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`Journal of Function SpacesVolume 2015, Article ID 376089, 6 pageshttp://dx.doi.org/10.1155/2015/376089`
Research Article

## Necessary and Sufficient Conditions for Inner Functions to Be in -Spaces

Department of Physics and Mathematics, University of Eastern Finland, P.O. Box 111, 80101 Joensuu, Finland

Received 17 July 2015; Accepted 8 November 2015

Copyright © 2015 Atte Reijonen. 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

Under some regularity conditions on , inner functions in -spaces are characterized in the following way: an inner function belongs to if and only if it is a Blaschke product associated with satisfying , where . The result generalizes earlier theorems in (Essén et al., 2006) and (Pérez-González and Rättyä, 2009).

#### 1. Introduction

Inner functions are bounded analytic functions in the unit disc such that their moduli are one almost everywhere on the boundary . Each inner function can be represented as a product of a Blaschke product and a singular inner function [1]. Singular inner functions are of the form where is a positive measure on which is singular with respect to the Lebesgue measure, and, for and , is a Blaschke product associated with if [2]. Information about inner functions can be found in [3]; see also [46], for example.

An analytic function in belongs to for and if Here is such that , is Green’s function, and is the Lebesgue area measure. Information about -spaces can be found in [7, 8]; see also the beginning of the next section.

Our purpose is to study the behaviour of inner functions in -spaces. In particular, we are interested in the Möbius invariant -spaces. Related to , it is proved that only inner functions there are Blaschke products if satisfies certain regularity conditions; see Proposition 7. Under corresponding assumptions, our main result, Theorem 1, gives a complete characterization of inner functions in the spaces. The results generalize [9, Theorems  1.3 and  1.4] and the essential content of [10, Theorem  5.1].

Theorem 1. Let , and assume that there exists such that a nondecreasing satisfies and for . Then an inner function belongs to if and only if it is a Blaschke product associated with a sequence satisfyingwhere .

Note that an assertion similar to Theorem 1 can be found in [11]. The proof there, however, contains some inaccuracies and does not seem to yield the claimed result. It is also worth noticing that the weight function is essentially so-called normal one in Theorem 1 [12].

The notation means that is essentially increasing; that is, for . The term essentially decreasing, shortly , is understood in an analogous manner. Here if and . The notation is used if there exists a constant such that , and is understood in a similar manner. Note that the constant may depend on fixed parameters or functions.

The remainder of the paper is organized as follows. In the next section, some auxiliary results are presented. Section 3 contains necessary conditions for singular inner functions to be in -spaces and the main purpose of the last section is to prove Theorem 1. Also Proposition 7 and an alternative version of Theorem 1 are stated and proved in the last section.

#### 2. Auxiliary Results

In this section, we present auxiliary results related to -spaces and inner functions. The weight function plays a crucial role in these results.

For simplicity, we denote the following conditions:(a) is continuous and nondecreasing.(b).(c) for .If (b) does not hold, then contains constant functions only [7]. Hence it is natural to assume (b), even though it would not be necessary. Nevertheless, it will always be mentioned if an assumption is made.

In the next lemma, we recall some basic properties of -spaces. It follows by combining results of [7]. For the lemma, we denote that belongs to for if . If , then we use the notation . Here means that is analytic in . Moreover, write if .

Lemma A. Let and , and assume that satisfies (a) and (b). Then the following statements are valid: (i). Moreover, if and only if (ii)If for and for , then .(iii) if and only if and (iv). Moreover, if and only if .

It is worth noticing that the statements (i)–(iii) of Lemma A are valid even if is discontinuous. This can be seen by looking the proofs of [7, Theorems  2.1 and  3.1].

Denote that , where is a subarc of such that . Then a positive measure on is a -Carleson measure if

Now we can characterize -spaces by using -Carleson measures. This result has been earlier presented in [11, Theorem  1], but practically, it follows by modifying the proof of [10, Theorem  3.1].

Lemma B. Let and , and assume that satisfies (a)–(c) andThen if and only if and is a -Carleson measure.

The next lemma shows that under certain conditions one may apply the Schwarz-Pick theorem without any essential loss of information. Before the lemma, we underline that the assumptions (a)–(c) are not necessary therein. In fact, it suffices to assume that but, of course, if (b) does not hold, then the statements of the lemma are trivial.

Lemma 2. Let and , and let be an inner function. If (i) and (ii) and thenis satisfied for all and almost all .

Proof. Assume first that (i) holds, and denote . Since for almost all , by Fubini’s theorem and (i), we obtain for almost all . If (ii) holds, then an application of [13, Theorem  2] yields for almost all . This completes the proof.

Corollary 3. Let , , and such that , and let be an inner function. If satisfies for , then (11) holds for all and almost all .

Proof. By the assumption, we have Similarly, for , we obtain Now the assertion follows from Lemma 2 applying the formulas above.

It is worth noticing that [9, Lemma  2.1] is a special case of Lemma 2. This is easy to see by choosing and in Corollary 3.

Denote that satisfies (A), if (8) and (a)–(c) are satisfied, and (B) if (9) holds for and (10) holds for . Related to (A) and (B), we end this section by proving the following consequence of Lemma 2.

Corollary 4. Let and , and assume that satisfies (A) and (B). Let , where is an inner function for all . Then if and only if for all .

Proof. Hölder’s inequality yields . Hence if for all . The other implication follows by applying Lemmas B and 2 together with the fact that for all .

#### 3. Singular Inner Functions and -Spaces

This section contains necessary conditions for singular inner functions to be in -spaces.

Write , where and .

Lemma 5. Let and , and assume that satisfies (A) and (B). If is the singular inner function associated with a measure and there exists such that and either or then .

Proof. Since , we may assume that . Then, for , Lemma 2 yields Since for all with fixed , we obtain Hence the assertion follows by Lemma B.

If for all , then by [14, Theorem  7.15]. In particular, there exists always such that . Hence the following result is a direct consequence of Lemma 5.

Corollary 6. Let and , and assume that satisfies (A) and (B). If is the singular inner function associated with a measure and then .

#### 4. Blaschke Products and -Spaces

The main purpose of this section is to prove Theorem 1. We begin by showing that all inner functions in are Blaschke products if satisfies certain regularity conditions. After that an alternative version of Theorem 1 is stated and proved; and finally, Theorem 1 follows by applying this result.

Proposition 7. Let and , and assume that a nondecreasing satisfies (b). Then the following statements are valid: (i)If , then only inner functions in are finite Blaschke products.(ii)If and , then only inner functions in are Blaschke products.

Proof. Since only inner functions in VMOA are finite Blaschke products [15], the statement (i) follows from the inclusions for .
By using [16, Lemma  2] twice together with [16, Corollary  3], [10, Lemma  2.3], and Lemma A(ii), we can assume that is second differentiable, for , as , and for . Hence it is clear that (A) holds. Moreover, the condition follows from [11, Lemma  3] and (B) with follows from the proof of Corollary 3 using [10, Lemma  2.2].
We may assume that because the inclusion yields for . Hence Corollary 6 implies that does not contain any singular inner functions. Now, since each inner function can be presented as a product of a Blaschke product and a singular inner function, the statement (ii) follows by Corollary 4.

Remark 8. In many cases, contains also nontrivial Blaschke products unlike for . The contrast between these spaces is strong also in the general case. More precisely, using the inclusion and [1, Theorem  5.1], we find that if , then ; that is, the boundary function satisfies the Lipschitz condition of order [1]. In particular, belongs to the disc algebra . On the other hand, by [10, Corollary  3.1], it is easy to find such that belongs to .

Next we prove an alternative version of Theorem 1. The proof uses some ideas from [17, 18]. For the result, denote .

Theorem 9. Let , and assume that satisfies the following conditions: (i) is nondecreasing.(ii).(iii).(iv).Then an inner function belongs to if and only if it is a Blaschke product associated with a sequence satisfying (4).

Proof. By a similar manner as in the proof of Proposition 7, we may assume that satisfies the basic assumption of [11], which means that, in addition to (i) and (ii), is continuous, for and for .
Assume first that an inner function belongs to for some . Then Proposition 7 implies that is a Blaschke product. Hence, using [11, Theorem  10], we obtain that the zero sequence of satisfies (4).
If is a Blaschke product associated with a sequence satisfying (4), then [11, Lemma  5] with parameters and yields Therefore by [11, Proposition  8], and hence the assertion follows.

We proceed to prove Theorem 1. First, it is known that satisfies (i) of Theorem 9. On the other hand, since for , Lemma A(ii) together with the assumption yields Therefore the condition (ii) holds. Moreover, we obtain and consequently (iii) is satisfied. Regarding (iv), we may assume, by Lemma A(ii), that there exists such that for all . Thus and hence, the assertion finally follows by Theorem 9.

We end this paper with the following remark.

Remark 10. We make the following observations about Theorem 1 and Proposition 7: (i)If with , then [19, 20]. Related to , Theorem 1 and Proposition 7 generalize [9, Theorems  1.3 and  1.4]. Moreover, by [10, Lemmas  2.1,  2.2 and Corollary  3.1], it is easy to see that Theorem 1 generalizes the essential content of [10, Theorem  5.1].(ii)The Möbius invariance of plays an important role in the proof of [11, Theorem  10]. Hence, if one want to characterize Blaschke products in -spaces where , using a similar technique as in this paper, it is necessary to assume that is Möbius invariant.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The research reported in this paper was supported in part by the Väisälä Foundation, the Academy of Finland, Project no. 268009, and the Faculty of Science and Forestry of the University of Eastern Finland, Project no. 930349. The author thanks the University of Malaga for hospitality during his visit there.

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