Abstract

The boundedness of the composition operator from the weighted Bergman space to the recently introduced by the author, the th weighted space on the unit disc, is characterized. Moreover, the norm of the operator in terms of the inducing function and weights is estimated.

1. Introduction

Let be the open unit disc in the complex plane , the Lebesgue area measure on , , , and the space of all analytic functions on the unit disc.

The weighted Bergman space , where and , 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

Let be a positive continuous function on a set (weight) and be fixed. The th weighted-type space on , denoted by consists of all such that

For the space becomes the weighted-type space , for the Bloch-type space , and for the Zygmund-type space .

For , the quantity is a seminorm on the th weighted-type space and a norm on , where is the set of all polynomials whose degrees are less than or equal to . A natural norm on the th weighted-type space can be introduced as follows: where is an element in . With this norm, the th weighted-type space becomes a Banach space.

For is obtained the space , on which a norm is introduced as follows: Some information on Zygmund-type spaces on the unit disc and some operators on them can be found, for example, in [16], for the case of the upper half-plane, see [7, 8], while some information in the setting of the unit ball can be found, for example, in [913]. This considerable interest in Zygmund-type spaces motivated us to introduce the th weighted-type space (see [8]).

Assume is a holomorphic self-map of . The composition operator induced by is defined on by

A typical problem is to provide function theoretic characterizations when induces bounded or compact composition operators between two given spaces of holomorphic functions. Some classical results on composition and weighted composition operators can be found, for example, in [14], while some recent results can be found in [1, 5, 7, 1534] (see also related references therein).

Here we characterize the boundedness of the composition operator from the weighted Bergman space to the th weighted space on the unit disc when . The case was previously treated in [16, 22, 24, 31, 35]. Hence we will not consider this case here. See also [36] for some good results on weighted composition operators between weighted-type spaces. The case was treated, for example, in [26, 32]. For some other results on weighted composition operators which map a space into a weighted or a Bloch-type space, see, for example, [15, 1721, 23, 25, 33, 34].

Let and be topological vector spaces whose topologies are given by translation-invariant metrics and , respectively, and be a linear operator. It is said that is metrically bounded if there exists a positive constant such that for all . When and are Banach spaces, the metrically boundedness coincides with the usual definition of bounded operators between Banach spaces.

If is a Banach space, then the quantity is defined as follows: It is easy to see that this quantity is finite if and only if the operator is metrically bounded. For the case this is the standard definition of the norm of the operator , between two Banach spaces. If we say that an operator is bounded, it means that it is metrically bounded.

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. Auxiliary Results

Here, we quote several auxiliary results. The first lemma is a direct consequence of a well-known estimate in [37, Proposition  1.4.10]. Hence, we omit its proof.

Lemma 2.1. Assume , , and . Then the function belongs to . Moreover, .

The next lemma is folklore and was essentially proved in [38]. We will sketch a proof of it for the completeness and the benefit of the reader.

Lemma 2.2. Assume , , and . Then there is a positive constant independent of such that

Proof. By the subharmonicity of the function , , applied on the disk: and since we have that
From (2.5) and in light of the following well-known asymptotic relation [38]: the lemma easily follows.

Lemma 2.3. Assume and Then .

Proof. By using elementary transformations, we have from which it follows that which along with the fact implies the lemma.

We will also need the classical Faà di Bruno's formula where and the sum is over all nonnegative integers satisfying . For a nice exposition related to this formula see, for example, [39].

By using Bell polynomials , (2.10) can be written in the following form:

Remark 2.4. Since the summation in (2.11) is from 1 to . Moreover, since and , (2.11) can be written in the following form:

3. Main Result

Here, we formulate and prove our main result.

Theorem 3.1. Assume , , , is a weight on and is a holomorphic self-map of . Then is bounded if and only if where for each fixed , the sum is over all nonnegative integers such that and .
Moreover, if the operator is bounded, then

Remark 3.2. Note that by (2.11) we see that the conditions in (3.1) can be written in the following form:

Proof. First assume that conditions in (3.1) hold. By formula (2.10) and Lemma 2.2 we have
From this, (2.2) with , and by conditions in (3.1), it follows that the operator is bounded. Moreover, if we consider the space , we have that
Now assume that the operator is bounded. For a fixed , and constants , set
Applying Lemma 2.1 we see that for every . Moreover, we have that
Now we show that for each , there are constants , such that Indeed, by differentiating function , for each , the system in (3.8) becomes
By using Lemma 2.3 with , we obtain that the determinant of system (3.9) is different from zero from which the claim follows.
Now for each , we choose the corresponding family of functions which satisfy (3.8) and denote it by .
For each , the boundedness of the operator along with (2.10) and (3.7) implies that for each : where (for each fixed ) the sum is over all nonnegative integers such that and .
From (3.10), it follows that for each ,
Now we use consecutively the test functions in order to deal with the case . Note that
By applying (2.11) to the function we get which along with the boundedness of the operator and (3.13) implies that or equivalently (see Remark 2.4).
Further, by applying formula (2.11) to the function we get From the boundedness of and (3.13), we get
From (3.16) and (3.17), and by using the triangle inequality it follows that
Using the fact and applying inequality (3.15) in (3.18) we get
Assume that we have proved the following inequalities: for and a .
Applying formula (2.11) to the function , , we have that From this, by using the boundedness of the operator , the boundedness of function , the triangle inequality, noticing that the coefficient at is independent of (it is equal ), and finally using hypothesis (3.20), we easily obtain Hence, by induction, we get that (3.22) holds for each .
From (3.22) and bearing in mind Remark 2.4, for each fixed , we have that where as usual for a fixed the sum is over all nonnegative integers such that and .
Hence from (3.11) and (3.23), we get
From (3.5) and (3.24), we obtain asymptotic relation (3.2).

Acknowledgment

The author would like to express his sincere thanks to the referees for numerous comments which improved the presentation of this paper.