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Research Article | Open Access

Volume 2009 |Article ID 161528 | https://doi.org/10.1155/2009/161528

Stevo Stević, "Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane", Abstract and Applied Analysis, vol. 2009, Article ID 161528, 8 pages, 2009. https://doi.org/10.1155/2009/161528

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

Accepted23 Feb 2009
Published29 Apr 2009

#### Abstract

Here we introduce the th weighted space on the upper half-plane in the complex plane . For the case , we call it the Zygmund-type space, and denote it by . The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator from the Hardy space on the upper half-plane, to the Zygmund-type space, where is an analytic self-map of the upper half-plane.

#### 1. Introduction

Let be the upper half-plane, that is, the set and the space of all analytic functions on . The Hardy space , consists of all such that With this norm is a Banach space when , while for it is a Fréchet space with the translation invariant metric , , .

We introduce here the th weighted space on the upper half-plane. The th weighted space consists of all such thatwhere . For the space is called the growth space and is denoted by and for it is called the Bloch space (for Bloch-type spaces on the unit disk, polydisk, or the unit ball and some operators on them, see, e.g.,  and the references therein).

When , we call the space the Zygmund-type space on the upper half-plane (or simply the Zygmund space) and denote it by . Recall that the space consists of all such that The quantity is a seminorm on the Zygmund space or a norm on , where 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 in the complex palne , Zygmund (see, e.g., [1, Theorem 5.3]) proved that a holomorphic function on continuous on the closed unit disk satisfies the following condition: if and only if

The family of all analytic functions on 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 . For some other information on this space and some operators on it, see, for example, .

Now note that is the distance from the point to the boundary of the unit disc, that is, , and that is the distance from the point to the real axis in which is the boundary of .

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

Theorem A. Assume and is a holomorphic self-map of . Then the following statements true hold.
(a) The operator is bounded if and only if(b)The operator is bounded if and only if

Motivated by Theorem A, here we investigate the boundedness of the operator . Some recent results on composition and weighted composition operators can be found, for example, in [4, 6, 7, 10, 12, 18, 2127].

Throughout this paper, constants are denoted by , they are positive and may differ from one occurrence to the other. The notation means that there is a positive constant such that . Moreover, if both and hold, then one says that .

#### 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 , , and . Then the function belongs to . Moreover

Proof. Let and . Then, we have where we have used the change of variables .

#### 3. Main Result

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

Theorem 3.1. Assume and is a holomorphic self-map of . Then is bounded if and only if
Moreover, if the operator is bounded, then

Proof. First assume that the operator is bounded.
For , set
By Lemma 2.1 (case ) we know that for every . Moreover, we have that
From (3.5) and since the operator is bounded, for every , we obtain
We also have that
Replacing (3.7) in (3.6) and taking , we obtain and consequently
Hence if we show that (3.1) holds then from the last inequality, condition (3.2) will follow.
For , set
Then it is easy to see that and by Lemma 2.1 (cases and ) it is easy to see that
From this, since is bounded and by taking , 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 for functions (note that ), we have

By differentiating formula (3.15), we obtain for each , from which it follows that

By using the change , 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 . 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 is bounded. Moreover, if we consider the space , we have that
From (3.14) and (3.23), we obtain the asymptotic relation (3.3).

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