Abstract and Applied Analysis

VolumeΒ 2011, Article IDΒ 626254, 26 pages

http://dx.doi.org/10.1155/2011/626254

## Inner Functions in Lipschitz, Besov, and Sobolev Spaces

^{1}Departamento de AnΓ‘lisis MatemΓ‘tico, Facultad de Ciencias, Universidad de MΓ‘laga, 29071 MΓ‘laga, Spain^{2}MatematiΔki Fakultet, University of Belgrade, PP. 550, 11000 Belgrade, Serbia

Received 23 February 2011; Accepted 12 April 2011

Academic Editor: WolfgangΒ Ruess

Copyright Β© 2011 Daniel Girela et al. 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

We study the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spaces with or any of the Besov spaces with and , except when , , and or when , , and are finite Blaschke products. Our assertion for the spaces , , follows from the fact that they are included in the space . We prove also that for , is not contained in and that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values of relating the membership of an inner function in the spaces under consideration with the distribution of the sequences of preimages , . In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.

#### 1. Introduction

One of the central questions about inner functions is that of their membership in some classical function spaces. This problem was studied in a number of papers in the 70's and 80's (see, e. g.,[1β10]) and also recently (see, e. g., [11β18]). In this paper, we shall be mainly concerned in studying the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces.

Denote by the open unit disk in the complex plane , and by the class of analytic functions on . The classical Hardy space , , consists of those functions for which where if , and .

We mention [19, 20] as references for the theory of Hardy spaces. The spaces form a decreasing chain as increases, and any function in has nontangential limits almost everywhere, with the additional property that the boundary function thus formed defines an isometry between and . The sequence of zeros of an function, counting multiplicities, satisfies the so-called *Blaschke condition*: . This condition characterizes the zero sequences of functions. By writing when , and , the Blaschke condition implies that the product converges absolutely and uniformly in each compact subset of , hence defining a function , called * the Blaschke product associated to the sequence *, which is analytic in the unit disk , and whose exact sequence of zeros, counting multiplicities, is . Also, , and its boundary function has modulus 1 almost everywhere.

An * inner function* in is an function whose nontangential boundary function has modulus 1 almost everywhere. Thus, Blaschke products are inner functions. Any inner function admits a factorization of the type , where is a real constant, is a Blaschke product (carrying all the zeros of ), and is what it is called a * singular inner function*, having the form
where * ΞΌ* is a positive Borel measure on , singular with respect to Lebesgue measure. From now on, a Blaschke product times a unimodular constant (that may be 1) will also be called a Blaschke product, just to simplify the language. Continuing with the terminology that may appear in the paper, a Blaschke product with a finite number of zeros will be called a

*finite Blaschke product*, while that with an infinite number of zeros will be called an

*infinite Blaschke product*.

The different classes of analytic functions that will be treated in this paper are presented now. Before, let us say a word about the notational conventions used in this paper. Constants will usually be denoted by the letter . Their dependence on other quantities, if specified, will appear as subindexes. In the same expression, the constant may change from one occurrence to the other. Two quantities or expressions, and , are said to be comparable (written ) if there exists a positive constant such that . If functions are involved in the quantities that are being compared, the constants relating them do not usually depend on those functions, neither on their variables.

The * weighted Bergman space *, , consists of those functions such that
where denotes the Lebesgue area measure in . We mention the books [21, 22] as general references for the theory of Bergman spaces.

If is analytic in and , * the fractional derivative of order ** of *, is defined as
(This definition also makes sense for , only that the name βderivativeβ would look a bit awkward.) For a positive integer , it is actually an β-th derivativeβ: . Besides, the integral means and are usually interchangeable, in the sense that their quotient is bounded away from 0 and (see, e.g., [23] for this fact and the formulas that appear below). In general, there is a formula to recover the function from its fractional derivative, which can be easily verified,
This yields the estimate

If and , then a function is said to belong to the * Lipschitz space * if
The subspace consists of those for which
Notice that is another name for the usual Bloch space , of functions such that . Analogously, is the little Bloch space , of functions such that .

Replacing the sup-norm with an -norm in the above definition gives way to the * Besov space *, , , , consisting of those functions for which
The space obtained when is precisely . Other classes treated in this paper are the *Hardy-Sobolev spaces *, , , consisting of those functions for which

The paper of Flett [23] gives many relations between the different types of integrals just mentioned above. Since they will occur recurrently along this paper, it may be convenient to state once for all some of them. Starting from the estimate (1.6), and working it out into the integral means of order gives The same estimate (1.6) combined with variants of Hardy's inequality gives

We continue displaying more estimates. The following one is due essentially to Hardy and Littlewood (see, e.g., [19, Theorem 5.9], or [24, Lemma 3.4]):

Finally, we provide also the following estimates, due to Littlewood and Paley [25, Theorems 5 and 6], for , and to Vinogradov [26, Lemma 1.4], for . which can be restated as , for , and , for .

With the above estimates, we can easily obtain some more relations of inclusion between the different spaces considered in this paper. We enumerate some of these relations, with the purpose of having them at hand. (1)If , : If , then as , so by using the increasing behavior of the means , obtain that , as , that is, that .(2)If , , because , as , for all .(3)If , , and , because for all and all .(4)If , . Indeed, for , (1.14) gives (5)If , , because of (1.11) and (1.12). (6)If , . (When , this says that increases with ). A proof of this lies in an application of (1.15) and (1.14),(7). This is again an application of (1.14), where . (8). Again an application of (1.14). (9)If , . This is (1.16). (10)If , . And this is (1.17).

In the following, we introduce some notation related to inner functions. Given an inner function and a point , its * Frostman shift * is defined as
A classical result of Frostman (see, e.g., [20, Section 2.6] asserts that if is an inner function in then the Frostman shifts are Blaschke products for all except for those in a set (depending on ) of logarithmic capacity zero. Even more, if cannot be analytically continued across one boundary point, that is, if is not a finite Blaschke product, then for all , with a set of logarithmic capacity zero, the Frostman shift is an infinite Blaschke product.

The fact that mixed norms of derivatives of an inner function are comparable to those of its Frostman shifts must be a well known result, which we have not found in the literature. Since it plays a key role to obtain some of our results, we include a proof just for the sake of completeness.

Lemma 1.1. *Let , , , and . Then , with constants depending only on , and , but not on or . In the case , the formulation is
*

*Proof. *Put , with a positive integer and . We shall only consider the case . The procedure for is rather similar. Also, it is enough to estimate the -norm of in terms of that of , for .

Writing , observe that the first derivative of can be written as , and, in general, for a positive integer , the th derivative is given by . This, together with the FaΓ di Bruno's formula for the th derivative of a composition, gives
where , and the sum runs over all -tuples of nonnegative integers such that . Observe that if , the quantities and are bounded away from 0 and by constants depending only on and , but not on .

Now, to estimate , we use, in order, (1.14), (1.23), twice HΓΆlder's inequality (HΓΆ) with indices , again (1.14), and finally we appeal to the fact (I) that (obtained as a result of using Cauchy's integral formula for the th derivative and Lemma 3 in Section 5.5 of Duren's book [19]),

#### 2. Inner Functions in the Spaces and ,

Ahern and JevtiΔ [5] proved that a Blaschke product lies in the space () if and only if its sequence of zeros is a finite union of exponential sequences, (see also VerbitskiΔ [27] for the case ). We refer the reader to the recent work of JevtiΔ [15] on this subject where references to previous works are given. In particular, we have Our results in this section imply that the opposite is true for all the spaces with and , except for the mentioned case, , and , and when , and .

Theorem 2.1. *
(a) Let and . Then the only inner functions in are finite Blaschke products. **
(b) If then the only inner functions in are finite Blaschke products.*

*Proof. *To prove (a) observe that . The first inclusion comes from the properties above and the last inclusion may be found in [28, Corollary 2.3], or directly using (1.13). Now, it is well known (see, e.g., [19, Theorem 5.1] that any function in , with , (even if it is forced to be a constant), belongs to the disk algebra (that is, it admits a continuous extension to ). Thus if we are in the conditions of part and is an inner function in then . Then it follows easily that is a finite Blaschke product. Indeed, write , where is a Blaschke product and is a singular inner function. The fact that readily implies that is a finite Blaschke product and then it follows that also belongs to . Then is a function in the disk algebra without zeros and with , for all . A simple application of the maximum-minimum principle readily yields that is a unimodular constant. Thus, is a finite Blaschke product as asserted.

Let us now turn to prove part . The following results come in our aid.

Theorem A (see [8, Corollary 1.6]). *Let . If is a Blaschke product in , then is a finite Blaschke product.*Theorem B (see [5, Theorem 3.2]). *For each there exists such that if is a Blaschke product and
**
then is a finite Blaschke product. In particular, the only Blaschke products in are the finite ones.*

Using Lemma 1.1, and the fact that the Frostman shifts of a finite Blaschke product are again finite Blaschke products: we deduce that Theorem A and Theorem B yield the following.

Proposition 2.2. *For , the only inner functions in either or are the finite Blaschke products.*Now Theorem 2.1(b) follows from this result and the fact that .

The fact that immediately implies the following.

Corollary 2.3. *Let and . If then admits a continuous extension to . Consequently, the only inner functions in , with , are finite Blaschke products.*

It remains to consider the case and . Of course, contains the whole class of inner functions.

Let us deal now with the spaces . First of all, is the Bloch space, and, hence, it contains all inner functions.

Bishop [29] proved that the little Bloch space, , contains infinite Blaschke products. Since, for , is a subspace of , the natural question rises as whether contains or not infinite Blaschke products. The answer depends on the result of intersecting with the subspace of , consisting of those functions whose boundary values have vanishing mean oscillation. The space was introduced by Sarason [30] and admits a number of equivalent definitions. Among them, we mention that a function is said to belong to if for some (or, equivalently, for all) finite positive . Here, is the typical involutive automorphism of interchanging the points 0 and . Using this definition and the fact that nonconstant inner functions take values as close to 0 as desired, Anderson [31] proved that contains no inner functions other than finite Blaschke products. (See also [32] for an extensive survey on and .) In the following result, we use another characterization of ; it is the space of functions such that where is an interval in , is its length, and is the Carleson square defined by .

Theorem 2.4. *
(a), If , then . Consequently, any inner function in is a finite Blaschke product. **
(b) There are infinite Blaschke products in .*

*Proof. *To prove (a) observe that, since for , it suffices to settle the result for .

Thus, take and take an interval in with , then
and observe that, since , the right hand side tends to 0 as .

To prove (b), observe that, by Theorem 5.2 of [33], there is a (singular) inner function such that
This implies that for all . Also, as explained also in [33, after (1.1)], such inner function cannot be analytically continued across any boundary point of . Therefore, we may choose such that the Frostman shift is an infinite Blaschke product (actually, this is true for all except for those in a set of zero logarithmic capacity). Now, Lemma 1.1 shows that is an infinite Blaschke product in .

Once Theorem 2.4 is proved, it is natural to ask whether or not the inclusion holds for . An argument based on duality shows that this is not so.

Theorem 2.5. *If , then the class is nonempty.*

*Proof. *Observe that the dual of is under the usual pairing: , (see [32, 34]). Also, using the same techniques as in [4], we get that the dual of is , , under the same pairing as before. Thus, the problem reduces to show that is nonempty.

It is shown in Theorem 3 of [35] that the function , is univalent in and . Also, an argument given in [24, page 61] shows that there exist and such that
It then follows that , whenever . This finishes the proof.

#### 3. The Case Max

For this range of values, we shall obtain a number of results relating the membership of an inner function in Besov or Hardy-Sobolev spaces with the distribution of the preimages , . We start introducing certain counting functions.

If is an inner function and , denote by the exact sequence of zeros, multiplicities included, of , placed in increasing modulus as the subindex increases (in other words, is the ordered sequence of preimages of ). Writing , the distribution of zeros in each annulus may be studied with the sequences and : Observe that always. When , just write , , , and . The following relations may be used in the text without further notice.

Lemma 3.1. *Under the previous settings, let . Then *(a)* if and only , and, in either case, their -norms are comparable.*(b)*. *

*Proof. *In order to keep up with readability, it is better to omit the letter value in what follows, that is, assume .

To prove (a), assume first that for all , then for each ,
In the other direction, assume that for all . Given find the unique such that . This implies that , and thus,

To prove (b), assume first that . In the case , use an easy integral estimate and the fact that , to obtain the desired result,

In the case , it is easy to arrive at . Now we imitate the proof of Hardy's inequality given in [36, Theorem 326 in page 239]. Write , for , and . Then , for , and by the inequality between the geometric and arithmetic means [36, Theorem 9 in page 17],
So the sum on of the left hand side is negative, and, therefore, using HΓΆlder's inequality with exponents and ,
giving as a result that , as desired.

In the other direction, assume that . In the case , it is easily verified that . To continue, use that and that the function is decreasing in ,
from where it follows that .

It remains to deal with the case under the assumption . Here, we use that, when , (because ), and use also the Mean Value Theorem,

Now we recall the following characterization, due to Ahern [37, Theorem 6].

Theorem C (see [37, Theorem 6]). *Assume that , that , and that is an inner function. Then the following quantities are comparable,
*

*Remark 3.2. *An examination of the proof in which the quantity (3.12) is controlled by that of (3.10), shows that it does not really require the function to be inner. Any bounded function would just work fine.

In [15], the corresponding characterization for is mentioned without proof ((3.2) of [15]). Its verification is done by following the same steps of the previous result (even easier, Hardy's inequality is not needed).

Theorem D. *If , , and is an inner function, then the following quantities are comparable, *

For Blaschke products, the third author [15, 38] gave sufficient (and in special cases, necessary) conditions for their membership in in terms of the distribution of their zeros. Recall that a * Carleson-Newman sequence* is a finite union of * interpolating sequences*, and a sequence in the unit disk is called * interpolating* if it is * uniformly separated*, that is,

Theorem E (see [15, 38]). *Let be such that . Assume that , and that is a Blaschke product. If , then and
**
On the other hand, if the zero sequence of is Carleson-Newman and , then and their respective norms are equivalent.*

As a consequence of this result, Lemma 1.1, and the fact that the Frostman shifts of inner functions are Blaschke products almost always, we have that if with , , and is an inner function satisfying for , and , then , and the norm is controlled by the integral in (3.19).

The crux of the matter here is that the above condition is also necessary for to belong to .

Theorem 3.3. *Let be such that . Assume that , and that is an inner function. Then if and only if (3.19) holds for some . In that case, both quantities, and the integral in (3.19), are comparable.*

In order to prove this theorem, certain homogeneity property is needed. See [38, Lemma 4.4], [7, Lemma 2.2] for similar statements on -functions, and also [15, Proposition 3.1] for the case .

Lemma 3.4. *If , , , and , then , that is, Bloch functions in are also in for any . Furthermore, the following relation holds:
*

*Proof of Lemma 3.4. *The case will not be treated due to its similarity with the other cases. Take , , , and . We need to show that . For that, use (1.14) to find an equivalent quantity to , and then separate it into two factors, the first will be controlled by , and the second by .

Two more lemmas are needed.

Lemma F (see [15, Corollary 4.5]). *If , is an inner function, and if , then
*

Lemma G (see [15, Corollary 4.7]). *If , , is an inner function, and if , then
*

*Proof of Theorem 3.3. *Again, we deal only with the case . The sufficiency of condition (3.19) has already been established. To prove its necessity, assume that and, rather than imposing the whole restriction , just assume . Observe that the integral in (3.19) (without the power ) remains unchanged if we replace with . Now choose such that , and . If the result holds in this situation, then, by the homogeneity property of Lemma 3.4, we have
So it suffices to prove the result for and . In what follows . First assume that . Then use, in order, Minkowski's inequality for , the fact that , and finally Lemmas F and G together with Theorem C to arrive at the desired estimate for (3.19),

The case follows the same procedure, only that instead of using Minkowski's inequality, we use HΓΆlder's with exponents and , and then, after applying Lemma F and before Lemma G, use again Minkowski's inequality with ,

*Remark 3.5. *Along the proof of this theorem, we have actually proved that if is an inner function in , with and , then (3.19) holds and

Also, as we observed before, the integral in (3.19), or (3.27), is unchanged if is replaced with , (). This allows us to extend the homogeneity property of Lemma 3.4 to other values of , provided that we can apply Theorem 3.3, that is, that we work with inner functions and that .

Corollary 3.6. *Let be such that . Assume that , and that is an inner function in . Then for all .*

*Remark 3.7. *Of course, when , the class of inner functions in coincides with that of for any (or any with , whatsoever), because they only contain the finite Blaschke products. The same reasoning applies when and . Is it the same for ? that is, is the class of inner functions in the same for all ? The answer is affirmative for the class of Blaschke products [5, Theorem 3.1] and then, using once more Lemma 1.1 and the fact that the Frostman shifts of inner functions are almost always Blaschke products, we arrive at an affirmative answer for the whole class of inner functions in . We should mention here that we shall prove later (see Remark 4.4 below) that the only inner functions in are Blaschke products.

*Remark 3.8. *In view of the previous remark, we could ask whether the result of the corollary remains true for the whole range of . The answer is negative. Ahern and Clark [3, Lemma 2] have constructed a Blaschke product in but not in . By property , we deduce that , and this is the space that would be obtained from by taking , which coincides with with the usual notation.

*Proof of Corollary 3.6. *Notice first that Remark 3.5 implies that (3.27) holds for , and thus the integral on the left hand side is unchanged for . Now, take . Then . If now , then , and so , proving that by Theorem 3.3. If, on the contrary, , then , and we still have , and again, by Theorem 3.3.

*Remark 3.9. *As an application of these results, we show how to recover a known result by Protas [9, Theorem 1]. Assume that is a Blaschke product satisfying for some . By Lemma 3.1(b), this condition is equivalent to , which implies that by Theorem E. (Notice that this is equivalent to by (1.14).) Observe now that we can apply Corollary 3.6 with , obtaining that , which is the aforementioned result by Protas, with our notation. On the other hand, notice also that if , for some , and the zero sequence of is Carleson-Newman then, by Theorem E, .

Next we will turn to study the membership of inner functions in the spaces . Properties and , that is, the Littlewood-Paley inequalities, relate quite well with . Our main result in this direction is that these relations of inclusion become equalities when the spaces are cut with the class of inner functions.

Theorem 3.10. *Let and . Then the class of inner functions in coincides with that of inner functions in .*

The proof of this result requires again an homogeneity property.

Proposition 3.11. *Let and let be a Bloch function in . Then for all .*

This is a straightforward consequence of the following result and the complex maximal theorem.

Lemma H(see [5, Lemma 2.1]). *For each , there is a constant such that if then
*

The original proof of this lemma runs with functions instead of Bloch () functions. However, the lemma can be proved for Bloch functions by just noticing the validity of the estimate for Bloch functions.

*Proof of Theorem 3.10. *First notice that when , the only inner functions in and are the finite Blaschke products. So we may assume without loss of generality the additional hypothesis . (This already implies .)

Now consider the case . Then by . To go in the other direction, take an inner function in . Then, by Proposition 3.11, for all . For , we have , and hence by . Using now that , we get , and we conclude that by Corollary 3.6.

Next consider the case . (Then .) By , . To go in the other direction, take an inner function in . Since then, by Corollary 3.6, for all . Choose . Thus, as , gives . Finally, as , Proposition 3.11 gives .

*Remark 3.12. *A careful reading of this proof shows that, in fact, any inner function in is also in , whenever and .

The analogous to Corollary 3.6 is the following.

Corollary 3.13. *Let be such that . Assume that is an inner function in . Then for all .*

*Proof. *Take . If then by Proposition 3.11. Otherwise we have , which implies , and, together with , it gives . So, by Theorem 3.10, and, by Corollary 3.6, and, finally, by Remark 3.12, .

*Remark 3.14. *Again, since it only contains finite Blaschke products, the class of inner functions in , with , coincides with that of for all , or for the case, with the class of inner functions of any , with . As for the accuracy of the corollary with regards to whether there is a possibility to establish the result for the whole range of , the same example given in Remark 3.8 shows that it is impossible. The Blaschke product constructed in [3, Lemma 2] is in but not in , which is the space that would be obtained from by taking ( with the usual notation). By Theorem 3.10, implies that .

*Remark 3.15. *As an application, we again regain a known result of Protas [9, Theorem 2], namely, * if ** is a Blaschke product such that **, for some **, then *. Indeed, by Lemma 3.1(b), , and so by Theorem E, and we are done with by Theorem 3.10. On the other hand, Theorem E also says that if the zero sequence of is Carleson-Newman then the condition is also necessary for .

#### 4. The Case

Ahern and Clark [2, Theorem 3] proved that the only inner functions in are Blaschke products. Later on, Ahern and Jevti obtained the following generalization:

Theorem I (see [5, Theorem 2.1]). *If is an inner function and
**
for some , then is a Blaschke product.*

Now, all functions in satisfy condition (4.1) and, by , the same is true for all -functions with . So the following is immediate:

Corollary 4.1. *Let with . Then the only inner functions in are Blaschke products, finite ones if .*

This result finds its analogue for Besov spaces. Its essence may be traced back to the last corollary in [37].

Proposition J (see [37]). *Let with . Then the only inner functions in are Blaschke products, finite ones if .*

*Remark 4.2. *These results are again accurate, for the βatomicβ singular inner function is in for all with . Indeed, in [39], it is shown that, for any positive integer ,
This implies that for all , because by (1.11),
and also for all and all , because by (1.14),
Hence, given with , take the smallest integer such that and then take and . In this way,