# Inner Functions in Lipschitz, Besov, and Sobolev Spaces

Authors: Daniel Girela; Cristóbal González; Miroljub Jevtić
Journal: Abstract and Applied Analysis
 (2011)

---

## 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 spacesHαp with 1/p≤α<∞ or any of the Besov spaces
Bαp,q with
0<p,q≤∞ and
α≥1/p, except when
p=∞,
α=0, and 2<q≤∞ or when
0<p<∞,
q=∞, and α=1/p are finite Blaschke products. Our assertion for the spaces
B0∞,q,
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 B0∞,q 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(a)},
|a|<1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.


---

## Body

## 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.,[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-Sobolev spaces.Denote by𝔻={z∈ℂ:|z|<1} the open unit disk in the complex plane ℂ, and by Hol(𝔻) the class of analytic functions on 𝔻. The classical Hardy space Hp, 0<p≤∞, consists of those functions f∈Hol(𝔻) for which(1.1)‖f‖Hp∶=sup⁡0<r<1Mp(r,f)<∞,
where Mp(r,f)=((1/2π)∫02π|f(reiθ)|pdθ)1/p if 0<p<∞, and M∞(r,f)=max|z|=r|f(z)|.We mention [19, 20] as references for the theory of Hardy spaces. The Hp spaces form a decreasing chain as p increases, and any function in Hp has nontangential limits almost everywhere, with the additional property that the boundary function thus formed defines an isometry between Hp and Lp(∂𝔻). The sequence {zk} of zeros of an Hp function, counting multiplicities, satisfies the so-called Blaschke condition:   ∑k(1-|zk|)<∞. This condition characterizes the zero sequences of Hp functions. By writing ba(z)=(|a|/a)((a-z)/(1-a¯z)) when a∈𝔻,a≠0, and b0(z)=z, the Blaschke condition implies that the product ∏kbzk(z) converges absolutely and uniformly in each compact subset of 𝔻, hence defining a function B(z), called the Blaschke product associated to the sequence{zk}, which is analytic in the unit disk 𝔻, and whose exact sequence of zeros, counting multiplicities, is {zk}. Also, ∥B∥H∞≤1, and its boundary function has modulus 1 almost everywhere.Aninner function in 𝔻 is an H∞ function whose nontangential boundary function has modulus 1 almost everywhere. Thus, Blaschke products are inner functions. Any inner function I admits a factorization of the type I(z)=eiγB(z)S(z), where γ is a real constant, B is a Blaschke product (carrying all the zeros of I), and S is what it is called a singular inner function, having the form(1.2)S(z)=exp⁡(-∫02πeiθ+zeiθ-zdμ(θ)),
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 letterC. 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-1B≤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.Theweighted Bergman spaceAp,α, 0<p<∞, -1<α<∞ consists of those functions f∈Hol(𝔻) such that(1.3)‖f‖Ap,α∶=(α+1π∫D|f(z)|p(1-|z|2)αdA(z))1/p<∞,
where dA(z)=dxdy denotes the Lebesgue area measure in 𝔻. We mention the books [21, 22] as general references for the theory of Bergman spaces.Iff(z)=∑k=0∞f̂(k)zk is analytic in 𝔻 and α>0, the fractional derivative of orderαoff, Dαf is defined as(1.4)Dαf(z)=∑k=0∞(k+1)αf̂(k)zk.
(This definition also makes sense for α≤0, only that the name “derivative” would look a bit awkward.) For a positive integer n, it is actually an “n-th derivative”: Dnf(z)=((d/dz)z)nf(z). Besides, the integral means Mp(r,Dnf) and Mp(r,f(n)) 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,(1.5)f(z)=1Γ(α)∫01Dαf(sz)log⁡α-11sds.
This yields the estimate(1.6)|f(z)|≤C∫01|Dαf(sz)|(1-s)α-1ds.If0<p≤∞ and 0≤α<∞, then a function f∈Hol(𝔻) is said to belong to the Lipschitz spaceΛp,α if(1.7)‖f‖Λp,α:=sup⁡0<r<1(1-r)Mp(r,D1+αf)<∞.
The subspace λp,α consists of those f∈Hol(𝔻) for which(1.8)lim⁡r→1(1-r)Mp(r,D1+αf)=0.
Notice that Λ∞,0 is another name for the usual Bloch space ℬ, of functions f∈Hol(𝔻) such that supz∈𝔻(1-|z|2)|f′(z)|<∞. Analogously, λ∞,0 is the little Bloch space ℬ0, of functions f∈Hol(𝔻) such that lim|z|→1(1-|z|2)|f′(z)|=0.Replacing the sup-norm with anLq-norm in the above definition gives way to the Besov spaceBαp,q, 0<p≤∞, 0<q<∞, 0≤α<∞, consisting of those functions f∈Hol(𝔻) for which(1.9)‖f‖Bαp,q∶=(∫01(1-r)q-1Mpq(r,D1+αf)dr)1/q<∞.
The space obtained when q=∞ is precisely Bαp,∞≡Λp,α. Other classes treated in this paper are the Hardy-Sobolev spacesHαp, 0<p≤∞, 0≤α<∞, consisting of those functions f∈Hol(𝔻) for which(1.10)‖f‖Hαp:=‖Dαf‖Hp<∞.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(1.11)Mpp(r,f)≤C∫01(1-s)pα-1Mpp(rs,Dαf)ds,(0<p<1,0<α),(1.12)Mp(r,f)≤C∫01(1-s)α-1Mp(rs,Dαf)ds,(1≤p≤∞,0<α).
The same estimate (1.6) combined with variants of Hardy's inequality gives(1.13)sup⁡0<ρ≤r(1-ρ)αMp(ρ,Dβf)≍sup⁡0<ρ≤r(1-ρ)α-β+γMp(ρ,Dγf),(0<p≤∞,α>max⁡{0,β-γ}),(1.14)∫01(1-r)α-1Mpq(r,Dβf)dr≍∫01(1-r)α-q(β-γ)-1Mpq(r,Dγf)dr,(0<q<∞,0<p≤∞,α>max⁡{0,q(β-γ)}).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]):(1.15)Mq(r,f)≤Cp,q(1-r)1/q-1/pMp(1+r2,f),(0<p<q≤∞).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.(1.16)sup⁡0≤r<1Mp(r,f)≤Cp∫01(1-r)p-1Mpp(r,D1f)dr,(0<p≤2),(1.17)∫01(1-r)p-1Mpp(r,D1f)dr≤Cpsup⁡0≤r<1Mp(r,f),(p≥2),
which can be restated as B0p,p⊆Hp, for 0<p≤2, and Hp⊆B0p,p, for p≥2.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.(P1)
Ifq<∞, Bαp,q⊆λp,α: If f∈Bαp,q, then ∫r1(1-ρ)q-1Mpq(ρ,D1+αf)dρ→0 as r→1, so by using the increasing behavior of the means Mp(r,D1+αf), obtain that (1-r)Mp(r,D1+αf)→0, as r→1, that is, that f∈λp,α.(P2)
Ifq1≤q2, Bαp,q1⊆Bαp,q2, because (1-r)Mp(r,D1+αf)→0, as r→1, for all f∈Bαp,q1.(P3)
Ifp1≤p2, Bαp2,q⊆Bαp1,q, and Hαp2⊆Hαp1, because Mp1(r,g)≤Mp2(r,g) for all r∈(0,1) and all g∈Hol(𝔻).(P4)
Ifα1≤α2, Bα2p,q⊆Bα1p,q. Indeed, for f∈Bα2p,q, (1.14) gives
(1.18)‖f‖Bα1p,qq≍∫01(1-r)q-1+q(α2-α1)Mpq(r,D1+α2f)dr<‖f‖Bα2p,qq<∞.(P5)
Ifα1≤α2, Hα2p⊆Hα1p, because of (1.11) and (1.12).(P6)
If0<p1≤p2, B1/p1p1,q⊆B1/p2p2,q. (When q=∞, this says that Λp,1/p increases with p). A proof of this lies in an application of (1.15) and (1.14),(1.19)‖f‖B1/p2p2,qq≍∫01(1-r)q-1+q(1/p1-1/p2)Mp2q(r,D1+1/p1f)dr≤C‖f‖B1/p1p1,qq.(P7)
Λp,α⊆∩{Bβp,q:β<α,0<q}. This is again an application of (1.14),
(1.20)‖f‖Bβp,qq≍∫01(1-r)q(α-β)-1+qMpq(r,D1+αf)dr≤C‖f‖Λp,α,
where C=∫01(1-r)q(α-β)-1dr<∞.(P8)
Hαp⊆∩{Bβp,q:β<α,0<q}. Again an application of (1.14).(P9)
If0<p≤2, Bαp,p⊆Hαp. This is (1.16).(P10)
If2≤p, Hαp⊆Bαp,p. And this is (1.17).In the following, we introduce some notation related to inner functions. Given an inner functionI and a point a∈𝔻, its Frostman shiftIa is defined as(1.21)Ia(z)=I(z)-a1-a¯I(z),z∈D.
A classical result of Frostman (see, e.g., [20, Section 2.6] asserts that if I is an inner function in 𝔻 then the Frostman shifts Ia are Blaschke products for all a∈𝔻 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∈𝔻∖E, with E a set of logarithmic capacity zero, the Frostman shift Ia 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.
Let0<p,q≤∞, 0≤α, 0<δ<1/2, and a∈Kδ={z∈𝔻:δ<|z|<1-δ}. Then ∥Ia∥Bαp,q≍∥I∥Bαp,q, with constants depending only on p,q,α, and δ, but not on I or a. In the case q=∞, the formulation is
(1.22)sup⁡0≤ρ<r(1-ρ)Mp(ρ,D1+αIa)≍sup⁡0≤ρ<r(1-ρ)Mp(ρ,D1+αI).Proof.
Put1+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 Bαp,q-norm of Ia in terms of that of I, for I=(Ia)-a.
Writingψa(z)=a¯/(1-a¯z), z∈𝔻 observe that the first derivative of Ia can be written as Ia′=((1-|a|2)/a¯2)(ψa∘I)′, and, in general, for a positive integer n, the nth derivative is given by Ia(n)=((1-|a|2)/a¯2)(ψa∘I)(n). This, together with the Faà di Bruno's formula for the nth derivative of a composition, gives
(1.23)Ia(n)=1-|a|2a¯2(ψa∘I)(n)=1-|a|2a¯2∑n!k1!⋯kn!ψa(k)∘I∏j=1n(I(j)j!)kj,
where k=∑j=1nkj, and the sum runs over all n-tuples k⃗=(k1,…,kn) of nonnegative integers such that ∑j=1njkj=n. Observe that if a∈Kδ, the quantities (1-|a|2)/|a¯|2 and |ψa(k)∘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∥Ia∥Bαp,qq, we use, in order, (1.14), (1.23), twice Hölder's inequality (Hö) with indices {n/(jkj):kj≠0}, again (1.14), and finally we appeal to the fact (I) that |I(n)(z)|≤Cn(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]),
(1.24)‖Ia‖Bαp,qq=∫01(1-r)q-1Mpq(r,Dn+βIa)dr≤(1.14)C∫01(1-r)q(1-β)-1Mpq(r,Ia(n))dr≤(1.23)C∑k⃗∫01(1-r)q(1-β)-1Mpq(r,∏j=1n(I(j))kj)dr≤(Hö)C∑k⃗∫01(1-r)q(1-β)-1∏j=1nMpn/jqkj(r,I(j))dr≤(Hö)C∑k⃗∏j=1n(∫01(1-r)q(1-β)-1Mpn/jqn/j(r,I(j))dr)jkj/n≤(1.14)C∑k⃗∏j=1n(∫01(1-r)q(1-β)-1+(qn(n-j))/jMpn/jqn/j(r,I(n))dr)jkj/n≤(I)C∑k⃗∏j=1n(∫01(1-r)q(1-β)-1Mpq(r,I(n))dr)jkj/n≤(1.14)C‖I‖Bαp,qq.


## 2. Inner Functions in the SpacesBαp,q and Hαp, α≥1/p

Ahern and Jevtić [5] proved that a Blaschke product lies in the space B1/pp,∞≡Λp,1/p (0<p<∞) if and only if its sequence of zeros is a finite union of exponential sequences, (see also Verbitskiĭ [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(2.1)for0<p<∞,the spaceB1/pp,∞contains infinite Blaschke products.
Our results in this section imply that the opposite is true for all the spaces Bαp,q 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≤∞.Theorem 2.1.
(a)  Let0<p,q≤∞ and α>1/p. Then the only inner functions in Bαp,q are finite Blaschke products.
(b) If0<p,q<∞ then the only inner functions in B1/pp,q are finite Blaschke products.Proof.
To prove (a) observe thatBαp,q⊆Bαp,∞≡Λp,α⊆Λ∞,α-1/p. 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 𝔻¯). Thus if we are in the conditions of part (a) and I is an inner function in Bαp,q 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∈∂𝔻. 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.
Let us now turn to prove part(b). The following results come in our aid.
Theorem A (see [8, Corollary 1.6]).
Let0<p<∞. If B is a Blaschke product in H1/pp, then B is a finite Blaschke product.Theorem B (see [5, Theorem 3.2]).
For each0<p<∞ there exists  ɛp>0 such that if  B is a Blaschke product and
(2.2)limsup⁡r→1(1-r)Mp(r,D1+1/pB)<ɛp,
then B is a finite Blaschke product. In particular, the only Blaschke products in λp,1/p are the finite ones.
Using Lemma1.1, 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.
For0<p<∞, the only inner functions in either H1/pp or λp,1/p are the finite Blaschke products.Now Theorem 2.1(b) follows from this result and the fact that B1/pp,q⊆λp,1/p.The fact thatHαp⊆∩{Bβp,q:β<α,0<q} immediately implies the following.Corollary 2.3.
Let0<p≤∞ and α>1/p. If f∈Hαp then f admits a continuous extension to 𝔻¯. Consequently, the only inner functions in Hαp, with α>1/p, are finite Blaschke products.It remains to consider the casep=∞ and α=0. Of course, H0∞≡H∞ contains the whole class of inner functions.Let us deal now with the spacesB0∞,q. First of all, B0∞,∞ is the Bloch space, and, hence, it contains all inner functions.Bishop [29] proved that the little Bloch space, λ∞,0, contains infinite Blaschke products. Since, for q<∞, B0∞,q is a subspace of λ∞,0, the natural question rises as whether B0∞,q contains or not infinite Blaschke products. The answer depends on the result of intersecting B0∞,q with the subspace VMOA of λ∞,0, consisting of those H1 functions whose boundary values have vanishing mean oscillation. The space VMOA was introduced by Sarason [30] and admits a number of equivalent definitions. Among them, we mention that a function f∈H1 is said to belong to VMOA if(2.3)lim⁡|a|→1‖f∘φa-f(a)‖Hp=0,
for some (or, equivalently, for all) finite positive p. Here, φa(z)=(a-z)/(1-a¯z) is the typical involutive automorphism of 𝔻 interchanging the points 0 and a∈𝔻. Using this definition and the fact that nonconstant inner functions take values as close to 0 as desired, Anderson [31] proved that VMOA contains no inner functions other than finite Blaschke products. (See also [32] for an extensive survey on BMOA and VMOA.) In the following result, we use another characterization of VMOA; it is the space of functions f∈Hol(𝔻) such that(2.4)1|J|∫S(J)(1-|z|2)|f′(z)|dA(z)⟶0,as|J|⟶0,
where J is an interval in ∂𝔻, |J| is its length, and S(J) is the Carleson square defined by S(J)={reiθ:eiθ∈J,1-|J|≤r<1}.Theorem 2.4.
(a),  If0<q≤2, then B0∞,q⊆VMOA. Consequently, any inner function in B0∞,q is a finite Blaschke product.
(b) There are infinite Blaschke products in∩2<q<∞B0∞,q.Proof.
To prove (a) observe that, sinceB0∞,q1⊆B0∞,q2 for 0<q1<q2<∞, it suffices to settle the result for q=2.
Thus, takef∈B0∞,2 and take an interval J in ∂𝔻 with |J|<1/2, then
(2.5)1|J|∫S(J)(1-|z|2)|f′(z)|2dA(z)≤21|J|∫1-|J|1(1-r)∫J|f′(reiθ)|2dθdr≤2∫1-|J|1(1-r)M∞2(r,f′)dr,
and observe that, since f∈B0∞,2, 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
(2.6)(1-|z|2)|I′(z)|1-|I(z)|2≤log⁡-1/2e1-|z|.
This implies that I∈B0∞,q 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 𝔻. Therefore, we may choose a∈𝔻 such that the Frostman shift Ia is an infinite Blaschke product (actually, this is true for all a∈𝔻 except for those in a set of zero logarithmic capacity). Now, Lemma 1.1 shows that Ia is an infinite Blaschke product in ∩2<q<∞B0∞,q.Once Theorem2.4 is proved, it is natural to ask whether or not the inclusion VMOA⊆B0∞,q holds for 2<q<∞. An argument based on duality shows that this is not so.Theorem 2.5.
If2<q<∞, then the class VMOA∖B0∞,q is nonempty.Proof.
Observe that the dual ofVMOA is H1 under the usual pairing: 〈f,g〉=limr→1∑kf̂(k)ĝ(k)¯rk, (see [32, 34]). Also, using the same techniques as in [4], we get that the dual of B0∞,q is B01,q′, 1/q+1/q′=1, under the same pairing as before. Thus, the problem reduces to show that B01,q′∖H1 is nonempty.
It is shown in Theorem 3 of [35] that the function f(z)=((1-z)log(2e/(1-z)))-1, z∈𝔻 is univalent in 𝔻 and f∉H1. Also, an argument given in [24, page 61] shows that there exist c>0 and r0∈(0,1) such that(2.7)M1(r,f′)≤c(1-r)(log⁡(2e/(1-r))),r0<r<1.
It then follows that f∈B01,q′, whenever 1<q′<∞. This finishes the proof.


## 3. The Case Max{0,1/p-1}<α<1/p

For this range of values, we shall obtain a number of results relating the membership of an inner functionI in Besov or Hardy-Sobolev spaces with the distribution of the preimages {I-1(a)}, a∈𝔻. We start introducing certain counting functions.IfI is an inner function and a∈𝔻, denote by {zk(a)} the exact sequence of zeros, multiplicities included, of Ia, placed in increasing modulus as the subindex k increases (in other words, {zk(a)} is the ordered sequence of preimages of a). Writing dk(a)=1-|zk(a)|, the distribution of zeros in each annulus may be studied with the sequences {kn(a)}n=0∞ and {νn(a)}n=0∞:(3.1)kn(a)=Card{k:2-n<dk(a)}=max⁡{k:2-n<dk(a)},νn(a)=Card{k:2-n-1<dk(a)≤2-n}=kn+1(a)-kn(a).
Observe that k0(a)=0 always. When a=0, just write {zk}, {dk}, {kn}, and {νn}. The following relations may be used in the text without further notice.Lemma 3.1.
Under the previous settings, letα,β>0. Then (a)
{2-nανnβ(a)}∈ℓ∞ if and only {dkα(a)kβ}∈ℓ∞, and, in either case, their ℓ∞-norms are comparable.(b)
∑n≥02-nανnβ(a)≍∑k≥1dkα(a)kβ-1.Proof.
In order to keep up with readability, it is better to omit the letter valuea in what follows, that is, assume a=0.
To prove (a), assume first thatdkαkβ≤C for all k, then for each n=0,1…,
(3.2)2-nανnβ=2-nα(kn+1-kn)β≤2αdkn+1αkn+1β≤C.
In the other direction, assume that 2-nανnβ≤C for all n. Given k find the unique n=n(k) such that 2-n-1<dk≤2-n. This implies that k≤kn+1, and thus,
(3.3)dkαkβ≤2-nαkn+1β=(∑j=0n2-αj/βνj2-α(n-j)/β)β,≤C(∑j=0n2-α(n-j)/β)β≤C(1-2-α/β)-β.
To prove (b), assume first that∑n≥02-nανnβ<∞. In the case β≤1, use an easy integral estimate and the fact that kn+1β-knβ≤(kn+1-kn)β, to obtain the desired result,
(3.4)∑k≥1dkαkβ-1=∑n≥0∑k=kn+1kn+1dkαkβ-1≤∑n≥02-nα∑k=kn+1kn+1kβ-1≤∑n≥02-nα∫knkn+1xβ-1dx=1β∑n≥02-nα(kn+1β-knβ)≤1β∑n≥02-nα(kn+1-kn)β=1β∑n≥02-nανnβ.
In the caseβ>1, it is easy to arrive at ∑k≥1dkαkβ-1≤∑n≥02-nαkn+1β. Now we imitate the proof of Hardy's inequality given in [36, Theorem 326 in page 239]. Write hn=2-nα/βkn+1=2-nα/β∑j=0nνj, for n≥0, and h-1=0. Then νn=2nα/β(hn-2-α/βhn-1), for n≥0, and by the inequality between the geometric and arithmetic means [36, Theorem 9 in page 17],
(3.5)hnβ-11-2-α/βhnβ-12-nα/βνn=12α/β-1(hnβ-1hn-1-hnβ)≤12α/β-1((β-1)hnβ+hn-1ββ-hnβ)=1β(2α/β-1)(hn-1β-hnβ).
So the sum on n of the left hand side is negative, and, therefore, using Hölder's inequality with exponents β/(β-1) and β,
(3.6)∑n≥02-nαkn+1β=∑n≥0hnβ≤11-2-α/β∑n≥0hnβ-12-nα/βνn≤11-2-α/β(∑n≥0hnβ)β-1/β(∑n≥02-nανnβ)1/β,
giving as a result that ∑n≥02-nαkn+1β≤(1-2-α/β)-β∑n≥02-nανnβ, as desired.
In the other direction, assume that∑k≥1dkαkβ-1<∞. In the case β<1, it is easily verified that ∑n≥02-nανnβ≤∑n≥02-nαkn+1β. To continue, use that k0=0 and that the function x↦xβ-1 is decreasing in (0,∞),
(3.7)∑n≥02-nαkn+1β=∑n≥02-nαkn+1β-1∑k=1kn+11≤∑n≥02-nα∑k=kn+1kn+1kβ-1+∑n≥12-nαknβ≤2α∑n≥0∑k=kn+1kn+1dkαkβ-1+2-α∑n≥02-nαkn+1β,
from where it follows that ∑n≥02-nαkn+1β≤2α(1-2-α)-1∑k≥1dkαkβ-1<∞.
It remains to deal with the caseβ≥1 under the assumption ∑k≥1dkαkβ-1<∞. Here, we use that, when 0≤a≤b, (b-a)β≤bβ-aβ (because ((b-a)β+aβ)1/β≤b), and use also the Mean Value Theorem,
(3.8)∑n≥02-nανnβ=∑n≥02-nα(kn+1-kn)β≤∑n≥02-nα(kn+1β-knβ)=2α∑n≥02-(n+1)α∑k=kn+1kn+1(kβ-(k-1)β)≤2αβ∑n≥0∑k=kn+1kn+1dkαkβ-1=2αβ∑k≥1dkαkβ-1.Now we recall the following characterization, due to Ahern [37, Theorem 6].Theorem C (see [37, Theorem 6]).
Assume that0<p,q<∞, that  0<α<1, and that  I is an inner function. Then the following quantities are comparable,
(3.9)‖I‖Bαp,qq=∫01(1-r)q-1Mpq(r,D1+αI)dr,(3.10)∫01(1-r)(1-α)q-1Mpq(r,I′)dr,(3.11)∫01(1-r)-αq-1(12π∫02π(1-|I(reiθ)|)pdθ)q/pdr,(3.12)∫01(1-r)-αq-1(12π∫02π|I(eiθ)-I(reiθ)|pdθ)q/pdr.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.
If0<p<∞,  0<α<1, and  I is an inner function, then the following quantities are comparable,(3.13)‖I‖Λp,α=sup⁡0≤r<1(1-r)Mp(r,D1+αI),(3.14)sup⁡0≤r<1(1-r)1-αMp(r,I′),(3.15)sup⁡0≤r<1(1-r)-α(12π∫02π(1-|I(reiθ)|)pdθ)1/p,(3.16)sup⁡0≤r<1(1-r)-α(12π∫02π|I(eiθ)-I(reiθ)|pdθ)1/p.For Blaschke products, the third author [15, 38] gave sufficient (and in special cases, necessary) conditions for their membership in Bαp,q in terms of the distribution of their zeros. Recall that a Carleson-Newman sequence is a finite union of interpolating sequences, and a sequence {zk} in the unit disk is called interpolating if it is uniformly separated, that is,
(3.17)inf⁡n∏{k:zk≠zn}|zk-zn1-zk¯zn|>0.Theorem E (see [15, 38]).
Let0<p,α<∞ be such that  max{0,1/p-1}<α<1/p. Assume that  0<q≤∞, and that  B is a Blaschke product. If  {(2-n(1-αp)νn)1/p}∈ℓq, then  B∈Bαp,q and
(3.18)‖B‖Bαp,q≤C‖{(2-n(1-αp)νn)1/p}‖lq.
On the other hand, if the zero sequence {zk} of B is Carleson-Newman  and B∈Bαp,q, then {(2-n(1-αp)νn)1/p}∈ℓq and their respective norms are equivalent.As a consequence of this result, Lemma1.1, and the fact that the Frostman shifts of inner functions are Blaschke products almost always, we have that if 0<p,α<∞ with max{0,1/p-1}<α<1/p, 0<q≤∞, and I is an inner function satisfying
(3.19)(∫Kδ‖{(2-n(1-αp)νn(a))1/p}‖lqpdA(a))1/p<∞,
for Kδ={a∈𝔻:δ≤|a|≤1-δ}, and 0<δ<1/2, then I∈Bαp,q, and the norm ∥I∥Bαp,q is controlled by the integral in (3.19).The crux of the matter here is that the above condition is also necessary forI to belong to Bαp,q.Theorem 3.3.
Let0<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∈Bαp,q if and only if (3.19) holds for some δ∈(0,1/2). In that case, both quantities, ∥I∥Bαp,q 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=∞.Lemma 3.4.
If0<p<∞, 0<q≤∞, 0≤α<∞, and 1≤t<∞, then Λ∞,0∩Bαp,q⊆Bα/tpt,qt, that is, Bloch functions in Bαp,q are also in Bα/tpt,qt for any t≥1. Furthermore, the following relation holds:
(3.20)‖f‖Bα/tpt,qt≤C‖f‖Λ∞,01-1/t‖f‖Bαp,q1/t.Proof of Lemma3.4.
The caseq=∞ will not be treated due to its similarity with the other cases. Take 0<p,q<∞, 0≤α<∞, 1≤t<∞, and f∈Λ∞,0∩Bαp,q. We need to show that f∈Bα/tpt,qt. For that, use (1.14) to find an equivalent quantity to ∥f∥Bα/tpt,qt, and then separate it into two factors, the first will be controlled by ∥f∥Λ∞,0, and the second by ∥f∥Bαp,q.
(3.21)‖f‖Bα/tpt,qt=(∫01(1-r)qt-1Mptqt(r,D1+α/tf)dr)1/qt≤(1.14)C(∫01(1-r)qt-1-qα+qαtMptqt(r,D1+αf)dr)1/qt=C(∫01(1-r)q(t-1)(1+α)+(q-1)(∫02π|D1+αf(reiθ)|ptdθ)q/pdr)1/qt≤Csup⁡0≤r<1((1-r)1+αM∞(r,D1+αf))1-1/t(∫01(1-r)q-1Mpq(r,D1+αf)dr)1/qt≤C‖f‖Λ∞,01-1/t‖f‖Bαp,q1/t.Two more lemmas are needed.Lemma F (see [15, Corollary 4.5]).
Ifp≥1,  I is an inner function, and if  1-2-n<r≤1-2-(n+1), then
(3.22)2-nνn≤Cp∫02πlog⁡p1|I(reiθ)|dθ.Lemma G (see [15, Corollary 4.7]).
If0<p<∞,  0<δ<1/2,  I is an inner function, and if  z∈𝔻, then
(3.23)∫Kδlog⁡p1|Ia(z)|dA(a)≤Cp,δ(1-|I(z)|)p.Proof of Theorem3.3.
Again, we deal only with the caseq<∞. The sufficiency of condition (3.19) has already been established. To prove its necessity, assume that I∈Bαp,q 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
(3.24)(∫Kδ‖{(2-n(1-αp)νn(a))1/p}‖lqpdA(a))1/p=(∫Kδ‖{(2-n(1-αp)νn(a))1/pt}‖lqtptdA(a))1/p≤C‖I‖Bα/tpt,qtt≤C‖I‖Λ∞,0t-1‖I‖Bαp,q.
So it suffices to prove the result for 0<α<1 and 1<p,q<∞. In what follows rn=1-2-n. First assume that p>q. Then use, in order, Minkowski's inequality for p/q>1, the fact that ∫rnrn+1r-αq-1dr≍2nαq, and finally Lemmas F and G together with Theorem C to arrive at the desired estimate for (3.19),
(3.25)(∫Kδ‖{(2-n(1-αp)νn(a))1/p}‖lqpdA(a))1/p=(∫Kδ(∑n≥02nαq(2-nνn(a))q/p)p/qdA(a))(q/p)(1/q)≤(∑n≥02nαq(∫Kδ2-nνn(a)dA(a))q/p)1/q≤C(∑n≥0∫rnrn+1(1-r)-αq-1(∫Kδ∫02πlog⁡p1|Ia(reiθ)|dθdA(a))q/pdr)1/q≤C(∫01(1-r)-αq-1(∫02π∫Kδlog⁡p1|Ia(reiθ)|dA(a)dθ)q/pdr)1/q≤C(∫01(1-r)-αq-1(∫02π(1-|I(reiθ)|)pdθ)q/pdr)1/q≤C‖I‖Bαp,q.
The casep<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,(3.26)(∫Kδ‖{(2-n(1-αp)νn(a))1/p}‖lqpdA(a))1/p=(∫Kδ(∑n≥02nαq(2-nνn(a))q/p)p/qdA(a))1/p≤(∫KδdA(a))1/p-1/q(∫Kδ∑n≥02nαq(2-nνn(a))q/pdA(a))1/q=C(∑n≥02nαq∫Kδ(2-nνn(a))q/pdA(a))1/q≤C(∑n≥0∫rnrn+1(1-r)-αq-1(∫Kδ(∫02πlog⁡p1|Ia(reiθ)|dθ)q/pdA(a))(p/q)(q/p)dr)1/q≤C(∫01(1-r)-αq-1(∫02π(∫Kδlog⁡q1|Ia(reiθ)|dA(a))p/qdθ)q/pd)1/q≤C(∫01r-αq-1(∫02π(1-|I(reiθ)|)pdθ)q/pdr)1/q≤C‖I‖Bαp,q.Remark 3.5.
Along the proof of this theorem, we have actually proved that ifI is an inner function in Bαp,q, with 0<α,p<∞ and 0<q≤∞, then (3.19) holds and
(3.27)(∫Kδ‖{(2-n(1-αp)νn(a))1/p}‖lqpdA(a))1/p≤C‖I‖Bαp,q.
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.
Let0<p,α<∞ be such that 0<α<1/p. Assume that 0<q≤∞, and that I is an inner function in Bαp,q. Then I∈Bα/tpt,qt for all t>1/p-α.Remark 3.7.
Of course, whenα>1/p, the class of inner functions in Bαp,q coincides with that of Bα/tpt,qt for any t>0 (or any Bα1p1,q1 with α1>1/p1, 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 Λp,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 Λp,1/p. We should mention here that we shall prove later (see Remark 4.4 below) that the only inner functions in Λp,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 oft>0. The answer is negative. Ahern and Clark [3, Lemma 2] have constructed a Blaschke product B in B1/21,1 but not in H11/2. By property (P9), we deduce that B∉B11/2,1/2, and this is the space that would be obtained from B1/21,1 by taking t=1/2, which coincides with 1/p-α with the usual notation.Proof of Corollary3.6.
Notice first that Remark3.5 implies that (3.27) holds for p,q,α, and thus the integral on the left hand side is unchanged for pt,qt,α/t. Now, take t>1/p-α>0. Then 1/(pt)-1<α/t<1/(pt). If now t≤1/p, then 0≤1/(pt)-1, and so max{0,1/(pt)-1}<α/t<1/(pt), proving that I∈Bα/tpt,qt by Theorem 3.3. If, on the contrary, t>1/p, then 1/(pt)-1<0, and we still have max{0,1/(pt)-1}=0<α/t<1/(pt), and again, I∈Bα/tpt,qt 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 B is a Blaschke product satisfying ∑kdk2-p<∞ for some p∈(1,2). By Lemma 3.1(b), this condition is equivalent to ∑n≥02-n(2-p)νn<∞, which implies that B∈B1-1/pp,p by Theorem E. (Notice that this is equivalent to B′∈Ap by (1.14).) Observe now that we can apply Corollary 3.6 with t=1/p, obtaining that B∈Bp-11,1, which is the aforementioned result by Protas, with our notation. On the other hand, notice also that if B′∈Ap, for some 1<p<2, and the zero sequence of B is Carleson-Newman then, by Theorem E, ∑kdk2-p<∞.
Next we will turn to study the membership of inner functions in the spacesHαp. Properties (P9) and (P10), that is, the Littlewood-Paley inequalities, relate quite well Hαp with Bαp,p. 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.
Let0<p≤∞ and α>max{0,1/p-1}. Then the class of inner functions in Hαp coincides with that of inner functions in Bαp,p.The proof of this result requires again an homogeneity property.Proposition 3.11.
Let0<p,α<∞ and let f be a Bloch function in Hαp. Then f∈Hα/tpt for all t≥1.This is a straightforward consequence of the following result and the complex maximal theorem.Lemma H(see [5, Lemma 2.1]).
For each0<α<β<∞, there is a constant  C=Cα,β such that if  f∈Λ∞,0 then
(3.28)|Dαf(z)|≤C(max⁡0≤t≤1|Dβf(tz)|)α/β.The original proof of this lemma runs withH∞ functions instead of Bloch (Λ∞,0) functions. However, the lemma can be proved for Bloch functions by just noticing the validity of the estimate |Dβf(z)|≤C(1-|z|)-β for Bloch functions.Proof of Theorem3.10.
First notice that whenα≥1/p, the only inner functions in Hαp and Bαp,p are the finite Blaschke products. So we may assume without loss of generality the additional hypothesis α<1/p. (This already implies p<∞.)
Now consider the case0<p≤2. Then Bαp,p⊆Hαp by (P9). To go in the other direction, take an inner function I in Hαp. Then, by Proposition 3.11, I∈Hα/tpt for all t≥1. For t≥2/p≥1, we have pt≥2, and hence Hα/tpt⊆Bα/tpt,pt by (P10). Using now that max{0,1/p-1}<α<1/p, we get 1/t>1/(pt)-α/t, and we conclude that I∈Bαp,p by Corollary 3.6.
Next consider the case2≤p<∞. (Then max{0,1/p-1}=0.) By (P10), Hαp⊆Bαp,p. To go in the other direction, take an inner function I in Bαp,p. Since 0<α<1/p then, by Corollary 3.6, I∈Bα/tpt,pt for all t>1/p-α. Choose t∈(1/p-α,2/p]. Thus, as pt≤2, (P9) gives Bα/tpt,pt⊆Hα/tpt. Finally, as 1/t≥p/2≥1, Proposition 3.11 gives I∈Hαp.Remark 3.12.
A careful reading of this proof shows that, in fact, any inner functionI in Bαp,p is also in Hαp, whenever 0<p≤∞ and 0<α.The analogous to Corollary3.6 is the following.Corollary 3.13.
Let0<p,α<∞ be such that 0<α<1/p. Assume that I is an inner function in Hαp. Then I∈Hα/tpt for all t>1/p-α.Proof.
Taket>1/p-α. If t≥1 then I∈Hα/tpt by Proposition 3.11. Otherwise we have 1/p-α<t<1, which implies 1/p-1<α, and, together with 0<α<1/p, it gives max{0,1/p-1}<α<1/p. So, by Theorem 3.10, I∈Bαp,p and, by Corollary 3.6, I∈Bα/tpt,pt and, finally, by Remark 3.12, I∈Hα/tpt.Remark 3.14.
Again, since it only contains finite Blaschke products, the class of inner functions inHαp, with α≥1/p, coincides with that of Hα/tpt for all t>0, or for the case, with the class of inner functions of any Hα1p1, with α1≥1/p1. As for the accuracy of the corollary with regards to whether there is a possibility to establish the result for the whole range of t>0, the same example given in Remark 3.8 shows that it is impossible. The Blaschke product B constructed in [3, Lemma 2] is in B1/21,1 but not in H11/2, which is the space that would be obtained from H1/21 by taking t=1/2(=1/p-α with the usual notation). By Theorem 3.10, B∈B1/21,1 implies that B∈H1/21.Remark 3.15.
As an application, we again regain a known result of Protas [9, Theorem 2], namely, ifBis a Blaschke product such that∑kdk1-p<∞, for some1/2<p<1, thenB∈H1p. Indeed, by Lemma 3.1(b), ∑kdk1-p≍∑n≥02-n(1-p)νn, and so B∈B1p,p by Theorem E, and we are done with B∈H1p by Theorem 3.10. On the other hand, Theorem E also says that if the zero sequence of B is Carleson-Newman then the condition ∑kdk1-p<∞ is also necessary for B∈H1p.


## 4. The Case1/(2p)<α<1/p

Ahern and Clark [2, Theorem 3] proved that the only inner functions in H11/2 are Blaschke products. Later on, Ahern and Jevtic' obtained the following generalization:Theorem I (see [5, Theorem 2.1]).
IfI is an inner function and
(4.1)Mpp(r,D1/(2p)I)=o(log⁡11-r),asr⟶1,
for some 0<p<∞, then I is a Blaschke product.Now, all functions inH1/(2p)p satisfy condition (4.1) and, by (P5), the same is true for all Hαp-functions with α≥1/(2p). So the following is immediate:Corollary 4.1.
Let0<p,α<∞ with α≥1/(2p). Then the only inner functions in Hαp are Blaschke products, finite ones if α≥1/p.This result finds its analogue for Besov spaces. Its essence may be traced back to the last corollary in [37].Proposition J (see [37]).
Let0<p,q,α<∞ with  α≥1/(2p). Then the only inner functions in  Bαp,q are Blaschke products, finite ones if  α≥1/p.Remark 4.2.
These results are again accurate, for the “atomic” singular inner functionS(z)=exp(-(1+z)/(1-z)) is in Bαp,q∩Hαp for all 0<p,q,α<∞ with α<1/(2p). Indeed, in [39], it is shown that, for any positive integer n,
(4.2)M1/(2n)(r,S(n))≍log⁡11-r.
This implies that S∈Hβ1/(2n) for all β<n, because by (1.11),
(4.3)M1/(2n)1/(2n)(r,DβS)≤C∫01(1-s)(n-β)/(2n)-1M1/(2n)1/(2n)(rs,DnS)ds≤C∫01(1-s)(n-β)/(2n)-1log⁡1/(2n)(11-rs)ds<∞,
and also S∈Bβ1/(2n),q̃ for all β<n and all 0<q̃<∞, because by (1.14),
(4.4)∫01(1-r)q̃-1M1/(2n)q̃(r,D1+βS)dr≤C∫01(1-r)q̃(n-β)-1M1/(2n)q̃(r,DnS)dr≤C∫01(1-r)q̃(n-β)-1log⁡q̃(11-r)dr<∞.
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 q̃=q/(2np). In this way, S∈Hβ1/(2n)∩Bβ1/(2n),q̃, so by the homogeneity properties of Lemma 3.4 and Proposition 3.11 with t=2np≥1, get that S∈Hβ/tt/(2n)∩Bβ/tt/(2n),q̃t=Hαp∩Bαp,q, as desired.Remark 4.3.
The casep=∞ would deal with the question of whether the inner functions in B0∞,q are just Blaschke products. From Theorem 2.4, we know that if q≤2, the inner functions in B0∞,q are finite Blaschke products, while the example considered to prove its part (b) is initially a singular inner function in ∩q>2B0∞,q.Remark 4.4.
The caseq=∞ corresponds to the Lipschitz spaces Λp,α. By property (P7), Λp,α⊆∩{Bβp,q:β<α,0<q}. Thus, if α>1/(2p), the only inner functions in Λp,α are Blaschke products (finite ones if α>1/p). Combining now the property (P1) and the fact that the atomic singular inner function S is in Bαp,q for all 0<p,q,α<∞ with α<1/(2p), we obtain that S∈λp,α for all 0<p,α<∞ with α<1/(2p). In fact, we claim that S∈λp,1/(2p) for all p>0: using the corresponding homogeneity property for λp,α (i.e., Bloch functions in λp,α are also in λpt,α/t for all t≥1, see [15, Proposition 4.1]), we obtain that the sequence of spaces Λ∞,0∩λp,1/(2p) increases with p. Therefore, to prove our claim, it suffices to see that S∈λ1/(2n),n for all positive integer n, and this is so by the result in [39],
(4.5)(1-r)M1/(2n)(r,D1+nS)≍(1-r)(1-r)1/2-(n+1)/(2n)=(1-r)1-1/(2n)⟶0.
Now we come to results relating the membership of a Blaschke product in the spaces under consideration for the range1/(2p)<α<1/p with summability properties of the associated counting sequence {νn}. We start mentioning the following results of Verbitskiĭ.Theorem K (see [27, Theorem3] and [10, Theorem 5]).
LetB be a Blaschke product with zeros  {zk} in a fixed Stolz angle, and let  p,q,α satisfy the relations  1≤p<∞,  1/(2p)<α<1/p, and  0<q≤∞. Then   (a)
B∈Λp,α=Bαp,∞⇔{2-n(1/p-α)νnα}∈ℓ∞⇔{dk1/p-αkα}∈ℓ∞,(b)
B∈Bαp,q⇔{2-n(1/p-α)νnα}∈ℓq⇔{dk1/p-αkα-1/q}∈ℓq.We will come back to Blaschke products with zeros in a Stolz angle in Section5, but we shall prove next that the implication B∈Bαp,q⇒{2-n(1/p-α)νnα}∈ℓq is true for general Blaschke products whenever 1/(2p)<α<1/p (even for 0<p<1).Theorem 4.5.
Let0<p,α<∞ with 1/(2p)<α<1/p, and let 0<q≤∞. Assume that I is an inner function in Bαp,q. Then (by Proposition J and Remark 4.4) I is a Blaschke product B and for its sequence of zeros {zk}, we have  (a)
Ifq=∞, that is, if B∈Λp,α, then {2-n(1/p-α)νnα}∈ℓ∞ or, equivalently, {dk1/p-αkα}∈ℓ∞.(b)
Ifq<∞, then {2-n(1/p-α)νnα}∈ℓq or, equivalently, {dk1/p-αkα-1/q}∈ℓq.Proof.
We will make use of the following inequality of Goldberg [40]:
(4.6)∫02π|B(reiθ)|dθ≤∫02π|B̂(reiθ)|dθ,0<r<1,
where B̂ is the Blaschke product whose zeros ẑk are the moduli |zk| of the corresponding zeros of B. Notice then that the sequence {d̂k} coincides with {dk}.  Assume first that q=∞. Since 1/p>1/p-α, Corollary 3.6 implies that B∈Λ1,αp. Now, since αp<1, Theorem D gives the following growth order:
(4.7)∫02π(1-|B(reiθ)|)dθ=O((1-r)αp),asr⟶1.
Combining this with (4.6), we obtain that B̂∈Λ1,αp. So, by Theorem K(a), we arrive at {2-n(1-αp)νnαp}∈ℓ∞, which is Theorem 4.5(a).
To prove part (b), we proceed analogously. Assume thatq<∞. Since 1/p>1/p-α, Corollary 3.6 implies that B∈Bαp1,q/p. Since αp<1, Theorem C gives
(4.8)∫01(1-r)-αq-1(12π∫02π(1-|B(reiθ)|)dθ)q/pdr<∞,
which combined with (4.6), gives the same estimate for B̂ in place of B, implying that B̂∈Bαp1,q/p. So, by Theorem K(b), {2-n(1-αp)νnαp}∈ℓq/p, and this is equivalent to the desired conclusion of Theorem 4.5(b).Theorem4.5 has some obvious consequences regarding the spaces Hαp.Corollary 4.6.
Let0<p,α<∞ with 1/(2p)<α<1/p. Assume that I is an inner function in Hαp. Then (by Corollary 4.1) I is a Blaschke product B whose sequence of zeros, {zk}, satisfies,  (a)
∑kdk1-αpkαp-1<∞ if 1/p-1<α,(b)
∑kdkt-αpkαp-1<∞ for all t>1 if 1/(2p)<α≤1/p-1.Proof.
Part (a) follows directly from the fact thatB∈Bαp,p by Corollary 4.6. As for part (b), it is because, by Property (P8), B∈Bβp,q for all β<α and all q>0.Results of this kind can be also obtained using Theorem 8 of [2], where it is proved that B∈H1p, 1/2<p<1 implies ∑kdk(1-p)/p<∞.Theorem 4.7.
Let0<p,α<∞ with 1/(2p)<α<1/p. Assume that I is an inner function in Hαp. Then (by Corollary 4.1) I is a Blaschke product B whose sequence of zeros, {zk}, satisfies ∑kdk(1-αp)/(αp)<∞.Proof.
Sinceα>1/p-α, then B∈H1αp by Corollary 3.13. Now, by Theorem 8 of [2], we conclude that ∑k=1∞dk(1-αp)/(αp)<∞.


## 5. Blaschke Products with Zeros in a Nontangential Region

This section will be mainly focussed on Blaschke products whose zeros lie in a fixed nontangential region, also called Stolz angle.Givenξ∈∂𝔻, a Stolz angle with vertex atξ is one of the following regions Ωσ(ξ)⊂𝔻, with σ∈[1,∞),(5.1)Ωσ(ξ)={z∈D:|1-ξ¯z|≤σ(1-|z|)}.
The Stolz angles with vertex at ξ=1 will be simply denoted by Ωσ.Let us start with the following result.Theorem L (see [7, Theorem 2.3]).
IfB is a Blaschke product whose zeros lie in a fixed Stolz angle, then  B∈Λp,1/(2p) for all  p>0.An improvement of this result for1/2<p<∞ is given in [15, Theorem 1.6], where it is actually proved that B∈λp,1/(2p). Anyhow, Theorem L provides the following result almost immediately.Theorem 5.1.
If0<p,α<∞ with α<1/(2p), 0<q≤∞, and B is a Blaschke product whose zeros lie in a fixed Stolz angle, then B∈Bαp,q∩Hαp.Proof.
ThatB∈Bαp,q is a direct application of Theorem L and property (P7). That B∈Hαp follows directly from the fact that B∈Bαp,p and Remark 3.12.This result is quite accurate, as the following one shows.Theorem 5.2.
If0<p,q<∞, then there exists a Blaschke product B whose zeros lie on the radius (0,1) and such that B∉B1/(2p)p,q. Also, there exists a Blaschke product B, with its sequence of zeros lying on the radius (0,1), and such that, independently of p, B∉H1/(2p)p.Proof.
By Corollary3.6, as 1/p>1/p-1/(2p), any inner function in B1/(2p)p,q is also in B1/21,q/p. So, for the first part of the proposition, it will suffice to give a Blaschke product B (depending on q/p) with its zeros lying on the radius (0,1) and such that B∉B1/21,q/p. On the other hand, since all inner functions in H1/(2p)p are also in H1/21 by Corollary 3.13, and since H1/21 and B1/21,1 contain the same inner functions by Theorem 3.10, it thus suffices to say that the Blaschke product sought in the second part of the proposition is one of the above that is not in B1/21,1.
Now letB be the Blaschke product whose sequence of zeros is given by zk=1-(klogβk)-1, k=2,3…, where β=1+2p/q>1. This example has been mentioned by Gluchoff [6, Example 2 after Corollary 1.15] to assert that, for all r sufficiently close to 1,
(5.2)∫02π(1-|B(reiθ)|)dθ≍(1-r)1/2log⁡-p/q11-r.
This implies that
(5.3)∫01(1-r)-q/(2p)-1(∫02π(1-|B(reiθ)|)dθ)q/pdr=∞.
which, by Theorem C, yields
(5.4)∫01(1-r)q/p-1M1q/p(r,D1+1/2B)dr=∞,
that is, B∉B1/21,q/p. This finishes the proof of the theorem.Remark 5.3.
The just mentioned example of a Blaschke productB not in B1/21,1 is that one that places its zeros at [1-(klog3k)-1], k≥2. This Blaschke product does not belong to any H1/(2p)p, 0<p<∞, neither to any B1/(2p)p,p. In particular, B does not belong to B1/33/2,3/2 which, by (1.14), is another way to say that B′∉A3/2. We remark that Girela and Peláez [12] proved that the Blaschke B with zeros ak=[1-(klog2k)-1], k≥2, also has this property. Actually, arguing as in the proof of [12], Theorem 2.1, we can deduce the following:
Ifβ≥2andBis the Blaschke product with zerosak=[1-(klogβk)-1],k≥2, thenB′∉A3/2.
Regarding Theorem K, we note that Jevtic' [7, Theorem 2.1] extended part (a) to the case 0<p<1. It would be interesting to know whether a similar extension holds for part (b). Of course, our Theorem 4.5 answers affirmatively the question for the implication  “B∈Bαp,q⇒{2-n(1/p-α)νnα}∈ℓq”. We have a partial result regarding the other implication.Theorem 5.4.
LetB be a Blaschke product with zeros {zk} in a fixed Stolz angle, and let 0<p,q,α<∞ be such that max{1/p-1,1/(2p)}<α<1/p. If {dk1/p-αkα-1/q}∈ℓq then B∈Bαp,q.Proof.
That{dk1/p-αkα-1/q}∈ℓq, it means that ∑kdkq/p-αqkαq-1<∞. Such sum remains unchanged if p,q,α are replaced, respectively, with pt,qt,α/t for any t>0. Thus, by Theorem K(b), B∈Bαp1,q/p because 1/2<αp<1. Now, since p>1-αp too, Corollary 3.6 can be applied to obtain that B∈Bαp,q.Using Theorem 9 of [2], we obtain the following result for Hardy-Sobolev spaces.Theorem 5.5.
Let0<p,α<∞ with max{1/p-1,1/(2p)}<α<1/p. Assume that B is a Blaschke product whose sequence of zeros {zk} lies in a fixed Stolz angle, and satisfies ∑kdk(1-αp)/(αp)<∞. Then B∈Hαp̃, for all 0<p̃<p.Proof.
Observe that the conditionmax{1/p-1,1/(2p)}<α<1/p is equivalent to saying that max{1/(2α),1/(1+α)}<p<1/α. Thus, by property (P3), the result will be proved if we show that B∈Hαp̃, for all max{1/(2α),1/(1+α)}<p̃<p.
For suchp̃, 1/2<αp̃<αp and, also, (1-αp)/(αp)<(1-αp̃)/(αp̃). So, by Theorem 9 of [2], B′∈Hαp̃, that is, B∈H1αp̃. Now, the fact that 1/(1+α)<p̃ implies that 1/α>1/(αp̃)-1. So, by Corollary 3.13, we conclude that B∈Hαp̃.



---

*Source: 626254-2011-06-16.xml*