Abstract and Applied AnalysisVolume 2011 (2011), Article ID 698038, 16 pagesdoi:10.1155/2011/698038

Research Article

Integral-Type Operators from Bloch-Type Spaces to Spaces

Stevo Stević¹ and Ajay K. Sharma²

¹Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia²School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, India

Received 16 May 2011; Accepted 29 June 2011

Academic Editor: Narcisa C. Apreutesei

Copyright © 2011 Stevo Stević and Ajay K. Sharma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The boundedness and compactness of the integral-type operator where , is a holomorphic self-map of the unit disk and is a holomorphic function on , from -Bloch spaces to spaces are characterized.

1. Introduction

Let be the open unit disk in the complex plane, be its boundary, be disk centered at of radius , and let be the class of all holomorphic functions on . Let be the involutive Möbius transformation which interchanges points 0 and . If is a Banach space, then by we will denote the closed unit ball in .

The -Bloch space, , consists of all such that The little -Bloch space consists of all functions holomorphic on such that . The norm on is defined by With this norm, is a Banach space, and the little -Bloch space is a closed subspace of the -Bloch space. Note that is the usual Bloch space.

Given a nonnegative Lebesgue measurable function on the space consists of those for which where is the normalized area measure on [1]. It is known that is a seminorm on which is Möbius invariant, that is, where is the group of all automorphisms of the unit disk . It is a Banach space with the norm defined by

The space consists of all such that It is known that is a closed subspace of . For classical spaces, see [2].

It is clear that each contains all constant functions. If consists of just constant functions, we say that it is trivial. is nontrivial if and only if Throughout this paper, we assume that condition (1.8) is satisfied, so that the space is nontrivial. An important tool in the study of spaces is the auxiliary function defined by where the domain of is extended to by setting when . The next two conditions play important role in the study of spaces. (a)There is a constant such that for all(b)The auxiliary function satisfies the following condition:

Let denote the class of all nondecreasing continuous functions on satisfying conditions (1.8), (1.10), and (1.11).

A positive Borel measure on is called a -Carleson measure [3] if where the supermum is taken over all subarcs , is the length of , and is the Carleson box defined by

A positive Borel measure is called a vanishing -Carleson measure if

We also need the following results of Wulan and Zhu in [3], in which spaces are characterized in terms of -Carleson measures.

Theorem 1.1. Let . Then a positive Borel measure on is a -Carleson measure if and only if Also, is a vanishing -Carleson measure if and only if

From Theorem 1.1 and the definition of the spaces and , we see that when , then if and only if the measure is a -Carleson measure, while if and only if this measure is a vanishing -Carleson measure.

Let be the family of all holomorphic self-maps of , , and . We define an integral-type operator as follows: Operator (1.17) extends several operators which has been introduced and studied recently (see, e.g., [4–9]). For related operators in -dimensional case, see, for example, [10–19]. For some classical operators see, for example, [20, 21] and the related references therein. For other product-type operators, see [22] and the references therein.

Motivated by [23, 24] (see also [25–29]), we characterize when and induce bounded and/or compact operators in (1.17) from -Bloch to spaces.

Throughout this paper, constants are denoted by ; they are positive and not necessarily the same at each occurrence. The notation means that there is a positive constant such that .

2. Auxiliary Results

Here, we quote several lemmas which will be used in the proofs of the main results in this paper. The following lemma is folklore (see, e.g., [30]).

Lemma 2.1. For any and , the following inequalities hold

The next lemma is obtained in [31, 32].

Lemma 2.2. Let . Then there are two functions such that Also, if , then there are two functions and , such that

The next Schwartz-type lemma [33] is proved in a standard way, so we omit the proof.

Lemma 2.3. Let , and . Then is compact if and only if for any bounded sequence in converging to zero on compacts of , we have .

Lemma 2.4. Let , and . Then is weakly compact if and only if it is compact.

Proof. By a known theorem is weakly compact if and only if is weakly compact. Since (the Bergman space) and has the Schur property, it follows that it is equivalent to , is compact, which is equivalent to , is compact, as claimed.

Lemma 2.5. Let , and . Then is compact if and only if is bounded.

Proof. By Lemma 2.4, is compact if and only if it is weakly compact, which, by Gantmacher's theorem ([34]), is equivalent to . Since and by a standard duality argument on , this can be written as , which by the closed graph theorem is equivalent to is bounded.

For , set

Lemma 2.6. Let , and . If is finite at some point , then it is continuous on .

Proof. We follow the lines of Lemma  2.3 in [24]. From the elementary inequality and since is nondecreasing and satisfies (1.10), we easily get From (2.7) and since is finite, it follows that is finite at each point . Let be fixed, and let be a sequence converging to .We have From (2.6), we have that for such that , say , holds and consequently for , it holds From this and since is finite at , by the Lebesgue dominated convergence theorem, we get that the integral in (2.8) converges to zero as which implies that as , from which the lemma follows.

3. Boundedness and Compactness of

In this section, we characterize the boundedness and compactness of the operators . Let

Theorem 3.1. Let , and , or and . Then the following statements are equivalent. (a) is bounded. (b) is bounded. (c). (d) is a -Carleson measure. Moreover, if is bounded, then the next asymptotic relations hold

Proof. By Theorem 1.1, it is clear that (c) and (d) are equivalent.(c) (a). Let . First note that for each and . From this and by Lemma 2.1, we have from which the boundedness of follows, and moreover . This implication is obvious.. By Lemma 2.2, if , there are two functions such that (2.3) holds, and if and such that (2.4) holds. Let It is known (see [30]) that for each and , we have From this, Lemma 2.2, and since , , we have that there is a such that Now note that for any , the functions , belong to , and moreover, .Applying (3.7), using an elementary inequality, the boundedness of , and the last inequality, we obtain Letting in (3.8) and using the monotone convergence theorem, we get On the other hand, for , we get which implies From (3.9) and (3.10), (c) follows. Moreover we get . From this, (3.4) and since the asymptotic relations in (3.2) follow, finishing the proof of the theorem.

Theorem 3.2. Let , and , or and . Let be bounded. Then the following statements are equivalent. (a) is compact. (b) is compact. (c) is weakly compact. (d), and

Proof. By Lemma 2.4, we have that (b) is equivalent to (c).(d) (a). Let be a bounded sequence in , say by , converging to zero uniformly on compacts of . Then also converges to zero uniformly on compacts of . From (3.11) we have that for every there is an such that for Therefore, by Lemma 2.1 and (3.12), we have that for Letting in (3.13), using the first condition in (d) and as , it follows that . Thus, by Lemma 2.3, is compact.(a) (b). The implication is trivial since .(b) (d). By choosing , , we have that the first condition in (d) holds. Let , . It is easy to see that is a bounded sequence in converging to zero uniformly on compacts of . Hence, by Lemma 2.3, it follows that as . Thus, for every , there is an , such that for From (3.14) we have that for each and Hence, for , we have that Let , and let . Then , , , and uniformly on compact subsets of as . The compactness of implies Hence, for every , there is a such that From this and (3.16), we have that for such and each From (3.19) we conclude that for every , there is a and such that for Since is compact, we have that for every there is a finite collection of functions such that, for each , there is a , such that On the other hand, from (3.20), it follows that if , then for and all , we have From (3.21) and (3.22), we have that for and every If we apply (3.23) to the delays of the functions in (3.5) which are normalized so that they belong to and then use the monotone convergence theorem, we easily get that for where is chosen as in (3.7) from which (3.11) follows, as desired.

Theorem 3.3. Let , and , or and . Then the next statements are equivalent. (a) is bounded. (b) is compact. (c) is compact. (d) is weakly compact. (e). (f) is a vanishing -Carleson measure.

Proof. By Theorem 1.1, (e) and (f) are equivalent; by Lemma 2.4, (c) is equivalent to (d), while, by Lemma 2.5, (a) is equivalent to (c). Also (b) obviously implies (a).(a) (e) Let and be as in (3.5). Then from (3.7) and an elementary inequality, we get For , we get . From this and since , by letting , we get that (e) holds.(e) (b). We have that for every there is a so that for On the other hand, by Lemma 2.6, is continuous on , so uniformly bounded therein, which along with (3.26) gives the boundedness of on . Hence, by Theorem 3.1, is bounded. By Lemma 2.1, we have so is bounded.Now assume that is a bounded sequence in , say by , converging to zero uniformly on compacta of as . To show that the operator is compact, it is enough to prove that there is a subsequence of such that converges in as . By Lemma 2.1 and Montel's theorem, it follows that there is a subsequence, which we may denote again by converging uniformly on compacta of to an , such that . Since , then clearly . We show that From (3.26), Lemma 2.1, and some simple calculation, we obtain For and , let Lemma 2.6 essentially shows that is continuous on . Hence, for each , there is a such that . Moreover, there is a neighborhood of such that, for every , . From this and since the set is compact, it follows that there is a such that when . This along with Lemma 2.1 implies that By the Weierstrass theorem uniformly on compacta as , from which along with (2.2) and since is compact, for , it follows thatFrom (3.29)–(3.32) and since for each , we easily get (3.28), from which (b) follows, finishing the proof of this theorem.

Theorem 3.4. Let , and , or and . Then the following statements are equivalent. (a) is bounded, (b), and

Proof. Suppose (b) holds and . Then by Theorem 3.1, is bounded. We show , for every . Since , we have that, for every , there is an such that (see, e.g., the idea in [35, Lemma 2.4]) Thus, We also have Combining (3.35) and (3.36), we get . Hence, is bounded.Conversely, if is bounded, then is bounded too. Thus, by Theorem 3.1, we get the first condition in (b). For , we get , which is equivalent to (3.33), finishing the proof of the theorem.

If , we simply denote the operator by .

Theorem 3.5. Let , and . Then the following statements are equivalent. (a) is bounded. (b) is bounded. (c). (d) is a -Carleson measure. (e) is compact. (f) is compact. (g) is weakly compact. Moreover, if is bounded, then the next asymptotic relations hold

Proof. The proof of the equivalence of statements (a)–(d) of this theorem is similar to the proof of Theorem 3.1; moreover, the implication (b) (c) is much simpler since it follows by using the test function . That (c) is equivalent to (e)–(g) is proved similarly as in Theorem 3.2, by using the well-known fact that if a bounded sequence in , converges to zero uniformly on compacts of , then it converges to zero uniformly on the whole . The details are omitted.

The proof of the next theorem is similar to the proofs of Theorems 3.3 and 3.4 and will be omitted.

Theorem 3.6. Let , and . Then the following statements are equivalent. (a) is bounded. (b) is bounded. (c) is compact. (d) is compact. (e) is weakly compact. (f). (g) is a vanishing -Carleson measure.

Acknowledgment

This work is partially supported by the National Board of Higher Mathematics (NBHM)/DAE, India (Grant no. 48/4/2009/R&D-II/426) and by the Serbian Ministry of Science (Projects III41025 and III44006).

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