Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane

Abstract

Here we introduce the 𝑛th weighted space on the upper half-plane Π₊={𝑧∈ℂ∶Im𝑧>0} in the complex plane ℂ. For the case 𝑛=2, we call it the Zygmund-type space, and denote it by 𝒵(Π₊). The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator 𝐶_𝜑𝑓(𝑧)=𝑓(𝜑(𝑧)) from the Hardy space 𝐻^𝑝(Π₊) on the upper half-plane, to the Zygmund-type space, where 𝜑 is an analytic self-map of the upper half-plane.

1. Introduction

Let Π₊ be the upper half-plane, that is, the set {𝑧∈ℂ∶Im𝑧>0} and 𝐻(Π₊) the space of all analytic functions on Π₊. The Hardy space 𝐻^𝑝(Π₊)=𝐻^𝑝, 𝑝>0, consists of all 𝑓∈𝐻(Π₊) such that With this norm 𝐻^𝑝(Π₊) is a Banach space when 𝑝≥1, while for 𝑝∈(0,1) it is a Fréchet space with the translation invariant metric 𝑑(𝑓,𝑔)=‖𝑓−𝑔‖^𝑝𝐻_𝑝, 𝑓,𝑔∈𝐻^𝑝(Π₊), [1].

We introduce here the 𝑛th weighted space on the upper half-plane. The 𝑛th weighted space consists of all 𝑓∈𝐻(Π₊) such thatwhere 𝑛∈ℕ₀. For 𝑛=0 the space is called the growth space and is denoted by 𝒜_∞(Π₊)=𝒜_∞ and for 𝑛=1 it is called the Bloch space ℬ_∞(Π₊)=ℬ_∞ (for Bloch-type spaces on the unit disk, polydisk, or the unit ball and some operators on them, see, e.g., [2–14] and the references therein).

When 𝑛=2, we call the space the Zygmund-type space on the upper half-plane (or simply the Zygmund space) and denote it by 𝒵(Π₊)=𝒵. Recall that the space consists of all 𝑓∈𝐻(Π₊) such that The quantity is a seminorm on the Zygmund space or a norm on 𝒵/ℙ₁, where ℙ₁ is the set of all linear polynomials. A natural norm on the Zygmund space can be introduced as follows: With this norm the Zygmund space becomes a Banach space.

To clarify the notation we have just introduced, we have to say that the main reason for this name is found in the fact that for the case of the unit disk 𝔻={𝑧∶|𝑧|<1} in the complex palne ℂ, Zygmund (see, e.g., [1, Theorem 5.3]) proved that a holomorphic function on 𝔻 continuous on the closed unit disk 𝔻 satisfies the following condition: if and only if

The family of all analytic functions on 𝔻 satisfying condition (1.6) is called the Zygmund class on the unit disk.

With the norm the Zygmund class becomes a Banach space. Zygmund class with this norm is called the Zygmund space and is denoted by 𝒵(𝔻). For some other information on this space and some operators on it, see, for example, [15–19].

Now note that 1−|𝑧| is the distance from the point 𝑧∈𝔻 to the boundary of the unit disc, that is, 𝜕𝔻, and that Im𝑧 is the distance from the point 𝑧∈Π₊ to the real axis in ℂ which is the boundary of Π₊.

In two main theorems in [20], the authors proved the following results, which we now incorporate in the next theorem.

Theorem A. Assume 𝑝≥1 and 𝜑 is a holomorphic self-map of Π₊. Then the following statements true hold.
(a) The operator 𝐶_𝜑∶𝐻^𝑝(Π₊)→𝒜_∞(Π₊) is bounded if and only if(b)The operator 𝐶_𝜑∶𝐻^𝑝(Π₊)→ℬ_∞(Π₊) is bounded if and only if

Motivated by Theorem A, here we investigate the boundedness of the operator 𝐶_𝜑∶𝐻^𝑝(Π₊)→𝒵(Π₊). Some recent results on composition and weighted composition operators can be found, for example, in [4, 6, 7, 10, 12, 18, 21–27].

Throughout 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. Moreover, if both 𝑎⪯𝑏 and 𝑏⪯𝑎 hold, then one says that 𝑎≍𝑏.

2. An Auxiliary Result

In this section we prove an auxiliary result which will be used in the proof of the main result of the paper.

Lemma 2.1. Assume that 𝑝≥1, 𝑛∈ℕ, and 𝑤∈Π₊. Then the function belongs to 𝐻^𝑝(Π₊). Moreover

Proof. Let 𝑧=𝑥+𝑖𝑦 and 𝑤=𝑢+𝑖𝜐. Then, we have where we have used the change of variables 𝑥=𝑢+𝑡(𝑦+𝑣).

3. Main Result

Here we formulate and prove the main result of the paper.

Theorem 3.1. Assume 𝑝≥1 and 𝜑 is a holomorphic self-map of Π₊. Then 𝐶_𝜑∶𝐻^𝑝(Π₊)→𝒵(Π₊) is bounded if and only if
Moreover, if the operator ‖‖𝐶_𝜑‖‖_{𝐻𝑝(Π+)→𝒵/ℙ1(Π+)}≍sup_𝑧∈Π+Im𝑧Im𝜑(𝑧)^2+1/𝑝||𝜑^(𝑧)||²+sup_𝑧∈Π+Im𝑧Im𝜑(𝑧)^1+1/𝑝||𝜑^(𝑧)||.(3.3) is bounded, then

Proof. First assume that the operator 𝑤∈Π₊ is bounded.
For 𝑓_𝑤(𝑧)=(Im𝑤)^2−1/𝑝𝜋^1/𝑝(𝑧−𝑤)².(3.4), set
By Lemma 2.1 (case 𝑓_𝑤∈𝐻^𝑝(Π₊)) we know that 𝑤∈Π₊ for every sup_𝑤∈Π+‖‖𝑓_𝑤‖‖_{𝐻𝑝(Π+)}≤1.(3.5). Moreover, we have that
From (3.5) and since the operator 𝑤∈Π₊ is bounded, for every sup_𝑧∈Π+Im𝑧||𝑓^𝑤𝜑(𝑧)𝜑^(𝑧)²+𝑓^𝑤𝜑(𝑧)𝜑^(𝑧)||=‖‖𝐶_𝜑𝑓_𝑤‖‖_𝒵(Π+)≤‖‖𝐶_𝜑‖‖_{𝐻𝑝(Π+)→𝒵(Π+)}.(3.6), we obtain
We also have that
Replacing (3.7) in (3.6) and taking Im𝑧||||38𝜑^(𝑧)²Im𝜑(𝑧)^2+1/𝑝−𝑖4𝜑^(𝑧)Im𝜑(𝑧)^1+1/𝑝||||≤𝜋_1/𝑝‖‖𝐶_𝜑‖‖_{𝐻𝑝(Π+)→𝒵(Π+)},(3.8), we obtain and consequently
Hence if we show that (3.1) holds then from the last inequality, condition (3.2) will follow.
For 𝑔_𝑤(𝑧)=3𝑖𝜋(Im_1/𝑝(𝑧𝑤−)𝑤)^2−1/𝑝2−4(Im𝑤)^3−1/𝑝𝜋^1/𝑝(𝑧−𝑤)³.(3.10), set
Then it is easy to see that and by Lemma 2.1 (cases 𝑛=3 and sup_𝑤∈Π+‖‖𝑔_𝑤‖‖_𝐻𝑝<∞.(3.12)) it is easy to see that
From this, since 𝑤=𝜑(𝑧) is bounded and by taking 𝐶Im𝑧Im𝜑(𝑧)^2+1/𝑝||𝜑^(𝑧)||²≤‖‖𝐶_𝜑𝑔_𝑤‖‖_𝒵(Π+)≤𝐶‖‖𝐶_𝜑‖‖_{𝐻𝑝(Π+)→𝒵(Π+)},(3.13), it follows that from which (3.1) follows, as desired.
Moreover, from (3.9) and (3.13) it follows that
Now assume that conditions (3.1) and (3.2) hold. By the Cauchy integral formula in 𝐻^𝑝(Π₊) for 𝑝≥1 functions (note that 𝑓(𝑧)=12𝜋𝑖^∞−∞𝑓(𝑡)𝑡−𝑧𝑑𝑡,𝑧∈Π⁺.(3.15)), we have

By differentiating formula (3.15), we obtain for each ||𝑓^(𝑛)(𝑧)||≤𝑛!2𝜋^∞−∞||𝑓(𝑡)||(𝑡−𝑥)²+𝑦²^(𝑛+1)/2𝑑𝑡,𝑧∈Π₊.(3.17), from which it follows that

By using the change ^∞−∞𝑦^𝑛(𝑡−𝑥)²+𝑦²^(𝑛+1)/2𝑑𝑡=^∞−∞𝑑𝑠𝑠²+1^(𝑛+1)/2=∶𝑐_𝑛<∞,𝑛∈ℕ.(3.18), we have that
From this, applying Jensen's inequality on (3.17) and an elementary inequality, we obtainwhere from which it follows that
Assume that ‖‖𝐶_𝜑𝑓‖‖_𝒵(Π+)==||||𝑓𝑓𝜑𝜑((𝑖𝑖))||||++||||𝑓𝑓_∘𝜑^(𝑖)||+sup_𝑧∈Π+Im𝑧||𝐶_𝜑𝑓^(𝑧)||≤𝐶‖𝑓‖_{𝐻𝑝(Π+)}𝜑(𝑖)||||𝜑^(𝑖)||+sup_𝑧∈Π+Im𝑧||𝑓^𝜑(𝑧)𝜑^(𝑧)²+𝑓^𝜑(𝑧)𝜑^(𝑧)||1+sup_𝑧∈Π+Im𝑧Im𝜑(𝑧)^2+1/𝑝||𝜑^(𝑧)||²+sup_𝑧∈Π+Im𝑧Im𝜑(𝑧)^1+1/𝑝||𝜑^(𝑧)||.(3.22). By applying (3.21) and Lemma 1 in [1, page 188], we have
From this and by conditions (3.1) and (3.2), it follows that the operator 𝒵/ℙ₁(Π₊) is bounded. Moreover, if we consider the space ‖‖𝐶_𝜑‖‖_{𝐻𝑝(Π+)→𝒵/ℙ1(Π+)}≤𝐶sup_𝑧∈Π+Im𝑧Im𝜑(𝑧)^2+1/𝑝||𝜑^(𝑧)||²+sup_𝑧∈Π+Im𝑧Im𝜑(𝑧)^1+1/𝑝||𝜑^(𝑧)||.(3.23), we have that
From (3.14) and (3.23), we obtain the asymptotic relation (3.3).