#### 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 FrΓ©chet space with the translation invariant metric , , [1].

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., [2β14] 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, [15β19].

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 [20], 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, 21β27].

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).