Also 𝑢𝜑=𝑢𝐶_𝜑𝑧∈𝒵_𝛼,0; thus 1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)+𝜑(𝑧)𝑢^′′(𝑧𝑧|)⟶0(|⟶1⁻). (35) Since |𝜑|≤1 and 𝑢∈𝒵_𝛼,0, we have lim_|𝑧|→1⁻(1−|𝑧|²)^𝛼|2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)|=0. Hence (32) holds.
Similarly, 𝑢𝐶_𝜑𝑧²∈𝒵_𝛼,0; then 1−|𝑧|^2𝛼×4𝜑(𝑧)𝜑^′(𝑧)𝑢^′(𝑧)+𝜑²(𝑧)𝑢^′′(𝑧)+2𝑢(𝑧)𝜑(𝑧)𝜑^′′(𝑧)+𝜑^′(𝑧)²𝑧|⟶0(|⟶1⁻). (36) By (32), |𝜑|≤1, and 𝑢∈𝒵_𝛼,0, we get that lim_|𝑧|→1⁻(1−|𝑧|²)^𝛼|𝑢(𝑧)(𝜑^′(𝑧))²|=0; that is, (31) holds.
On the other hand, from Lemma 3 and Theorem 4, we obtain that (11), (12), and (13) hold.

4. Compactness of 𝑢𝐶_𝜑

In order to prove the compactness of 𝑢𝐶_𝜑 from 𝐻^𝑝 to the Zygmund spaces 𝒵_𝛼, we require the following lemmas.

Lemma 6. Let 𝛼>0,  0<𝑝<∞, and 𝑢 be an analytic function on the unit disc 𝐷 and 𝜑 an analytic self-map of 𝐷. Suppose that 𝑢𝐶_𝜑 is a bounded operator from 𝐻^𝑝 to 𝒵_𝛼. Then 𝑢𝐶_𝜑 is compact if and only if, for any bounded sequence {𝑓_𝑛} in 𝐻^𝑝 which converges to 0 uniformly on compact subsets of 𝐷, one has ‖𝑢𝐶_𝜑(𝑓_𝑛)‖_∗→0 as 𝑛→∞.

The proof is similar to that of Proposition  3.11 in [32]. The details are omitted.

Theorem 7. Let 𝛼>0,0<𝑝<∞, 𝑢 be an analytic function on the unit disc 𝐷 and 𝜑 an analytic self-map of 𝐷. Then 𝑢𝐶_𝜑 is a compact operator from 𝐻^𝑝 to 𝒵_𝛼 if and only if 𝑢𝐶_𝜑 is a bounded operator and the following are satisfied: (i)lim_{𝜑(𝑧)→1−}1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝()=0,iilim_{𝜑(𝑧)→1−}1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2()=0,iiilim_{|𝜑(𝑧)|→1−}1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1=0. (37)

Proof. Suppose that 𝑢𝐶_𝜑 is compact from 𝐻^𝑝 to the Zygmund type space 𝒵_𝛼. Let {𝑧_𝑛} be a sequence in 𝐷 such that |𝜑(𝑧_𝑛)|→1 as 𝑛→∞. If such a sequence does not exist, then (37) are automatically satisfied. Without loss of generality we may suppose that |𝜑(𝑧_𝑛)|>1/2 for all 𝑛. We take the test functions 𝑓_𝑛(𝑧)𝜑𝑧=−1−_𝑛²1−𝜑𝑧_𝑛𝑧^1/𝑝+1𝜑𝑧+21−_𝑛²²1−𝜑𝑧_𝑛𝑧^1/𝑝+2−1−𝜑𝑧_𝑛²³1−𝜑𝑧_𝑛𝑧^1/𝑝+3. (38) By a direct calculation, we may easily prove that {𝑓_𝑛} converges to 0 uniformly on compact subsets of 𝐷 and sup_𝑛‖𝑓_𝑛‖_𝑝≤𝐶(𝑝)<∞. Then {𝑓_𝑛} is a bounded sequence in 𝐻^𝑝 which converges to 0 uniformly on compact subsets of 𝐷. Then lim_𝑛→∞‖𝑢𝐶_𝜑(𝑓_𝑛)‖_∗=0 by Lemma 6. Note that 𝑓_𝑛𝜑𝑧_𝑛=0,𝑓^′_𝑛𝜑𝑧_𝑛𝑓=0,_′′𝑛𝜑𝑧_𝑛2=−𝜑𝑧_𝑛²1−𝜑𝑧_𝑛^21/𝑝+2. (39) It follows that 𝑢𝐶_𝜑𝑓_𝑛∗≥𝑢𝐶_𝜑𝑓_𝑛𝒵𝛼≥1−𝑧_𝑛^2𝛼×2𝑢^′𝑧_𝑛𝜑^′𝑧_𝑛+𝜑^′′𝑧_𝑛𝑢𝑧_𝑛×𝑓^′_𝑛𝜑𝑧_𝑛+𝑢𝑧_𝑛𝑓^′′_𝑛𝜑𝑧_𝑛𝜑^′𝑧_𝑛²+𝑢^′′𝑧_𝑛𝑓_𝑛𝜑𝑧_𝑛=1−𝑧_𝑛^2𝛼×𝜑^′𝑧_𝑛²𝑢𝑧_𝑛2𝜑𝑧_𝑛²1−𝜑𝑧_𝑛^21/𝑝+2. (40) Then lim_𝑛→∞1−𝑧_𝑛^2𝛼𝑢𝑧_𝑛𝜑^′𝑧_𝑛²1−𝜑𝑧_𝑛^21/𝑝+2=0. (41)
Next, let 𝑔_𝑛(𝑧)𝜑𝑧=−31−_𝑛²1−𝜑𝑧_𝑛𝑧^1/𝑝+1𝜑𝑧+51−_𝑛²²1−𝜑𝑧_𝑛𝑧^1/𝑝+2𝜑𝑧−21−_𝑛²³1−𝜑𝑧_𝑛𝑧^1/𝑝+3. (42) By a direct calculation we obtain that 𝑔_𝑛⇉0  (𝑛→∞) on compact subsets of 𝐷 and sup_𝑛‖𝑔_𝑛‖_𝑝≤𝐶(𝑝)<+∞. Consequently, {𝑔_𝑛} is a bounded sequence in 𝐻^𝑝 which converges to 0 uniformly on compact subsets of 𝐷. Then lim_𝑛→∞‖𝑢𝐶_𝜑(𝑔_𝑛)‖_∗=0 by Lemma 6. Note that 𝑔_𝑛(𝜑(𝑧_𝑛))≡0, 𝑔^′′_𝑛(𝜑(𝑧_𝑛))≡0 and 𝑔^′_𝑛(𝜑(𝑧_𝑛))=𝜑(𝑧_𝑛)/(1−|𝜑(𝑧_𝑛)|²)^1/𝑝+1; it follows that 𝑢𝐶_𝜑𝑔_𝑛∗≥𝑢𝐶_𝜑𝑔_𝑛𝒵𝛼≥1−𝑧_𝑛^2𝛼×2𝑢^′𝑧_𝑛𝜑^′𝑧_𝑛+𝜑^′′𝑧_𝑛𝑢𝑧_𝑛×𝑔^′_𝑛𝜑𝑧_𝑛+𝑢𝑧_𝑛𝑔^′′_𝑛𝜑𝑧_𝑛𝜑^′𝑧_𝑛²+𝑢^′′𝑧_𝑛𝑔_𝑛𝜑𝑧_𝑛=1−𝑧_𝑛^2𝛼×2𝑢^′𝑧_𝑛𝜑^′𝑧_𝑛+𝜑^′′𝑧_𝑛𝑢𝑧_𝑛×𝜑𝑧_𝑛1−𝜑𝑧_𝑛^21/𝑝+1. (43) Then lim_𝑛→∞(1−|𝑧_𝑛|²)^𝛼(2𝑢^′(𝑧_𝑛)𝜑^′(𝑧_𝑛)+𝜑^′′(𝑧_𝑛)𝑢(𝑧_𝑛))/(1−|𝜑(𝑧_𝑛)|²)^1/𝑝+1=0.
Finally, let ℎ_𝑛(𝑧)=−𝑝+3𝜑𝑧𝑝+11−_𝑛²1−𝜑𝑧_𝑛𝑧^1/𝑝+1+2𝑝+3𝜑𝑧𝑝+21−_𝑛²²1−𝜑𝑧_𝑛𝑧^1/𝑝+2−1−𝜑𝑧_𝑛²³1−𝜑𝑧_𝑛𝑧^1/𝑝+3. (44) By a direct calculation we obtain that ℎ_𝑛⇉0  (𝑛→∞) on compact subsets of 𝐷 and sup_𝑛‖ℎ_𝑛‖_𝑝<∞. Consequently, {ℎ_𝑛} is a bounded sequence in 𝐻^𝑝 which converges to 0 uniformly on compact subsets of 𝐷. Then lim_𝑛→∞‖𝑢𝐶_𝜑(ℎ_𝑛)‖_∗=0 by Lemma 6. Note that ℎ_𝑛(𝜑(𝑧_𝑛))=2/(𝑝+1)(𝑝+2)(1−|𝜑(𝑧_𝑛)|²)^1/𝑝,  ℎ^′_𝑛(𝜑(𝑧_𝑛))≡0, and ℎ^′′_𝑛(𝜑(𝑧_𝑛))≡0; it follows that 𝑢𝐶_𝜑ℎ_𝑛∗≥𝑢𝐶_𝜑ℎ_𝑛𝒵𝛼≥1−𝑧_𝑛^2𝛼×2𝑢^′𝑧_𝑛𝜑^′𝑧_𝑛+𝜑^′′𝑧_𝑛𝑢𝑧_𝑛×ℎ^′_𝑛𝜑𝑧_𝑛+𝑢𝑧_𝑛ℎ^′′_𝑛𝜑𝑧_𝑛𝜑^′𝑧_𝑛²+𝑢^′′𝑧_𝑛ℎ_𝑛𝜑𝑧_𝑛=21−𝑧_𝑛^2𝛼𝑢^′′𝑧_𝑛𝑝+1𝑝+21−𝜑𝑧_𝑛^21/𝑝. (45) Then lim_𝑛→∞((1−|𝑧_𝑛|²)^𝛼|𝑢^′′(𝑧_𝑛)|/(1−|𝜑(𝑧_𝑛)|²)^1/𝑝)=0. The proof of the necessary is completed.
Conversely, Suppose that (37) hold. Since 𝑢𝐶_𝜑 is a bounded operator, from Theorem 4, we have 𝑀₁≜sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²𝑀<∞,₂≜sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢^′′(𝑧𝑀)<∞,₃≜sup_𝑧∈𝐷1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)<∞. (46) Let {𝑓_𝑛} be a bounded sequence in 𝐻^𝑝 with ‖𝑓_𝑛‖_𝑝≤1 and 𝑓_𝑛→0 uniformly on compact subsets of 𝐷. We only prove lim_𝑛→∞‖𝑢𝐶_𝜑(𝑓_𝑛)‖_∗=0 by Lemma 6. By the assumption, for any 𝜖>0, there is a constant 𝛿,  0<𝛿<1, such that 𝛿<|𝜑(𝑧)|<1 implies 1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2|𝑧|<𝜖,1−^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝|𝑧|<𝜖,1−^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1<𝜖. (47) Let 𝐾={𝑤∈𝐷∶|𝑤|≤𝛿}. Note that 𝐾 is a compact subset of 𝐷. Then from Lemma 2 it follows that 𝑢𝐶_𝜑𝑓_𝑛𝒵𝛼=sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓_𝑛^′′(𝑧=|𝑧|)1−^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′_𝑛𝜑(𝑧)+𝑓^′′_𝑛𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+𝑢^′′(𝑧)𝑓_𝑛𝜑(𝑧)≤sup_𝑧∈𝐷1−|𝑧|^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′_𝑛𝜑(𝑧)+sup_𝑧∈𝐷1−|𝑧|^2𝛼𝑓^′′_𝑛𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+sup_𝑧∈𝐷1−|𝑧|²𝑢^′′(𝑧)𝑓_𝑛𝜑(𝑧)≤3𝜖+sup_{(𝑧)|𝜑|≤𝛿}1−|𝑧|^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′_𝑛𝜑(𝑧)+sup_{(𝑧)|𝜑|≤𝛿}1−|𝑧|^2𝛼𝑓^′′_𝑛𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧)+sup_{(𝑧)|𝜑|≤𝛿}1−|𝑧|^2𝛼𝑢^′′(𝑧)𝑓_𝑛𝜑(𝑧)≤3𝜖+𝑀₃sup_𝑤∈𝐾𝑓^′_𝑛(𝑤)+𝑀₁sup_𝑤∈𝐾𝑓^′′_𝑛(𝑤)+𝑀₂sup_𝑤∈𝐾𝑓_𝑛(𝑤). (48) As 𝑛→∞, 𝑢𝐶_𝜑𝑓_𝑛𝒵𝛼⟶0. (49) Hence 𝑢𝐶_𝜑 is compact. This completes the proof of Theorem 7.

In order to prove the compactness of 𝑢𝐶_𝜑 on the little Zygmund spaces 𝒵_𝛼,0, we require the following lemma.

Lemma 8. Let 𝑈⊂𝒵_𝛼,0. Then 𝑈 is compact if and only if it is closed, bounded and satisfies lim_{|𝑧|→1−}sup_𝑓∈𝑈1−|𝑧|²𝑓^′′(𝑧)=0. (50)

The proof is similar to that of Lemma  1 in [1], but we omit it.

Theorem 9. Let 𝛼>0, 0<𝑝<∞, 𝑢 be an analytic function on the unit disc 𝐷 and 𝜑 an analytic self-map of 𝐷. Then 𝑢𝐶_𝜑 is compact from 𝐻^𝑝 to the little Zygmund type spaces 𝒵_𝛼,0 if and only if (28), (29), and (30) hold.

Proof. Assume that (28), (29), and (30) hold. By Theorem 5, we know that 𝑢𝐶_𝜑 is bounded from 𝐻^𝑝 to the little Zygmund type spaces 𝒵_𝛼,0. Suppose that 𝑓∈𝐻^𝑝 with ‖𝑓‖_𝑝≤1. From Lemmas 1 and 2 we obtain that 1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓^′′(𝑧≤|𝑧|)1−^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)𝑓^′𝜑(𝑧)+1−|𝑧|^2𝛼𝑓^′′𝜑(𝑧)𝜑^′(𝑧)²𝑢(𝑧+|𝑧|)1−²𝑢^′′(𝑧)𝑓𝜑(𝑧)≤1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧×1)1−𝜑(𝑧)^21/𝑝+1𝑓_𝑝+1−|𝑧|^2𝛼𝜑^′(𝑧)²𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+2𝑓_𝑝+1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝𝑓_𝑝, (51) thus sup1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓^′′(𝑧)∶𝑓∈𝐻^𝑝,𝑓_𝑝≤|𝑧|≤11−^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧1)1−𝜑(𝑧)²+1−|𝑧|^2𝛼𝜑^′(𝑧)²𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+2+1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝, (52) and it follows that lim_{|𝑧|→1−}sup1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓^′′(𝑧)∶𝑓∈𝐻^𝑝,𝑓_𝑝≤1=0. (53) Hence 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is compact by Lemma 8.
Conversely, suppose that 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is compact.
Firstly, it is obvious 𝑢𝐶_𝜑 is bounded, from Theorem 5 we have 𝑢∈𝒵_𝛼,0, and (31), (32) hold. On the other hand, we have lim_{|𝑧|→1−}sup1−|𝑧|^2𝛼𝑢𝐶_𝜑𝑓^′′(𝑧)∶𝑓∈𝐻^𝑝,𝑓_𝑝≤𝑀=0, (54) for some 𝑀>0 by Lemma 6.
Next, note that the proof of Theorem 4 and the fact that the functions given in (18) are in 𝐻^𝑝 and have norms bounded independently of 𝑎; we obtain that lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2=0 (55) for |𝜑(𝑧)|>1/2. However, if |𝜑(𝑧)|≤1/2, by (31), we easily have lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2≤lim_{|𝑧|→1−}43^1/𝑝+21−|𝑧|^2𝛼𝑢(𝑧)𝜑^′(𝑧)²=0. (56)
Similarly, note that the functions given in (22) and (25) are in 𝐻^𝑝 and have norms bounded independently of 𝑎, we obtain that lim_{|𝑧|→1−}1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1=0,lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝=0 (57) for |𝜑(𝑧)|>1/2. However, if |𝜑(𝑧)|≤1/2, from 𝑢∈𝒵_𝛼,0 and (32), we easily have lim_{|𝑧|→1−}1−|𝑧|^2𝛼2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)1−𝜑(𝑧)^21/𝑝+1≤43^1/𝑝+1lim_{|𝑧|→1−}1−|𝑧|^2𝛼×2𝜑^′(𝑧)𝑢^′(𝑧)+𝜑^′′(𝑧)𝑢(𝑧)=0,lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝑢^′′(𝑧)1−𝜑(𝑧)^21/𝑝≤lim_{|𝑧|→1−}43^1/𝑝1−|𝑧|^2𝛼𝑢^′′(𝑧)=0. (58) This completes the proof of Theorem 9.

Remark 10. From Theorems 5 and 9, we conjecture that 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is compact if and only if 𝑢𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is bounded.

Taking 𝑢(𝑧)=1 from Theorems 4, 7, and 9, we obtain the following results about the characterization of the boundedness and compactness of the composition operator 𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼(or𝒵_𝛼,0).

Corollary 11. Let 𝛼>0,  0<𝑝<∞,  and  𝜑 be an analytic self-map of 𝐷. Then 𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼 is a bounded operator if and only if the following are satisfied: sup_𝑧∈𝐷1−|𝑧|^2𝛼𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2<∞,sup_𝑧∈𝐷1−|𝑧|^2𝛼𝜑^′′(𝑧)1−𝜑(𝑧)^21/𝑝+1<∞. (59)

Corollary 12. Let 𝛼>0,  0<𝑝<∞,  and  𝜑 be an analytic self-map of 𝐷. Then 𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼 is a compact operator if and only if 𝐶_𝜑 is bounded and the following are satisfied: lim_{|𝜑(𝑧)|→1−}1−|𝑧|^2𝛼𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2=0,lim_{|𝜑(𝑧)|→1−}1−|𝑧|^2𝛼𝜑^′′(𝑧)1−𝜑(𝑧)^21/𝑝+1=0. (60)

Corollary 13. Let 𝛼>0,  0<𝑝<∞,  and  𝜑 be an analytic self-map of 𝐷. Then 𝐶_𝜑∶𝐻^𝑝→𝒵_𝛼,0 is a compact operator if and only if lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝜑^′(𝑧)²1−𝜑(𝑧)^21/𝑝+2=0,lim_{|𝑧|→1−}1−|𝑧|^2𝛼𝜑^′′(𝑧)1−𝜑(𝑧)^21/𝑝+1=0. (61)

In the formulation of corollary, we use the notation 𝑀_𝑢 on 𝐻(𝐷) defined by 𝑀_𝑢𝑓=𝑢𝑓 for 𝑓∈𝐻(𝐷). Taking 𝜑(𝑧)=𝑧 from Theorems 4, 5, 7, and 9, we obtain the following results about the characterization of the boundedness and compactness of pointwise multiplier 𝑀_𝑢∶𝐻^𝑝→𝒵_𝛼(𝑜𝑟𝒵_𝛼,0).

Corollary 14. Let 𝛼>0,  0<𝑝<∞. Then the pointwise multiplier 𝑀_𝑢∶𝐻^𝑝→𝒵_𝛼 is a bounded operator if and only if(i)𝑢=0 if 𝛼<2+1/𝑝;(ii)𝑢∈𝐻^∞ if 𝛼=2+1/𝑝;(iii)sup_𝑧∈𝐷(1−|𝑧|²)^{𝛼−2−1/𝑝}|𝑢(𝑧)|<∞ if 𝛼>2+1/𝑝.

Corollary 15. Let 𝛼>0,  0<𝑝<∞. Then the pointwise multiplier 𝑀_𝑢∶𝐻^𝑝→𝒵_𝛼,0 is a bounded operator if and only if 𝑀_𝑢∶𝐻^𝑝→𝒵_𝛼,0 is a compact operator if and only if 𝑀_𝑢∶𝐻^𝑝→𝒵_𝛼 is a compact operator if and only if(i)𝑢=0 if 𝛼≤2+1/𝑝,(ii)lim_|𝑧|→1⁻(1−|𝑧|²)^{𝛼−2−1/𝑝}|𝑢(𝑧)|=0 if 𝛼>2+1/𝑝.

Acknowledgments

The research was supported by Special Fund of Colleges and Universities in Fujian Province (no. JK2012010) and Natural Science Foundation of Fujian Province, China (no. 2009J01004).