Inner Functions in Lipschitz, Besov, and Sobolev Spaces

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 di ﬀ erent authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spaces H pα with 1 /p ≤ α < ∞ or any of the Besov spaces B p,qα with 0 < p, q ≤ ∞ and α ≥ 1 /p , except when p (cid:3) ∞ , α (cid:3) 0, and 2 < q ≤ ∞ or when 0 < p < ∞ , q (cid:3) ∞ , and α (cid:3) 1 /p are ﬁnite Blaschke products. Our assertion for the spaces B ∞ ,q 0 , 0 < q ≤ 2, follows from the fact that they are included in the space VMOA . We prove also that for 2 < q < ∞ , VMOA is not contained in B ∞ ,q 0 and that this space contains inﬁnite Blaschke products. Furthermore, we obtain distinct results for other values of α relating the membership of an inner function I in the spaces under consideration with the distribution of the sequences of preimages { I − 1 (cid:4) a (cid:5) } , | a | < 1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.


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., 1-10 and also recently see, e. g., 11-18 . In this paper, we shall be mainly concerned in studying the membership of inner functions in Besov, Lipschitz, and Hardy- where μ is a positive Borel measure on 0, 2π , 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 C. Their dependence on other quantities, if specified, will appear as subindexes. In the same expression, the constant C may change from one occurrence to the other. Two quantities or expressions, A and B, are said to be comparable written A B if there exists a positive constant C such that C −1 B ≤ A ≤ CB. 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 A p,α , 0 < p < ∞, −1 < α < ∞ consists of those functions f ∈ Hol such that f A p,α : α 1 π f z p 1 − |z| 2 α dA z D α f sz log α−1 1 s ds. 1.5 This yields the estimate If 0 < p ≤ ∞ and 0 ≤ α < ∞, then a function f ∈ Hol is said to belong to the Lipschitz space Λ p,α if f Λ p,α : sup 0<r<1 1 − r M p r, D 1 α f < ∞. 1.7 The subspace λ p,α consists of those f ∈ Hol for which lim r → 1 1 − r M p r, D 1 α f 0. 1.8 Notice that Λ ∞,0 is another name for the usual Bloch space B, of functions f ∈ Hol such that sup z∈ 1 − |z| 2 |f z | < ∞. Analogously, λ ∞,0 is the little Bloch space B 0 , of functions f ∈ Hol such that lim |z| → 1 1 − |z| 2 |f z | 0. Replacing the sup-norm with an L q -norm in the above definition gives way to the Besov space B p,q α , 0 < p ≤ ∞, 0 < q < ∞, 0 ≤ α < ∞, consisting of those functions f ∈ Hol 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 p gives The same estimate 1.6 combined with variants of Hardy's inequality gives

1.14
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 p ≥ 1, and to Vinogradov 26, Lemma 1.4 , for 0 < p < 1. 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.
so by using the increasing behavior of the means M p r, D 1 α f , obtain that 1 − r M p r, D 1 α f → 0, as r → 1, that is, that f ∈ λ p,α .
α , because M p 1 r, g ≤ M p 2 r, g for all r ∈ 0, 1 and all g ∈ Hol .
When q ∞, this says that Λ p,1/p increases with p . A proof of this lies in an application of 1.15 and 1.14 , 1.19 P 7 Λ p,α ⊆ ∩{B p,q β : β < α, 0 < q}. This is again an application of 1.14 , In the following, we introduce some notation related to inner functions. Given an inner function I and a point a ∈ , its Frostman shift I a is defined as A classical result of Frostman see, e.g., 20, Section 2.6 asserts that if I is an inner function in then the Frostman shifts I a are Blaschke products for all a ∈ except for those in a set E depending on I of logarithmic capacity zero. Even more, if I cannot be analytically continued across one boundary point, that is, if I is not a finite Blaschke product, then for all a ∈ \ E, with E a set of logarithmic capacity zero, the Frostman shift I a 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.
Then I a B p,q α I B p,q α , with constants depending only on p, q, α, and δ, but not on I or a. In the case q ∞, the formulation is Proof. Put 1 α n β, with n a positive integer and β ∈ 0, 1 . We shall only consider the case q < ∞. The procedure for q ∞ is rather similar. Also, it is enough to estimate the B p,q α -norm of I a in terms of that of I, for I I a −a . Writing ψ a z a/ 1 − az , z ∈ observe that the first derivative of I a can be written as I a 1 − |a| 2 /a 2 ψ a • I , and, in general, for a positive integer n, the nth derivative is given by I n a 1 − |a| 2 /a 2 ψ a • I n . This, together with the Faà di Bruno's formula for the nth derivative of a composition, gives where k n j 1 k j , and the sum runs over all n-tuples k k 1 , . . . , k n of nonnegative integers such that n j 1 jk j n. Observe that if a ∈ K δ , the quantities 1 − |a| 2 /|a| 2 and |ψ k a • I| |ψ • I| k 1 are bounded away from 0 and ∞ by constants depending only on k and δ, but not on a. Now, to estimate I a q B p,q α , we use, in order, 1.14 , 1.23 , twice Hölder's inequality Hö with indices {n/ jk j : k j / 0}, again 1.14 , and finally we appeal to the fact I that |I n z | ≤ C n 1 − |z| −n obtained as a result of using Cauchy's integral formula for the nth derivative and Lemma 3 in Section 5.5 of Duren's book 19 , and only if its sequence of zeros is a finite union of exponential sequences, see also Verbitskiȋ 27 for the case 1 ≤ p < ∞ . We refer the reader to the recent work of Jevtić 15 on this subject where references to previous works are given. In particular, we have for 0 < p < ∞, the space B p,∞ 1/p contains infinite Blaschke products.

2.1
Our results in this section imply that the opposite is true for all the spaces B p,q α with 0 < p, q ≤ ∞ and α ≥ 1/p, except for the mentioned case, 0 < p < ∞, q ∞ and α 1/p, and when p ∞, α 0 and 2 < q ≤ ∞.
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 Λ ∞,β , with β > 0, even if it is forced to be a constant , belongs to the disk algebra A that is, it admits a continuous extension to . Thus if we are in the conditions of part a and I is an inner function in B p,q α then I ∈ A. Then it follows easily that I is a finite Blaschke product. Indeed, write I z S z B z , where B is a Blaschke product and S is a singular inner function. The fact that I ∈ A readily implies that B is a finite Blaschke product and then it follows that S also belongs to A. Then S is a function in the disk algebra without zeros and with |S ξ | 1, for all ξ ∈ ∂ . A simple application of the maximum-minimum principle readily yields that S is a unimodular constant. Thus, I is a finite Blaschke product as asserted.

Abstract and Applied Analysis
Let us now turn to prove part b . The following results come in our aid.
for some or, equivalently, for all finite positive p. Here, ϕ a z a − z / 1 − az is the typical involutive automorphism of interchanging the points 0 and a ∈ . Using this definition and the fact that nonconstant inner functions take values as close to 0 as desired, Anderson 31 proved that V MOA contains no inner functions other than finite Blaschke products. See also Abstract and Applied Analysis 9 32 for an extensive survey on BMOA and V MOA. In the following result, we use another characterization of V MOA; it is the space of functions f ∈ Hol such that where J is an interval in ∂ , |J| is its length, and S J is the Carleson square defined by Proof. To prove a observe that, since B Thus, take f ∈ B ∞,2 0 and take an interval J in ∂ with |J| < 1/2, then and observe that, since f ∈ B ∞,2 0 , the right hand side tends to 0 as |J| → 0. To prove b , observe that, by Theorem 5.2 of 33 , there is a singular inner function I such that This implies that I ∈ B ∞,q 0 for all q > 2. 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 a ∈ such that the Frostman shift I a is an infinite Blaschke product actually, this is true for all a ∈ except for those in a set of zero logarithmic capacity . Now, Lemma 1.1 shows that I a is an infinite Blaschke product in ∩ 2<q<∞ B ∞,q 0 .
Once Theorem 2.4 is proved, it is natural to ask whether or not the inclusion V MOA ⊆ B ∞,q 0 holds for 2 < q < ∞. An argument based on duality shows that this is not so. It is shown in Theorem 3 of 35 that the function f z 1 − z log 2e/ 1 − z −1 , z ∈ is univalent in and f / ∈ H 1 . Also, an argument given in 24, page 61 shows that there exist c > 0 and r 0 ∈ 0, 1 such that It then follows that f ∈ B 1,q 0 , whenever 1 < q < ∞. This finishes the proof. If I is an inner function and a ∈ , denote by {z k a } the exact sequence of zeros, multiplicities included, of I a , placed in increasing modulus as the subindex k increases in other words, {z k a } is the ordered sequence of preimages of a . Writing d k a 1 − |z k a |, the distribution of zeros in each annulus may be studied with the sequences {k n a } ∞ n 0 and {ν n a } ∞ n 0 : k n a Card k : 2 −n < d k a max k : 2 −n < d k a , ν n a Card k : 2 −n−1 < d k a ≤ 2 −n k n 1 a − k n a .

3.1
Observe that k 0 a 0 always. When a 0, just write {z k }, {d k }, {k n }, and {ν n }. The following relations may be used in the text without further notice. Proof. In order to keep up with readability, it is better to omit the letter value a in what follows, that is, assume a 0.
To prove a , assume first that d α k k β ≤ C for all k, then for each n 0, 1 . . ., In the other direction, assume that 2 −nα ν β n ≤ C for all n. Given k find the unique n n k such that 2 −n−1 < d k ≤ 2 −n . This implies that k ≤ k n 1 , and thus,

3.3
To prove b , assume first that n≥0 2 −nα ν β n < ∞. In the case β ≤ 1, use an easy integral estimate and the fact that k β n 1 − k β n ≤ k n 1 − k n β , to obtain the desired result,
In the other direction, assume that k≥1 d α k k β−1 < ∞. In the case β < 1, it is easily verified that n≥0 2 −nα ν β n ≤ n≥0 2 −nα k β n 1 . To continue, use that k 0 0 and that the function It remains to deal with the case β ≥ 1 under the assumption k≥1 d α k k β−1 < ∞. Here, we use that, when 0 ≤ a ≤ b, b − a β ≤ b β − a β because b − a β a β 1/β ≤ b , and use also the Mean Value Theorem,

3.8
Now we recall the following characterization, due to Ahern 37, Theorem 6 .
Theorem C see 37, Theorem 6 . Assume that 0 < p, q < ∞, that 0 < α < 1, and that I is an inner function. Then the following quantities are comparable,

3.12
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 I to be inner. Any bounded function would just work fine. In 15 , the corresponding characterization for q ∞ 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 0 < p < ∞, 0 < α < 1, and I is an inner function, then the following quantities are comparable,

3.18
On the other hand, if the zero sequence {z k } of B is Carleson-Newman and B ∈ B p,q α , then { 2 −n 1−αp ν n 1/p } ∈ q and their respective norms are equivalent.
The crux of the matter here is that the above condition is also necessary for I to belong to B p,q α . Theorem 3.3. Let 0 < p, α < ∞ be such that max{0, 1/p − 1} < α < 1/p. Assume that 0 < q ≤ ∞, and that I is an inner function. Then I ∈ B p,q α if and only if 3.19 holds for some δ ∈ 0, 1/2 . In that case, both quantities, I B p,q α 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 H ∞ -functions, and also 15, Proposition 3.1 for the case q ∞.
α are also in B pt,qt α/t for any t ≥ 1. Furthermore, the following relation holds: 3.20 Proof of Lemma 3.4. The case q ∞ will not be treated due to its similarity with the other cases. Take 0 < p, q < ∞, 0 ≤ α < ∞, 1 ≤ t < ∞, and f ∈ Λ ∞,0 ∩ B p,q α . We need to show that f ∈ B pt,qt α/t . For that, use 1.14 to find an equivalent quantity to f B pt,qt α/t , and then separate it into two factors, the first will be controlled by f Λ ∞,0 , and the second by f B p,q α .

3.21
Abstract and Applied Analysis 15 Two more lemmas are needed.

3.23
Proof of Theorem 3.3. Again, we deal only with the case q < ∞. The sufficiency of condition 3.19 has already been established. To prove its necessity, assume that I ∈ B p,q α and, rather than imposing the whole restriction max{0, 1/p − 1} < α < 1/p, just assume 0 < α, p, q < ∞. Observe that the integral in 3.19 without the power 1/p remains unchanged if we replace p, q, α with pt, qt, α/t. Now choose t ≥ 1 such that α/t < 1, pt > 1 and qt > 1. 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 0 < α < 1 and 1 < p, q < ∞. In what follows r n 1 − 2 −n . First assume that p > q. Then use, in order, Minkowski's inequality for p/q > 1, the fact that r n 1 r n r −αq−1 dr 2 nαq , and finally Lemmas F and G together with Theorem C to arrive at the desired estimate for 3.19 , The case p < q follows the same procedure, only that instead of using Minkowski's inequality, we use Hölder's with exponents q/ q − p and q/p, and then, after applying Lemma F and before Lemma G, use again Minkowski's inequality with q/p > 1,  Also, as we observed before, the integral in 3.19 , or 3.27 , is unchanged if p, q, α is replaced with pt, qt, α/t, t > 0 . This allows us to extend the homogeneity property of Lemma 3.4 to other values of t, provided that we can apply Theorem 3.3, that is, that we work with inner functions and that max{0, 1/ pt − 1} < α/t < 1/ pt . Corollary 3.6. Let 0 < p, α < ∞ be such that 0 < α < 1/p. Assume that 0 < q ≤ ∞, and that I is an inner function in B p,q α . Then I ∈ B pt,qt α/t for all t > 1/p − α. with α 1 > 1/p 1 , whatsoever , because they only contain the finite Blaschke products. The same reasoning applies when α 1/p and 0 < q < ∞. Is it the same for q ∞? that is, is the class of inner functions in Λ p,1/p the same for all p > 0? 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 Λ p,1/p . We should mention here that we shall prove later see Remark 4.4 below that the only inner functions in Λ p,1/p 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 t > 0. The answer is negative. Ahern and Clark 3, Lemma 2 have constructed a Blaschke product B in B 1,1 1/2 but not in H 1/2 1 . By property P 9 , we deduce that B / ∈ B 1/2,1/2 1 , and this is the space that would be obtained from B 1,1 1/2 by taking t 1/2, which coincides with 1/p − α with the usual notation.

4.4
Hence, given 0 < p, q, α < ∞ with α < 1/ 2p , take the smallest integer n such that 1/ 2p ≤ n and then take β 2npα < n and q q/ 2np . In this way, S ∈ H  Remark 4.4. The case q ∞ corresponds to the Lipschitz spaces Λ p,α . By property P 7 , Λ p,α ⊆ ∩{B p,q β : β < α, 0 < q}. Thus, if α > 1/ 2p , the only inner functions in Λ p,α are Blaschke products finite ones if α > 1/p . Combining now the property P 1 and the fact that the atomic singular inner function S is in B p,q α for all 0 < p, q, α < ∞ with α < 1/ 2p , we obtain that S ∈ λ p,α for all 0 < p, α < ∞ with α < 1/ 2p . In fact, we claim that S ∈ λ p,1/ 2p for all p > 0: using the corresponding homogeneity property for λ p,α i.e., Bloch functions in λ p,α are also in λ pt,α/t for all t ≥ 1, see 15, Proposition 4.1 , we obtain that the sequence of spaces Λ ∞,0 ∩λ p,1/ 2p increases with p. Therefore, to prove our claim, it suffices to see that S ∈ λ 1/ 2n ,n for all positive integer n, and this is so by the result in 39 , 1 − r M 1/ 2n r, D 1 n S 1 − r 1 − r 1/2− n 1 / 2n 1 − r 1−1/ 2n −→ 0. 4.5 Now we come to results relating the membership of a Blaschke product in the spaces under consideration for the range 1/ 2p < α < 1/p with summability properties of the associated counting sequence {ν n }. We start mentioning the following results of Verbitskiȋ.
Theorem K see 27, Theorem3 and 10, Theorem 5 . Let B be a Blaschke product with zeros {z k } in a fixed Stolz angle, and let p, q, α satisfy the relations 1 ≤ p < ∞, 1/ 2p < α < 1/p, and 0 < q ≤ ∞. Then We will come back to Blaschke products with zeros in a Stolz angle in Section 5, but we shall prove next that the implication B ∈ B p,q α ⇒ {2 −n 1/p−α ν α n } ∈ q is true for general Blaschke products whenever 1/ 2p < α < 1/p even for 0 < p < 1 . b If q < ∞, then {2 −n 1/p−α ν α n } ∈ q or, equivalently, {d Proof. We will make use of the following inequality of Goldberg 40 : where B is the Blaschke product whose zeros z k are the moduli |z k | of the corresponding zeros of B. Notice then that the sequence { d k } coincides with {d k }. Assume first that q ∞. Since 1/p > 1/p − α, Corollary 3.6 implies that B ∈ Λ 1,αp . Now, since αp < 1, Theorem D gives the following growth order: Combining this with 4.6 , we obtain that B ∈ Λ 1,αp . So, by Theorem K a , we arrive at {2 −n 1−αp ν αp n } ∈ ∞ , which is Theorem 4.5 a . To prove part b , we proceed analogously. Assume that q < ∞. Since 1/p > 1/p − α, Corollary 3.6 implies that B ∈ B 1,q/p αp . Since αp < 1, Theorem C gives