Inner Functions in Lipschitz, Besov, and Sobolev Spaces

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 𝐻^𝑝𝛼 with 1/𝑝≤𝛼<∞ or any of the Besov spaces 𝐵^{𝛼𝑝,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 𝐵^0∞,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 𝐵^0∞,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.

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 by 𝔻={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 𝐻^𝑝, 0<𝑝≤∞, consists of those functions 𝑓∈Hol(𝔻) for which‖𝑓‖_𝐻^𝑝∶=sup_0<𝑟<1𝑀_𝑝(𝑟,𝑓)<∞,(1.1) where 𝑀_𝑝(𝑟,𝑓)=((1/2𝜋)∫^02𝜋|𝑓(𝑟𝑒^𝑖𝜃)|^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 𝐻^𝑝 spaces form a decreasing chain as 𝑝 increases, and any function in 𝐻^𝑝 has nontangential limits almost everywhere, with the additional property that the boundary function thus formed defines an isometry between 𝐻^𝑝 and 𝐿^𝑝(𝜕𝔻). The sequence {𝑧_𝑘} of zeros of an 𝐻^𝑝 function, counting multiplicities, satisfies the so-called Blaschke condition: ∑_𝑘(1−|𝑧_𝑘|)<∞. This condition characterizes the zero sequences of 𝐻^𝑝 functions. By writing 𝑏_𝑎(𝑧)=(|𝑎|/𝑎)((𝑎−𝑧)/(1−𝑎𝑧)) when 𝑎∈𝔻,𝑎≠0, and 𝑏₀(𝑧)=𝑧, the Blaschke condition implies that the product ∏_𝑘𝑏_𝑧𝑘(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 {𝑧_𝑘}, which is analytic in the unit disk 𝔻, and whose exact sequence of zeros, counting multiplicities, is {𝑧_𝑘}. Also, ‖𝐵‖_𝐻^∞≤1, and its boundary function has modulus 1 almost everywhere.

An inner 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)=𝑒^𝑖𝛾𝐵(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 form𝑆(𝑧)=exp−^02𝜋𝑒^𝑖𝜃+𝑧𝑒^𝑖𝜃−𝑧𝑑𝜇(𝜃),(1.2) 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⁻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.

The weighted Bergman space 𝐴^𝑝,𝛼, 0<𝑝<∞, −1<𝛼<∞ consists of those functions 𝑓∈Hol(𝔻) such that‖𝑓‖_𝐴^𝑝,𝛼∶=𝛼+1𝜋_𝔻||𝑓(𝑧)||^𝑝1−|𝑧|²^𝛼𝑑𝐴(𝑧)^1/𝑝<∞,(1.3) 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.

If 𝑓(𝑧)=∑^∞𝑘=0𝑓(𝑘)𝑧^𝑘 is analytic in 𝔻 and 𝛼>0, the fractional derivative of order 𝛼 of 𝑓, 𝐷^𝛼𝑓 is defined as𝐷^𝛼𝑓(𝑧)=^∞_𝑘=0(𝑘+1)^𝛼𝑓(𝑘)𝑧^𝑘.(1.4) (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”: 𝐷^𝑛𝑓(𝑧)=((𝑑/𝑑𝑧)𝑧)^𝑛𝑓(𝑧). Besides, the integral means 𝑀_𝑝(𝑟,𝐷^𝑛𝑓) and 𝑀_𝑝(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,𝑓(𝑧)=1Γ(𝛼)¹⁰𝐷^𝛼𝑓(𝑠𝑧)log^𝛼−11𝑠𝑑𝑠.(1.5) This yields the estimate||𝑓(𝑧)||≤𝐶¹⁰||𝐷^𝛼𝑓(𝑠𝑧)||(1−𝑠)^𝛼−1𝑑𝑠.(1.6)

If 0<𝑝≤∞ and 0≤𝛼<∞, then a function 𝑓∈Hol(𝔻) 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+𝛼𝑓<∞.(1.7) The subspace 𝜆^𝑝,𝛼 consists of those 𝑓∈Hol(𝔻) for whichlim_𝑟→1(1−𝑟)𝑀_𝑝𝑟,𝐷^1+𝛼𝑓=0.(1.8) Notice that Λ^∞,0 is another name for the usual Bloch space ℬ, of functions 𝑓∈Hol(𝔻) such that sup_𝑧∈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 an 𝐿^𝑞-norm in the above definition gives way to the Besov space 𝐵^{𝛼𝑝,𝑞}, 0<𝑝≤∞, 0<𝑞<∞, 0≤𝛼<∞, consisting of those functions 𝑓∈Hol(𝔻) for which‖𝑓‖_𝐵^{𝛼𝑝,𝑞}∶=¹⁰(1−𝑟)^𝑞−1𝑀^𝑞𝑝𝑟,𝐷^1+𝛼𝑓𝑑𝑟^1/𝑞<∞.(1.9) The space obtained when 𝑞=∞ is precisely 𝐵^𝛼𝑝,∞≡Λ^𝑝,𝛼. Other classes treated in this paper are the Hardy-Sobolev spaces 𝐻^𝑝𝛼, 0<𝑝≤∞, 0≤𝛼<∞, consisting of those functions 𝑓∈Hol(𝔻) for which‖𝑓‖_𝐻^𝑝𝛼∶=‖𝐷^𝛼𝑓‖_𝐻^𝑝<∞.(1.10)

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 gives𝑀^𝑝𝑝(𝑟,𝑓)≤𝐶¹⁰(1−𝑠)^𝑝𝛼−1𝑀^𝑝𝑝(𝑟𝑠,𝐷^𝛼𝑓)𝑑𝑠,(0<𝑝<1,0<𝛼),(1.11)𝑀_𝑝(𝑟,𝑓)≤𝐶¹⁰(1−𝑠)^𝛼−1𝑀_𝑝(𝑟𝑠,𝐷^𝛼𝑓)𝑑𝑠,(1≤𝑝≤∞,0<𝛼).(1.12) The same estimate (1.6) combined with variants of Hardy's inequality givessup_{0<𝜌≤𝑟}(1−𝜌)^𝛼𝑀_𝑝𝜌,𝐷^𝛽𝑓(0<≍𝑝su≤p∞_{0<𝜌≤𝑟}(1,−𝛼𝜌)>ma^{𝛼−𝛽+𝛾}x𝑀{_𝑝0(,𝜌𝛽,−𝐷^𝛾𝛾𝑓})),,(1.13)₁₀(1−𝑟)^𝛼−1𝑀(0^𝑞𝑝<𝑟,𝑞𝐷<^𝛽𝑓∞,𝑑𝑟0≍<𝑝¹⁰≤(1∞−,𝑟)𝛼>max^{𝛼−𝑞(𝛽−𝛾)−1}𝑀{0^𝑞𝑝,𝑞(𝑟(,𝛽𝐷−^𝛾𝑓𝛾))}𝑑)𝑟.,(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]):𝑀_𝑞(𝑟,𝑓)≤𝐶_𝑝,𝑞(1−𝑟)^{1/𝑞−1/𝑝}𝑀_𝑝1+𝑟2,𝑓,(0<𝑝<𝑞≤∞).(1.15)

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.sup_0≤𝑟<1𝑀_𝑝(𝑟,𝑓)≤𝐶_𝑝¹⁰(1−𝑟)^𝑝−1𝑀^𝑝𝑝𝑟,𝐷¹𝑓𝑑𝑟,(0<𝑝≤2),(1.16)¹⁰(1−𝑟)^𝑝−1𝑀^𝑝𝑝𝑟,𝐷¹𝑓𝑑𝑟≤𝐶_𝑝sup_0≤𝑟<1𝑀_𝑝(𝑟,𝑓),(𝑝≥2),(1.17) which can be restated as 𝐵^0𝑝,𝑝⊆𝐻^𝑝, for 0<𝑝≤2, and 𝐻^𝑝⊆𝐵^0𝑝,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)If 𝑞<∞, 𝐵^{𝛼𝑝,𝑞}⊆𝜆^𝑝,𝛼: If 𝑓∈𝐵^{𝛼𝑝,𝑞}, then ∫^1𝑟(1−𝜌)^𝑞−1𝑀^𝑞𝑝(𝜌,𝐷^1+𝛼𝑓)𝑑𝜌→0 as 𝑟→1, so by using the increasing behavior of the means 𝑀_𝑝(𝑟,𝐷^1+𝛼𝑓), obtain that (1−𝑟)𝑀_𝑝(𝑟,𝐷^1+𝛼𝑓)→0, as 𝑟→1, that is, that 𝑓∈𝜆^𝑝,𝛼.(𝑃2)If 𝑞₁≤𝑞₂, 𝐵^𝑝,𝑞1_𝛼⊆𝐵^𝑝,𝑞2_𝛼, because (1−𝑟)𝑀_𝑝(𝑟,𝐷^1+𝛼𝑓)→0, as 𝑟→1, for all 𝑓∈𝐵^𝑝,𝑞1_𝛼.(𝑃3)If 𝑝₁≤𝑝₂, 𝐵^{𝑝2𝛼,𝑞}⊆𝐵^{𝑝1𝛼,𝑞}, and 𝐻^𝑝2_𝛼⊆𝐻^𝑝1_𝛼, because 𝑀_𝑝1(𝑟,𝑔)≤𝑀_𝑝2(𝑟,𝑔) for all 𝑟∈(0,1) and all 𝑔∈Hol(𝔻).(𝑃4)If 𝛼₁≤𝛼₂, 𝐵^{𝛼𝑝,𝑞}₂⊆𝐵^{𝛼𝑝,𝑞}₁. Indeed, for 𝑓∈𝐵^{𝛼𝑝,𝑞}₂, (1.14) gives ‖𝑓‖^𝑞𝐵_{𝛼1𝑝,𝑞}≍¹⁰(1−𝑟)^{𝑞−1+𝑞(𝛼2−𝛼1)}𝑀^𝑞𝑝𝑟,𝐷^1+𝛼2𝑓𝑑𝑟<‖𝑓‖^𝑞𝐵_{𝛼2𝑝,𝑞}<∞.(1.18)(𝑃5)If 𝛼₁≤𝛼₂, 𝐻^𝑝𝛼₂⊆𝐻^𝑝𝛼₁, because of (1.11) and (1.12). (𝑃6)If 0<𝑝₁≤𝑝₂, 𝐵^{𝑝1,𝑞1/𝑝}₁⊆𝐵^{𝑝2,𝑞1/𝑝}₂. (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),‖𝑓‖^𝑞𝐵_{𝑝22,𝑞1/𝑝}≍¹⁰(1−𝑟)^{𝑞−1+𝑞(1/𝑝1−1/𝑝2)}𝑀^𝑞𝑝₂𝑟,𝐷^1+1/𝑝1𝑓𝑑𝑟≤𝐶‖𝑓‖^𝑞𝐵_{𝑝11,𝑞1/𝑝}.(1.19)(𝑃7)Λ^𝑝,𝛼⊆∩{𝐵^{𝛽𝑝,𝑞}∶𝛽<𝛼,0<𝑞}. This is again an application of (1.14), ‖𝑓‖^𝑞𝐵_{𝛽𝑝,𝑞}≍¹⁰(1−𝑟)^{𝑞(𝛼−𝛽)−1+𝑞}𝑀^𝑞𝑝𝑟,𝐷^1+𝛼𝑓𝑑𝑟≤𝐶‖𝑓‖_Λ^𝑝,𝛼,(1.20) where 𝐶=∫¹⁰(1−𝑟)^{𝑞(𝛼−𝛽)−1}𝑑𝑟<∞. (𝑃8)𝐻^𝑝𝛼⊆∩{𝐵^{𝛽𝑝,𝑞}∶𝛽<𝛼,0<𝑞}. Again an application of (1.14). (𝑃9)If 0<𝑝≤2, 𝐵^{𝛼𝑝,𝑝}⊆𝐻^𝑝𝛼. This is (1.16). (𝑃10)If 2≤𝑝, 𝐻^𝑝𝛼⊆𝐵^{𝛼𝑝,𝑝}. And this is (1.17).

In the following, we introduce some notation related to inner functions. Given an inner function 𝐼 and a point 𝑎∈𝔻, its Frostman shift 𝐼_𝑎 is defined as𝐼_𝑎(𝑧)=𝐼(𝑧)−𝑎1−𝑎𝐼(𝑧),𝑧∈𝔻.(1.21) 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 𝐼_𝑎 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 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.

Lemma 1.1. Let 0<𝑝,𝑞≤∞, 0≤𝛼, 0<𝛿<1/2, and 𝑎∈𝐾_𝛿={𝑧∈𝔻∶𝛿<|𝑧|<1−𝛿}. Then ‖𝐼_𝑎‖_𝐵^{𝛼𝑝,𝑞}≍‖𝐼‖_𝐵^{𝛼𝑝,𝑞}, with constants depending only on 𝑝,𝑞,𝛼, and 𝛿, but not on 𝐼 or 𝑎. In the case 𝑞=∞, the formulation is sup_{0≤𝜌<𝑟}(1−𝜌)𝑀_𝑝𝜌,𝐷^1+𝛼𝐼_𝑎≍sup_{0≤𝜌<𝑟}(1−𝜌)𝑀_𝑝𝜌,𝐷^1+𝛼𝐼.(1.22)

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^{𝛼𝑝,𝑞}-norm of 𝐼_𝑎 in terms of that of 𝐼, for 𝐼=(𝐼_𝑎)_−𝑎.
Writing 𝜓_𝑎(𝑧)=𝑎/(1−𝑎𝑧), 𝑧∈𝔻 observe that the first derivative of 𝐼_𝑎 can be written as 𝐼^𝑎=((1−|𝑎|²)/𝑎²)(𝜓_𝑎∘𝐼)^, and, in general, for a positive integer 𝑛, the 𝑛th derivative is given by 𝐼^𝑎(𝑛)=((1−|𝑎|²)/𝑎²)(𝜓_𝑎∘𝐼)^(𝑛). This, together with the Faà di Bruno's formula for the 𝑛th derivative of a composition, gives 𝐼^𝑎(𝑛)=1−|𝑎|²𝑎²𝜓_𝑎∘𝐼^(𝑛)=1−|𝑎|²𝑎²𝑘₁!⋯𝑛!𝑘_𝑛!𝜓^𝑎(𝑘)∘𝐼^𝑛_𝑗=1𝐼𝑗!^(𝑗)^𝑘𝑗,(1.23) where 𝑘=∑^𝑛𝑗=1𝑘_𝑗, and the sum runs over all 𝑛-tuples 𝑘⃗=(𝑘₁,…,𝑘_𝑛) of nonnegative integers such that ∑^𝑛𝑗=1𝑗𝑘_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‖^𝑞𝐵_{𝛼𝑝,𝑞}, 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]), ‖‖𝐼_𝑎‖‖^𝑞𝐵_{𝛼𝑝,𝑞}=¹⁰(1−𝑟)^𝑞−1𝑀^𝑞𝑝𝑟,𝐷^𝑛+𝛽𝐼_𝑎𝑑𝑟≤^(1.14)𝐶¹⁰(1−𝑟)^{𝑞(1−𝛽)−1}𝑀^𝑞𝑝𝑟,𝐼^𝑎(𝑛)𝑑𝑟_(1.23)≤𝐶_⃗𝑘¹⁰(1−𝑟)^{𝑞(1−𝛽)−1}𝑀^𝑞𝑝𝑟,^𝑛_𝑗=1𝐼^(𝑗)^𝑘𝑗𝑑𝑟_(Ḧ‌o)≤𝐶_⃗𝑘¹⁰(1−𝑟)^{𝑛𝑞(1−𝛽)−1}_𝑗=1𝑀^𝑞𝑘𝑗_{𝑝𝑛/𝑗}𝑟,𝐼^(𝑗)𝑑𝑟_(Ḧ‌o)≤𝐶_⃗𝑘𝑛_𝑗=1¹⁰(1−𝑟)^{𝑞(1−𝛽)−1}𝑀^{𝑞𝑛/𝑗𝑝𝑛/𝑗}𝑟,𝐼^(𝑗)𝑑𝑟^{𝑗𝑘𝑗/𝑛(1.14)}≤𝐶_⃗𝑘𝑛_𝑗=1¹⁰(1−𝑟)^{𝑞(1−𝛽)−1+(𝑞𝑛(𝑛−𝑗))/𝑗}𝑀^{𝑞𝑛/𝑗𝑝𝑛/𝑗}𝑟,𝐼^(𝑛)𝑑𝑟^{𝑗𝑘𝑗(/𝑛}_I)≤𝐶_⃗𝑘𝑛_𝑗=1¹⁰(1−𝑟)^{𝑞(1−𝛽)−1}𝑀^𝑞𝑝𝑟,𝐼^(𝑛)𝑑𝑟≤^{𝑗𝑘𝑗/𝑛(1.14)}𝐶‖𝐼‖^𝑞𝐵_{𝛼𝑝,𝑞}.(1.24)

2. Inner Functions in the Spaces 𝐵^{𝛼𝑝,𝑞} and 𝐻^𝑝𝛼, 𝛼≥1/𝑝

Ahern and Jevtić [5] proved that a Blaschke product lies in the space 𝐵^{𝑝,∞1/𝑝}≡Λ^𝑝,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 havefor0<𝑝<∞,thespace𝐵^{𝑝,∞1/𝑝}containsinfiniteBlaschkeproducts.(2.1) Our results in this section imply that the opposite is true for all the spaces 𝐵^{𝛼𝑝,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) Let 0<𝑝,𝑞≤∞ and 𝛼>1/𝑝. Then the only inner functions in 𝐵^{𝛼𝑝,𝑞} are finite Blaschke products.
(b) If 0<𝑝,𝑞<∞ then the only inner functions in 𝐵^{𝑝,𝑞1/𝑝} are finite Blaschke products.

Proof. To prove (a) observe that 𝐵^{𝛼𝑝,𝑞}⊆𝐵^𝛼𝑝,∞≡Λ^𝑝,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 𝐵^{𝛼𝑝,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]). Let 0<𝑝<∞. If B is a Blaschke product in H^p_1/p, then B is a finite Blaschke product.Theorem B (see [5, Theorem 3.2]). For each 0<𝑝<∞ there exists 𝜀_𝑝>0 such that if 𝐵 is a Blaschke product and limsup_𝑟→1(1−𝑟)𝑀_𝑝𝑟,𝐷^1+1/𝑝𝐵<𝜀_𝑝,(2.2) then 𝐵 is a finite Blaschke product. In particular, the only Blaschke products in 𝜆^𝑝,1/𝑝 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 0<𝑝<∞, the only inner functions in either 𝐻^𝑝1/𝑝 or 𝜆^𝑝,1/𝑝 are the finite Blaschke products.Now Theorem 2.1(b) follows from this result and the fact that 𝐵^{𝑝,𝑞1/𝑝}⊆𝜆^𝑝,1/𝑝.

The fact that 𝐻^𝑝𝛼⊆∩{𝐵^{𝛽𝑝,𝑞}∶𝛽<𝛼,0<𝑞} immediately implies the following.

Corollary 2.3. Let 0<𝑝≤∞ and 𝛼>1/𝑝. If 𝑓∈𝐻^𝑝𝛼 then 𝑓 admits a continuous extension to 𝔻. Consequently, the only inner functions in 𝐻^𝑝𝛼, with 𝛼>1/𝑝, are finite Blaschke products.

It remains to consider the case 𝑝=∞ and 𝛼=0. Of course, 𝐻^∞0≡𝐻^∞ contains the whole class of inner functions.

Let us deal now with the spaces 𝐵^0∞,𝑞. First of all, 𝐵^0∞,∞ 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<∞, 𝐵^0∞,𝑞 is a subspace of 𝜆^∞,0, the natural question rises as whether 𝐵^0∞,𝑞 contains or not infinite Blaschke products. The answer depends on the result of intersecting 𝐵^0∞,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 iflim_|𝑎|→1‖‖𝑓∘𝜑_𝑎−𝑓(𝑎)‖‖_𝐻𝑝=0,(2.3) for some (or, equivalently, for all) finite positive 𝑝. Here, 𝜑_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧) 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 that1||𝐽||_𝑆(𝐽)1−|𝑧|²||𝑓^(𝑧)||𝑑𝐴(𝑧)⟶0,as||𝐽||⟶0,(2.4) where 𝐽 is an interval in 𝜕𝔻, |𝐽| is its length, and 𝑆(𝐽) is the Carleson square defined by 𝑆(𝐽)={𝑟𝑒^𝑖𝜃∶𝑒^𝑖𝜃∈𝐽,1−|𝐽|≤𝑟<1}.

Theorem 2.4. (a), If 0<𝑞≤2, then 𝐵^0∞,𝑞⊆𝑉𝑀𝑂𝐴. Consequently, any inner function in 𝐵^0∞,𝑞 is a finite Blaschke product.
(b) There are infinite Blaschke products in ∩_2<𝑞<∞𝐵^0∞,𝑞.

Proof. To prove (a) observe that, since 𝐵^∞,𝑞1₀⊆𝐵^∞,q2₀ for 0<𝑞₁<𝑞₂<∞, it suffices to settle the result for 𝑞=2.
Thus, take 𝑓∈𝐵^0∞,2 and take an interval 𝐽 in 𝜕𝔻 with |𝐽|<1/2, then 1||𝐽||_𝑆(𝐽)1−|𝑧|²||𝑓^(𝑧)||²𝑑𝐴(𝑧)≤21||𝐽||^{1||𝐽||1−}≤2_{1||𝐽||1−}(1−𝑟)_𝐽||𝑓^𝑟𝑒^𝑖𝜃||²𝑑𝜃𝑑𝑟(1−𝑟)𝑀^2∞𝑟,𝑓^𝑑𝑟,(2.5) and observe that, since 𝑓∈𝐵^0∞,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 1−|𝑧|²||𝐼^(𝑧)||1−||𝐼(𝑧)||²≤log^−1/2𝑒1−|𝑧|.(2.6) This implies that 𝐼∈𝐵^0∞,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 𝐼_𝑎 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 𝐼_𝑎 is an infinite Blaschke product in ∩_2<𝑞<∞𝐵^0∞,𝑞.

Once Theorem 2.4 is proved, it is natural to ask whether or not the inclusion 𝑉𝑀𝑂𝐴⊆𝐵^0∞,𝑞 holds for 2<𝑞<∞. An argument based on duality shows that this is not so.

Theorem 2.5. If 2<𝑞<∞, then the class 𝑉𝑀𝑂𝐴⧵𝐵^0∞,𝑞 is nonempty.

Proof. Observe that the dual of 𝑉𝑀𝑂𝐴 is 𝐻¹ under the usual pairing: ⟨𝑓,𝑔⟩=lim_𝑟→1∑_𝑘𝑓(𝑘)𝑔̂(𝑘)𝑟^𝑘, (see [32, 34]). Also, using the same techniques as in [4], we get that the dual of 𝐵^0∞,𝑞 is 𝐵^1,𝑞′₀, 1/𝑞+1/𝑞^=1, under the same pairing as before. Thus, the problem reduces to show that 𝐵^1,𝑞′₀⧵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 that𝑀₁𝑟,𝑓^≤𝑐(1−𝑟)(log(2𝑒/(1−𝑟))),𝑟⁰<𝑟<1.(2.7) It then follows that 𝑓∈𝐵^1,𝑞′₀, 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 function 𝐼 in Besov or Hardy-Sobolev spaces with the distribution of the preimages {𝐼⁻¹(𝑎)}, 𝑎∈𝔻. We start introducing certain counting functions.

If 𝐼 is an inner function and 𝑎∈𝔻, 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:𝜈𝑘_𝑛(𝑎)=Card𝑘∶2^−𝑛<𝑑_𝑘(𝑎)=max𝑘∶2^−𝑛<𝑑_𝑘(𝑎)_𝑛,(𝑎)=Card𝑘∶2^−𝑛−1<𝑑_𝑘(𝑎)≤2^−𝑛=𝑘_𝑛+1(𝑎)−𝑘_𝑛(𝑎).(3.1) Observe that 𝑘₀(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.

Lemma 3.1. Under the previous settings, let 𝛼,𝛽>0. Then (a){2^−𝑛𝛼𝜈^𝛽𝑛(𝑎)}∈ℓ^∞ if and only {𝑑^𝛼𝑘(𝑎)𝑘^𝛽}∈ℓ^∞, and, in either case, their ℓ^∞-norms are comparable.(b)∑_𝑛≥02^−𝑛𝛼𝜈^𝛽𝑛(𝑎)≍∑_𝑘≥1𝑑^𝛼𝑘(𝑎)𝑘^𝛽−1.

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^𝛼𝑘𝑘^𝛽≤𝐶 for all 𝑘, then for each 𝑛=0,1…, 2^−𝑛𝛼𝜈^𝛽𝑛=2^−𝑛𝛼𝑘_𝑛+1−𝑘_𝑛^𝛽≤2^𝛼𝑑^𝛼𝑘_𝑛+1𝑘^𝛽𝑛+1≤𝐶.(3.2) In the other direction, assume that 2^−𝑛𝛼𝜈^𝛽𝑛≤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, 𝑑^𝛼𝑘𝑘^𝛽≤2^−𝑛𝛼𝑘^𝛽𝑛+1=^𝑛_𝑗=02^{−𝛼𝑗/𝛽}𝜈_𝑗2^{−𝛼(𝑛−𝑗)/𝛽}^𝛽≤𝐶_𝑛,_𝑗=02^{−𝛼(𝑛−𝑗)/𝛽}^𝛽≤𝐶1−2^−𝛼/𝛽^−𝛽.(3.3)
To prove (b), assume first that ∑_𝑛≥02^−𝑛𝛼𝜈^𝛽𝑛<∞. In the case 𝛽≤1, use an easy integral estimate and the fact that 𝑘^𝛽𝑛+1−𝑘^𝛽𝑛≤(𝑘_𝑛+1−𝑘_𝑛)^𝛽, to obtain the desired result, _𝑘≥1𝑑^𝛼𝑘𝑘^𝛽−1=_𝑘𝑛≥0^𝑛+1_𝑘=_𝑘𝑛+1𝑑^𝛼𝑘𝑘^𝛽−1≤_𝑛≥02^{𝑘−𝑛𝛼𝑛+1}_𝑘=_𝑘𝑛+1𝑘^𝛽−1≤_𝑛≥02^−𝑛𝛼^𝑘𝑛+1_𝑘𝑛𝑥^𝛽−1𝑑𝑥=1𝛽_𝑛≥0≤12^−𝑛𝛼𝑘^𝛽𝑛+1−𝑘^𝛽𝑛𝛽_𝑛≥02^−𝑛𝛼𝑘_𝑛+1−𝑘_𝑛^𝛽=1𝛽_𝑛≥02^−𝑛𝛼𝜈^𝛽𝑛.(3.4)
In the case 𝛽>1, it is easy to arrive at ∑_𝑘≥1𝑑^𝛼𝑘𝑘^𝛽−1≤∑_𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1. Now we imitate the proof of Hardy's inequality given in [36, Theorem 326 in page 239]. Write ℎ_𝑛=2^{−𝑛𝛼/𝛽}𝑘_𝑛+1=2^{−𝑛𝛼/𝛽}∑^𝑛𝑗=0𝜈_𝑗, for 𝑛≥0, and ℎ₋₁=0. Then 𝜈_𝑛=2^{𝑛𝛼/𝛽}(ℎ_𝑛−2^−𝛼/𝛽ℎ_𝑛−1), for 𝑛≥0, and by the inequality between the geometric and arithmetic means [36, Theorem 9 in page 17], ℎ^𝛽𝑛−11−2^−𝛼/𝛽ℎ^𝑛𝛽−12^{−𝑛𝛼/𝛽}𝜈_𝑛=12^𝛼/𝛽−1ℎ^𝑛𝛽−1≤1ℎ_n−1−ℎ^𝛽𝑛2^𝛼/𝛽−1(𝛽−1)ℎ^𝛽𝑛+ℎ^𝛽𝑛−1=1𝛽−ℎ^𝛽𝑛𝛽2^𝛼/𝛽−1ℎ^𝛽𝑛−1−ℎ^𝛽𝑛.(3.5) 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, _𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1=_𝑛≥0ℎ^𝛽𝑛≤11−2^−𝛼/𝛽_𝑛≥0ℎ^𝑛𝛽−12^{−𝑛𝛼/𝛽}𝜈_𝑛≤11−2^−𝛼/𝛽_𝑛≥0ℎ^𝛽𝑛^{𝛽−1/𝛽}_𝑛≥02^−𝑛𝛼𝜈^𝛽𝑛^1/𝛽,(3.6) giving as a result that ∑_𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1≤(1−2^−𝛼/𝛽)^−𝛽∑_𝑛≥02^−𝑛𝛼𝜈^𝛽𝑛, as desired.
In the other direction, assume that ∑_𝑘≥1𝑑^𝛼𝑘𝑘^𝛽−1<∞. In the case 𝛽<1, it is easily verified that ∑_𝑛≥02^−𝑛𝛼𝜈^𝛽𝑛≤∑_𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1. To continue, use that 𝑘₀=0 and that the function 𝑥↦𝑥^𝛽−1 is decreasing in (0,∞), _𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1=_𝑛≥02^−𝑛𝛼𝑘^{𝑘𝛽−1𝑛+1𝑛+1}_𝑘=11≤_𝑛≥02^{𝑘−𝑛𝛼𝑛+1}_𝑘=_𝑘𝑛+1𝑘^𝛽−1+_𝑛≥12^−𝑛𝛼𝑘^𝛽𝑛≤2^𝛼_𝑘𝑛≥0^𝑛+1_{𝑘=𝑘𝑛+1}𝑑^𝛼𝑘𝑘^𝛽−1+2^−𝛼_𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1,(3.7) from where it follows that ∑_𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1≤2^𝛼(1−2^−𝛼)⁻¹∑_𝑘≥1𝑑^𝛼𝑘𝑘^𝛽−1<∞.
It remains to deal with the case 𝛽≥1 under the assumption ∑_𝑘≥1𝑑^𝛼𝑘𝑘^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, _𝑛≥02^−𝑛𝛼𝜈^𝛽𝑛=_𝑛≥02^−𝑛𝛼𝑘_𝑛+1−𝑘_𝑛^𝛽≤_𝑛≥02^−𝑛𝛼𝑘^𝛽𝑛+1−𝑘^𝛽𝑛=2^𝛼_𝑛≥02^{𝑘−(𝑛+1)𝛼𝑛+1}_𝑘=_𝑘𝑛+1𝑘^𝛽≤2_𝛼−(𝑘−1)^𝛽𝛽_𝑘𝑛≥0^𝑛+1_𝑘=_𝑘𝑛+1𝑑^𝛼𝑘𝑘^𝛽−1=2^𝛼𝛽_𝑘≥1𝑑^𝛼𝑘𝑘^𝛽−1.(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, ‖𝐼‖^𝑞𝐵_{𝛼𝑝,𝑞}=¹⁰(1−𝑟)^𝑞−1𝑀^𝑞𝑝𝑟,𝐷^1+𝛼𝐼𝑑𝑟,(3.9)¹⁰(1−𝑟)^{(1−𝛼)𝑞−1}𝑀^𝑞𝑝𝑟,𝐼^𝑑𝑟,(3.10)¹⁰(1−𝑟)^{−𝛼𝑞−1}21𝜋^02𝜋1−||𝐼𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃^𝑞/𝑝𝑑𝑟,(3.11)¹⁰(1−𝑟)^{−𝛼𝑞−1}21𝜋^02𝜋||𝐼𝑒^𝑖𝜃−𝐼𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃^𝑞/𝑝𝑑𝑟.(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, ‖𝐼‖_Λ^𝑝,𝛼=sup_0≤𝑟<1(1−𝑟)𝑀_𝑝𝑟,𝐷^1+𝛼𝐼,(3.13)sup_0≤𝑟<1(1−𝑟)^1−𝛼𝑀_𝑝𝑟,𝐼^,(3.14)sup_0≤𝑟<1(1−𝑟)^−𝛼21𝜋^02𝜋1−||𝐼𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃^1/𝑝,(3.15)sup_0≤𝑟<1(1−𝑟)^−𝛼21𝜋^02𝜋||𝐼𝑒^𝑖𝜃−𝐼𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃^1/𝑝.(3.16)

For Blaschke products, the third author [15, 38] gave sufficient (and in special cases, necessary) conditions for their membership in 𝐵^{𝛼𝑝,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 {𝑧_𝑘} in the unit disk is called interpolating if it is uniformly separated, that is, inf_{𝑛𝑘∶}_{𝑧𝑘≠𝑧𝑛}||||𝑧_k−𝑧_𝑛1−𝑧_𝑘𝑧_𝑛||||>0.(3.17)

Theorem E (see [15, 38]). Let 0<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)}𝜈_𝑛)^1/𝑝}∈ℓ^𝑞, then 𝐵∈𝐵^{𝛼𝑝,𝑞} and ‖𝐵‖_𝐵^{𝛼𝑝,𝑞}≤𝐶‖‖2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛^1/𝑝‖‖_ℓ𝑞.(3.18) On the other hand, if the zero sequence {𝑧_𝑘} of 𝐵 is Carleson-Newman and 𝐵∈𝐵^{𝛼𝑝,𝑞}, then {(2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛)^1/𝑝}∈ℓ^𝑞 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 0<𝑝,𝛼<∞ with max{0,1/𝑝−1}<𝛼<1/𝑝, 0<𝑞≤∞, and 𝐼 is an inner function satisfying _𝐾𝛿‖‖2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛(𝑎)^1/𝑝‖‖^𝑝ℓ_𝑞𝑑𝐴(𝑎)^1/𝑝<∞,(3.19) for 𝐾_𝛿={𝑎∈𝔻∶𝛿≤|𝑎|≤1−𝛿}, and 0<𝛿<1/2, then 𝐼∈𝐵^{𝛼𝑝,𝑞}, and the norm ‖𝐼‖_𝐵^{𝛼𝑝,𝑞} is controlled by the integral in (3.19).

The crux of the matter here is that the above condition is also necessary for 𝐼 to belong to 𝐵^{𝛼𝑝,𝑞}.

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^{𝛼𝑝,𝑞} if and only if (3.19) holds for some 𝛿∈(0,1/2). In that case, both quantities, ‖𝐼‖_𝐵^{𝛼𝑝,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. If 0<𝑝<∞, 0<𝑞≤∞, 0≤𝛼<∞, and 1≤𝑡<∞, then Λ^∞,0∩𝐵^{𝛼𝑝,𝑞}⊆𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡}, that is, Bloch functions in 𝐵^{𝛼𝑝,𝑞} are also in 𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡} for any 𝑡≥1. Furthermore, the following relation holds: ‖𝑓‖_𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡}≤𝐶‖𝑓‖^Λ1−1/𝑡_∞,0‖𝑓‖^𝐵1/𝑡_{𝛼𝑝,𝑞}.(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^{𝛼𝑝,𝑞}. We need to show that 𝑓∈𝐵^{𝑝𝑡,𝑞𝑡𝛼/}_t. For that, use (1.14) to find an equivalent quantity to ‖𝑓‖_𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡}, and then separate it into two factors, the first will be controlled by ‖𝑓‖_Λ^∞,0, and the second by ‖𝑓‖_𝐵^{𝛼𝑝,𝑞}. ‖𝑓‖_𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡}=¹⁰(1−𝑟)^𝑞𝑡−1𝑀^{𝑞𝑡𝑝𝑡}𝑟,𝐷^1+𝛼/𝑡𝑓𝑑𝑟^{1/𝑞𝑡(1.14)}≤𝐶¹⁰(1−𝑟)^{𝑞𝑡−1−𝑞𝛼+𝑞𝛼𝑡}𝑀^{𝑞𝑡𝑝𝑡}𝑟,𝐷^1+𝛼𝑓𝑑𝑟^1/𝑞𝑡=𝐶¹⁰(1−𝑟)^{𝑞(𝑡−1)(1+𝛼)+(𝑞−1)}^02𝜋||𝐷^1+𝛼𝑓𝑟𝑒^𝑖𝜃||^𝑝𝑡𝑑𝜃^𝑞/𝑝𝑑𝑟^1/𝑞𝑡≤𝐶sup_0≤𝑟<1(1−𝑟)^1+𝛼𝑀_∞𝑟,𝐷^1+𝛼𝑓^1−1/𝑡¹⁰(1−𝑟)^𝑞−1𝑀^𝑞𝑝𝑟,𝐷^1+𝛼𝑓𝑑𝑟^1/𝑞𝑡≤𝐶‖𝑓‖^Λ1−1/𝑡_∞,0‖𝑓‖^𝐵1/𝑡_{𝛼𝑝,𝑞}.(3.21)

Two more lemmas are needed.

Lemma F (see [15, Corollary 4.5]). If 𝑝≥1, 𝐼 is an inner function, and if 1−2^−𝑛<𝑟≤1−2^−(𝑛+1), then 2^−𝑛𝜈_𝑛≤𝐶_𝑝^02𝜋log^𝑝1||𝐼𝑟𝑒^𝑖𝜃||𝑑𝜃.(3.22)

Lemma G (see [15, Corollary 4.7]). If 0<𝑝<∞, 0<𝛿<1/2, 𝐼 is an inner function, and if 𝑧∈𝔻, then _𝐾𝛿log^𝑝1||𝐼_𝑎(𝑧)||𝑑𝐴(𝑎)≤𝐶^𝑝,𝛿1−||𝐼(𝑧)||^𝑝.(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^{𝛼𝑝,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 _𝐾𝛿‖‖2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛(𝑎)^1/𝑝‖‖^𝑝ℓ_𝑞𝑑𝐴(𝑎)^1/𝑝=_𝐾𝛿‖‖2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛(𝑎)^1/𝑝𝑡‖‖^ℓ𝑝𝑡_𝑞𝑡𝑑𝐴(𝑎)^1/𝑝≤𝐶‖𝐼‖^𝑡𝐵_{𝑝𝑡,𝑞𝑡𝛼/𝑡}≤𝐶‖𝐼‖^Λ𝑡−1_∞,0‖𝐼‖_𝐵^{𝛼𝑝,𝑞}.(3.24) 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 ∫^𝑟𝑛+1_𝑟𝑛𝑟^{−𝛼𝑞−1}𝑑𝑟≍2^𝑛𝛼𝑞, and finally Lemmas F and G together with Theorem C to arrive at the desired estimate for (3.19), _𝐾𝛿‖‖2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛(𝑎)^1/𝑝‖‖^𝑝ℓ_𝑞𝑑𝐴(𝑎)^1/𝑝=⎛⎜⎜⎝_𝐾𝛿_𝑛≥02^𝑛𝛼𝑞2^−𝑛𝜈_𝑛(𝑎)^𝑞/𝑝^𝑝/𝑞𝑑𝐴(𝑎)⎞⎟⎟⎠^{(𝑞/𝑝)(1/𝑞)}≤_𝑛≥02^𝑛𝛼𝑞_𝐾𝛿2^−𝑛𝜈_𝑛(𝑎)𝑑𝐴(𝑎)^𝑞/𝑝^1/𝑞≤𝐶⎛⎜⎜⎝_𝑛≥0^𝑟𝑛+1_r𝑛(1−𝑟)^{−𝛼𝑞−1}_𝐾𝛿^02𝜋log^𝑝1||𝐼_𝑎𝑟𝑒^𝑖𝜃||𝑑𝜃𝑑𝐴(𝑎)^𝑞/𝑝𝑑𝑟⎞⎟⎟⎠^1/𝑞≤𝐶⎛⎜⎜⎝¹⁰(1−𝑟)^{−𝛼𝑞−1}^02𝜋_𝐾𝛿log^𝑝1||𝐼_𝑎𝑟𝑒^𝑖𝜃||𝑑𝐴(𝑎)𝑑𝜃^𝑞/𝑝𝑑𝑟⎞⎟⎟⎠^1/𝑞≤𝐶¹⁰(1−𝑟)^{−𝛼𝑞−1}^02𝜋1−||𝐼𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃^𝑞/𝑝𝑑𝑟^1/𝑞≤𝐶‖𝐼‖_𝐵^{𝛼𝑝,𝑞}.(3.25)
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,_𝐾𝛿‖‖2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛(𝑎)^1/𝑝‖‖^𝑝ℓ_𝑞𝑑𝐴(𝑎)^1/𝑝=⎛⎜⎜⎝_𝐾𝛿_𝑛≥02^𝑛𝛼𝑞2^−𝑛𝜈_𝑛(𝑎)^𝑞/𝑝^𝑝/𝑞𝑑𝐴(𝑎)⎞⎟⎟⎠^1/𝑝≤_𝐾𝛿𝑑𝐴(𝑎)^{1/𝑝−1/𝑞}_𝐾𝛿_𝑛≥02^𝑛𝛼𝑞2^−𝑛𝜈_𝑛(𝑎)^𝑞/𝑝𝑑𝐴(𝑎)^1/𝑞=𝐶_𝑛≥02^𝑛𝛼𝑞_𝐾𝛿2^−𝑛𝜈_𝑛(𝑎)^𝑞/𝑝𝑑𝐴(𝑎)^1/𝑞≤𝐶⎛⎜⎜⎝_𝑛≥0^𝑟𝑛+1_𝑟𝑛(1−𝑟)^{−𝛼𝑞−1}⎛⎜⎜⎝_𝐾𝛿^02𝜋log^𝑝1||𝐼_𝑎𝑟𝑒^𝑖𝜃||𝑑𝜃^𝑞/𝑝𝑑𝐴(𝑎)⎞⎟⎟⎠^{(𝑝/𝑞)(𝑞/𝑝)}𝑑𝑟⎞⎟⎟⎠^1/𝑞≤𝐶⎛⎜⎜⎝¹⁰(1−𝑟)^{−𝛼𝑞−1}⎛⎜⎜⎝^02𝜋_𝐾𝛿log^𝑞1||𝐼_𝑎𝑟𝑒^𝑖𝜃||𝑑𝐴(𝑎)^𝑝/𝑞𝑑𝜃⎞⎟⎟⎠^𝑞/𝑝𝑑⎞⎟⎟⎠^1/𝑞≤𝐶¹⁰𝑟^{−𝛼𝑞−1}^02𝜋1−||𝐼𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃^𝑞/𝑝𝑑𝑟^1/𝑞≤𝐶‖𝐼‖_𝐵^{𝛼𝑝,𝑞}.(3.26)

Remark 3.5. Along the proof of this theorem, we have actually proved that if 𝐼 is an inner function in 𝐵^{𝛼𝑝,𝑞}, with 0<𝛼,𝑝<∞ and 0<𝑞≤∞, then (3.19) holds and _𝐾𝛿‖‖2^{−𝑛(1−𝛼𝑝)}𝜈_𝑛(𝑎)^1/𝑝‖‖^𝑝ℓ_𝑞𝑑𝐴(𝑎)^1/𝑝≤𝐶‖𝐼‖_𝐵^{𝛼𝑝,𝑞}.(3.27)
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^{𝛼𝑝,𝑞}. Then 𝐼∈𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡} for all 𝑡>1/𝑝−𝛼.

Remark 3.7. Of course, when 𝛼>1/𝑝, the class of inner functions in 𝐵^{𝛼𝑝,𝑞} coincides with that of 𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡} for any 𝑡>0 (or any 𝐵^𝑝1,𝑞1_𝛼1 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 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^1,11/2 but not in 𝐻^11/2. By property (𝑃9), we deduce that 𝐵∉𝐵^11/2,1/2, and this is the space that would be obtained from 𝐵^1,11/2 by taking 𝑡=1/2, which coincides with 1/𝑝−𝛼 with the usual notation.

Proof of Corollary 3.6. Notice first that Remark 3.5 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∈𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡} 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∈𝐵^{𝑝𝑡,𝑞𝑡𝛼/𝑡} 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 ∑_𝑘𝑑^𝑘2−𝑝<∞ for some 𝑝∈(1,2). By Lemma 3.1(b), this condition is equivalent to ∑_𝑛≥02^{−𝑛(2−𝑝)}𝜈_𝑛<∞, which implies that 𝐵∈𝐵^{𝑝,𝑝1−1/𝑝} by Theorem E. (Notice that this is equivalent to 𝐵′∈𝐴^𝑝 by (1.14).) Observe now that we can apply Corollary 3.6 with 𝑡=1/𝑝, obtaining that 𝐵∈𝐵^1,1𝑝−1, which is the aforementioned result by Protas, with our notation. On the other hand, notice also that if 𝐵^∈𝐴^𝑝, for some 1<𝑝<2, and the zero sequence of 𝐵 is Carleson-Newman then, by Theorem E, ∑_𝑘𝑑^𝑘2−𝑝<∞.
Next we will turn to study the membership of inner functions in the spaces 𝐻^𝑝𝛼. Properties (𝑃9) and (𝑃10), that is, the Littlewood-Paley inequalities, relate quite well 𝐻^𝑝𝛼 with 𝐵^{𝛼𝑝,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. Let 0<𝑝≤∞ and 𝛼>max{0,1/𝑝−1}. Then the class of inner functions in 𝐻^𝑝𝛼 coincides with that of inner functions in 𝐵^{𝛼𝑝,𝑝}.

The proof of this result requires again an homogeneity property.

Proposition 3.11. Let 0<𝑝,𝛼<∞ and let 𝑓 be a Bloch function in 𝐻^𝑝𝛼. Then 𝑓∈𝐻^{𝑝𝑡𝛼/𝑡} 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 each 0<𝛼<𝛽<∞, there is a constant 𝐶=𝐶_𝛼,𝛽 such that if 𝑓∈Λ^∞,0 then ||𝐷^𝛼𝑓(𝑧)||≤𝐶max_0≤𝑡≤1||𝐷^𝛽𝑓(𝑡𝑧)||^𝛼/𝛽.(3.28)

The original proof of this lemma runs with 𝐻^∞ functions instead of Bloch (Λ^∞,0) functions. However, the lemma can be proved for Bloch functions by just noticing the validity of the estimate |𝐷^𝛽𝑓(𝑧)|≤𝐶(1−|𝑧|)^−𝛽 for Bloch functions.

Proof of Theorem 3.10. First notice that when 𝛼≥1/𝑝, the only inner functions in 𝐻^𝑝𝛼 and 𝐵^{𝛼𝑝,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 case 0<𝑝≤2. Then 𝐵^{𝛼𝑝,𝑝}⊆𝐻^𝑝𝛼 by (𝑃9). To go in the other direction, take an inner function 𝐼 in 𝐻^𝑝𝛼. Then, by Proposition 3.11, 𝐼∈𝐻^{𝑝𝑡𝛼/𝑡} for all 𝑡≥1. For 𝑡≥2/𝑝≥1, we have 𝑝𝑡≥2, and hence 𝐻^{𝑝𝑡𝛼/𝑡}⊆𝐵^{𝑝𝑡,𝑝𝑡𝛼/𝑡} 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∈𝐵^{𝛼𝑝,𝑝} by Corollary 3.6.
Next consider the case 2≤𝑝<∞. (Then max{0,1/𝑝−1}=0.) By (𝑃10), 𝐻^𝑝𝛼⊆𝐵^{𝛼𝑝,𝑝}. To go in the other direction, take an inner function 𝐼 in 𝐵^{𝛼𝑝,𝑝}. Since 0<𝛼<1/𝑝 then, by Corollary 3.6, 𝐼∈𝐵^{𝑝𝑡,𝑝𝑡𝛼/𝑡} for all 𝑡>1/𝑝−𝛼. Choose 𝑡∈(1/𝑝−𝛼,2/𝑝]. Thus, as 𝑝𝑡≤2, (𝑃9) gives 𝐵^{𝑝𝑡,𝑝𝑡𝛼/𝑡}⊆𝐻^{𝑝𝑡𝛼/𝑡}. Finally, as 1/𝑡≥𝑝/2≥1, Proposition 3.11 gives 𝐼∈𝐻^𝑝𝛼.

Remark 3.12. A careful reading of this proof shows that, in fact, any inner function 𝐼 in 𝐵^{𝛼𝑝,𝑝} is also in 𝐻^𝑝𝛼, whenever 0<𝑝≤∞ and 0<𝛼.

The analogous to Corollary 3.6 is the following.

Corollary 3.13. Let 0<𝑝,𝛼<∞ be such that 0<𝛼<1/𝑝. Assume that 𝐼 is an inner function in 𝐻^𝑝𝛼. Then 𝐼∈𝐻^{𝑝𝑡𝛼/𝑡} for all 𝑡>1/𝑝−𝛼.

Proof. Take 𝑡>1/𝑝−𝛼. If 𝑡≥1 then 𝐼∈𝐻^{𝑝𝑡𝛼/𝑡} 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∈𝐵^{𝛼𝑝,𝑝} and, by Corollary 3.6, 𝐼∈𝐵^{𝑝𝑡,𝑝𝑡𝛼/𝑡} and, finally, by Remark 3.12, 𝐼∈𝐻^{𝑝𝑡𝛼/𝑡}.

Remark 3.14. Again, since it only contains finite Blaschke products, the class of inner functions in 𝐻^𝑝𝛼, with 𝛼≥1/𝑝, coincides with that of 𝐻^{𝑝𝑡𝛼/𝑡} for all 𝑡>0, or for the case, with the class of inner functions of any 𝐻^𝑝1_𝛼1, 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 𝐵^1,11/2 but not in 𝐻^11/2, which is the space that would be obtained from 𝐻^11/2 by taking 𝑡=1/2(=1/𝑝−𝛼 with the usual notation). By Theorem 3.10, 𝐵∈𝐵^1,11/2 implies that 𝐵∈𝐻^11/2.

Remark 3.15. As an application, we again regain a known result of Protas [9, Theorem 2], namely, if 𝐵 is a Blaschke product such that ∑_𝑘𝑑^𝑘1−𝑝<∞, for some 1/2<𝑝<1, then 𝐵∈𝐻^𝑝1. Indeed, by Lemma 3.1(b), ∑_𝑘𝑑^𝑘1−𝑝≍∑_𝑛≥02^{−𝑛(1−𝑝)}𝜈_𝑛, and so 𝐵∈𝐵^1𝑝,𝑝 by Theorem E, and we are done with 𝐵∈𝐻^𝑝1 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 ∑_𝑘𝑑^𝑘1−𝑝<∞ is also necessary for 𝐵∈𝐻^𝑝1.

4. The Case 1/(2𝑝)<𝛼<1/𝑝

Ahern and Clark [2, Theorem 3] proved that the only inner functions in 𝐻^11/2 are Blaschke products. Later on, Ahern and Jevtic' obtained the following generalization:

Theorem I (see [5, Theorem 2.1]). If 𝐼 is an inner function and 𝑀^𝑝𝑝𝑟,𝐷^1/(2𝑝)𝐼=olog11−𝑟,as𝑟⟶1,(4.1) for some 0<𝑝<∞, then 𝐼 is a Blaschke product.

Now, all functions in 𝐻^{𝑝1/(2𝑝)} satisfy condition (4.1) and, by (𝑃5), the same is true for all 𝐻^𝑝𝛼-functions with 𝛼≥1/(2𝑝). So the following is immediate:

Corollary 4.1. Let 0<𝑝,𝛼<∞ with 𝛼≥1/(2𝑝). Then the only inner functions in 𝐻^𝑝𝛼 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]). Let 0<𝑝,𝑞,𝛼<∞ with 𝛼≥1/(2𝑝). Then the only inner functions in 𝐵^{𝛼𝑝,𝑞} are Blaschke products, finite ones if 𝛼≥1/𝑝.

Remark 4.2. These results are again accurate, for the “atomic” singular inner function 𝑆(𝑧)=exp(−(1+𝑧)/(1−𝑧)) is in 𝐵^{𝛼𝑝,𝑞}∩𝐻^𝑝𝛼 for all 0<𝑝,𝑞,𝛼<∞ with 𝛼<1/(2𝑝). Indeed, in [39], it is shown that, for any positive integer 𝑛, 𝑀_1/(2𝑛)𝑟,𝑆^(𝑛)≍log11−𝑟.(4.2) This implies that 𝑆∈𝐻^{𝛽1/(2𝑛)} for all 𝛽<𝑛, because by (1.11), 𝑀^{1/(2𝑛)1/(2𝑛)}𝑟,𝐷^𝛽𝑆≤≤𝐶𝐶¹⁰₁₀(1−𝑠)^{(𝑛−𝛽)/(2𝑛)−1}𝑀^{1/(2𝑛)1/(2𝑛)}(𝑟𝑠,𝐷^𝑛𝑆)𝑑𝑠(1−𝑠)^{(𝑛−𝛽)/(2𝑛)−1}log^1/(2𝑛)11−𝑟𝑠𝑑𝑠<∞,(4.3) and also 𝑆∈𝐵^1/(2𝑛),_𝛽𝑞̃ for all 𝛽<𝑛 and all 0<𝑞̃<∞, because by (1.14), ¹⁰(1−𝑟)𝑞̃⁻¹𝑀𝑞̃_1/(2𝑛)𝑟,𝐷^1+𝛽𝑆𝑑𝑟≤≤𝐶𝐶¹⁰₁₀(1−𝑟)𝑞̃^{(𝑛−𝛽)−1}𝑀𝑞̃_1/(2𝑛)(𝑟,𝐷^𝑛𝑆)𝑑𝑟(1−𝑟)𝑞̃^{(𝑛−𝛽)−1}log𝑞̃11−𝑟𝑑𝑟<∞.(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 𝑞̃=𝑞/(2𝑛𝑝). In this way, 𝑆∈𝐻^{𝛽1/(2𝑛)}∩𝐵^1/(2𝑛),_𝛽𝑞̃, so by the homogeneity properties of Lemma 3.4 and Proposition 3.11 with 𝑡=2𝑛𝑝≥1, get that 𝑆∈𝐻^{𝑡/(2n)𝛽/𝑡}∩𝐵^{𝑡/(2𝑛),}_𝑡𝛽/𝑞_𝑡̃=𝐻^𝑝𝛼∩𝐵^{𝛼𝑝,𝑞}, as desired.

Remark 4.3. The case 𝑝=∞ would deal with the question of whether the inner functions in 𝐵^0∞,𝑞 are just Blaschke products. From Theorem 2.4, we know that if 𝑞≤2, the inner functions in 𝐵^0∞,𝑞 are finite Blaschke products, while the example considered to prove its part (𝑏) is initially a singular inner function in ∩_𝑞>2𝐵^0∞,𝑞.

Remark 4.4. The case 𝑞=∞ corresponds to the Lipschitz spaces Λ^𝑝,𝛼. By property (𝑃7), Λ^𝑝,𝛼⊆∩{𝐵^{𝛽𝑝,𝑞}∶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 (𝑃1) and the fact that the atomic singular inner function 𝑆 is in 𝐵^{𝛼𝑝,𝑞} 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+𝑛𝑆≍(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 Verbitskiĭ.

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 (a)𝐵∈Λ^𝑝,𝛼=𝐵^𝛼𝑝,∞⇔{2^{−𝑛(1/𝑝−𝛼)}𝜈^𝛼𝑛}∈ℓ^∞⇔{𝑑^{𝑘1/𝑝−𝛼}𝑘^𝛼}∈ℓ^∞,(b)𝐵∈𝐵^{𝛼𝑝,𝑞}⇔{2^{−𝑛(1/𝑝−𝛼)}𝜈^𝛼𝑛}∈ℓ^𝑞⇔{𝑑^{𝑘1/𝑝−𝛼}𝑘^{𝛼−1/𝑞}}∈ℓ^𝑞.

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^{𝛼𝑝,𝑞}⇒{2^{−𝑛(1/𝑝−𝛼)}𝜈^𝛼𝑛}∈ℓ^𝑞 is true for general Blaschke products whenever 1/(2𝑝)<𝛼<1/𝑝 (even for 0<𝑝<1).

Theorem 4.5. Let 0<𝑝,𝛼<∞ with 1/(2𝑝)<𝛼<1/𝑝, and let 0<𝑞≤∞. Assume that 𝐼 is an inner function in 𝐵^{𝛼𝑝,𝑞}. Then (by Proposition J and Remark 4.4) 𝐼 is a Blaschke product 𝐵 and for its sequence of zeros {𝑧_𝑘}, we have (a)If 𝑞=∞, that is, if 𝐵∈Λ^𝑝,𝛼, then {2^{−𝑛(1/𝑝−𝛼)}𝜈^𝛼𝑛}∈ℓ^∞ or, equivalently, {𝑑^{𝑘1/𝑝−𝛼}𝑘^𝛼}∈ℓ^∞.(b)If 𝑞<∞, then {2^{−𝑛(1/𝑝−𝛼)}𝜈^𝛼𝑛}∈ℓ^𝑞 or, equivalently, {𝑑^{𝑘1/𝑝−𝛼}𝑘^{𝛼−1/𝑞}}∈ℓ^𝑞.

Proof. We will make use of the following inequality of Goldberg [40]: ^02𝜋||𝐵𝑟𝑒^𝑖𝜃||𝑑𝜃≤^02𝜋||𝐵𝑟𝑒^𝑖𝜃||𝑑𝜃,0<𝑟<1,(4.6) where 𝐵 is the Blaschke product whose zeros 𝑧_𝑘̂ 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: ^02𝜋1−||𝐵𝑟𝑒^𝑖𝜃||𝑑𝜃=O((1−𝑟)^𝛼𝑝),as𝑟⟶1.(4.7) 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^𝑛𝛼𝑝}∈ℓ^∞, 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/𝑝𝛼𝑝}. Since 𝛼𝑝<1, Theorem C gives ¹⁰(1−𝑟)^{−𝛼𝑞−1}21𝜋^02𝜋1−||𝐵𝑟𝑒^𝑖𝜃||𝑑𝜃^𝑞/𝑝𝑑𝑟<∞,(4.8) which combined with (4.6), gives the same estimate for 𝐵 in place of 𝐵, implying that 𝐵∈𝐵^{1,𝑞/𝑝𝛼𝑝}. So, by Theorem K(b), {2^{−𝑛(1−𝛼𝑝)}𝜈^𝑛𝛼𝑝}∈ℓ^𝑞/𝑝, and this is equivalent to the desired conclusion of Theorem 4.5(b).

Theorem 4.5 has some obvious consequences regarding the spaces 𝐻^𝑝𝛼.

Corollary 4.6. Let 0<𝑝,𝛼<∞ with 1/(2𝑝)<𝛼<1/𝑝. Assume that 𝐼 is an inner function in 𝐻^𝑝𝛼. Then (by Corollary 4.1) 𝐼 is a Blaschke product 𝐵 whose sequence of zeros, {𝑧_𝑘}, satisfies, (a)∑_𝑘𝑑^{𝑘1−𝛼𝑝}𝑘^𝛼𝑝−1<∞ if 1/𝑝−1<𝛼,(b)∑_𝑘𝑑^{𝑘𝑡−𝛼𝑝}𝑘^𝛼𝑝−1<∞ for all 𝑡>1 if 1/(2𝑝)<𝛼≤1/𝑝−1.

Proof. Part (a) follows directly from the fact that 𝐵∈𝐵^{𝛼𝑝,𝑝} by Corollary 4.6. As for part (b), it is because, by Property (𝑃8), 𝐵∈𝐵^{𝛽𝑝,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∈𝐻^𝑝1, 1/2<𝑝<1 implies ∑_𝑘𝑑^{𝑘(1−𝑝)/𝑝}<∞.

Theorem 4.7. Let 0<𝑝,𝛼<∞ with 1/(2𝑝)<𝛼<1/𝑝. Assume that 𝐼 is an inner function in H^𝑝𝛼. Then (by Corollary 4.1) 𝐼 is a Blaschke product 𝐵 whose sequence of zeros, {𝑧_𝑘}, satisfies ∑_𝑘𝑑^{𝑘(1−𝛼𝑝)/(𝛼𝑝)}<∞.

Proof. Since 𝛼>1/𝑝−𝛼, then 𝐵∈𝐻^1𝛼𝑝 by Corollary 3.13. Now, by Theorem 8 of [2], we conclude that ∑^∞𝑘=1𝑑^{𝑘(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.

Given 𝜉∈𝜕𝔻, a Stolz angle with vertex at 𝜉 is one of the following regions Ω_𝜎(𝜉)⊂𝔻, with 𝜎∈[1,∞),Ω_𝜎(𝜉)=𝑧∈𝔻∶|||1−𝜉𝑧|||≤𝜎(1−|𝑧|).(5.1) The Stolz angles with vertex at 𝜉=1 will be simply denoted by Ω_𝜎.

Let us start with the following result.

Theorem L (see [7, Theorem 2.3]). If 𝐵 is a Blaschke product whose zeros lie in a fixed Stolz angle, then 𝐵∈Λ^{𝑝,1/(2𝑝)} for all 𝑝>0.

An improvement of this result for 1/2<𝑝<∞ is given in [15, Theorem 1.6], where it is actually proved that 𝐵∈𝜆^{𝑝,1/(2𝑝)}. Anyhow, Theorem L provides the following result almost immediately.

Theorem 5.1. If 0<𝑝,𝛼<∞ with 𝛼<1/(2𝑝), 0<𝑞≤∞, and 𝐵 is a Blaschke product whose zeros lie in a fixed Stolz angle, then 𝐵∈𝐵^{𝛼𝑝,𝑞}∩𝐻^𝑝𝛼.

Proof. That 𝐵∈𝐵^{𝛼𝑝,𝑞} is a direct application of Theorem L and property (𝑃7). That 𝐵∈𝐻^𝑝𝛼 follows directly from the fact that 𝐵∈𝐵^{𝛼𝑝,𝑝} and Remark 3.12.

This result is quite accurate, as the following one shows.

Theorem 5.2. If 0<𝑝,𝑞<∞, then there exists a Blaschke product 𝐵 whose zeros lie on the radius (0,1) and such that 𝐵∉𝐵^{𝑝,𝑞1/(2𝑝)}. 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∉𝐻^{𝑝1/(2𝑝)}.

Proof. By Corollary 3.6, as 1/𝑝>1/𝑝−1/(2𝑝), any inner function in 𝐵^{𝑝,𝑞1/(2𝑝)} is also in 𝐵^{1,𝑞/𝑝1/2}. 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∉𝐵^{1,𝑞/𝑝1/2}. On the other hand, since all inner functions in 𝐻^{𝑝1/(2𝑝)} are also in 𝐻^11/2 by Corollary 3.13, and since 𝐻^11/2 and 𝐵^1,11/2 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 𝐵^1,11/2.
Now let 𝐵 be the Blaschke product whose sequence of zeros is given by 𝑧_𝑘=1−(𝑘log^𝛽𝑘)⁻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, ^02𝜋1−||𝐵𝑟𝑒^𝑖𝜃||𝑑𝜃≍(1−𝑟)^1/2log^−𝑝/𝑞11−𝑟.(5.2) This implies that ¹⁰(1−𝑟)^{−𝑞/(2𝑝)−1}^02𝜋1−||𝐵𝑟𝑒^𝑖𝜃||𝑑𝜃^𝑞/𝑝𝑑𝑟=∞.(5.3) which, by Theorem C, yields ¹⁰(1−𝑟)^{𝑞/𝑝−1}𝑀^1𝑞/𝑝𝑟,𝐷^1+1/2𝐵𝑑𝑟=∞,(5.4) that is, 𝐵∉𝐵^{1,𝑞/𝑝1/2}. This finishes the proof of the theorem.

Remark 5.3. The just mentioned example of a Blaschke product 𝐵 not in 𝐵^1,11/2 is that one that places its zeros at [1−(𝑘log³𝑘)⁻¹], 𝑘≥2. This Blaschke product does not belong to any 𝐻^{𝑝1/(2𝑝)}, 0<𝑝<∞, neither to any 𝐵^{𝑝,𝑝1/(2𝑝)}. In particular, 𝐵 does not belong to 𝐵^3/2,3/21/3 which, by (1.14), is another way to say that 𝐵′∉𝐴^3/2. We remark that Girela and Peláez [12] proved that the Blaschke 𝐵 with zeros 𝑎_𝑘=[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:
If 𝛽≥2 and 𝐵 is the Blaschke product with zeros 𝑎_𝑘=[1−(𝑘log^𝛽𝑘)⁻¹], 𝑘≥2, then 𝐵^∉𝐴^3/2.
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∈𝐵^{𝛼𝑝,𝑞}⇒{2^{−𝑛(1/𝑝−𝛼)}𝜈^𝛼𝑛}∈ℓ^𝑞”. We have a partial result regarding the other implication.

Theorem 5.4. Let 𝐵 be a Blaschke product with zeros {𝑧_𝑘} in a fixed Stolz angle, and let 0<𝑝,𝑞,𝛼<∞ be such that max{1/𝑝−1,1/(2𝑝)}<𝛼<1/𝑝. If {𝑑^{𝑘1/𝑝−𝛼}𝑘^{𝛼−1/𝑞}}∈ℓ^𝑞 then 𝐵∈𝐵^{𝛼𝑝,𝑞}.

Proof. That {𝑑^{𝑘1/𝑝−𝛼}𝑘^{𝛼−1/𝑞}}∈ℓ^𝑞, it means that ∑_𝑘𝑑^{𝑘𝑞/𝑝−𝛼𝑞}𝑘^𝛼𝑞−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∈𝐵^{1,𝑞/𝑝𝛼𝑝} because 1/2<𝛼𝑝<1. Now, since 𝑝>1−𝛼𝑝 too, Corollary 3.6 can be applied to obtain that 𝐵∈𝐵^{𝛼𝑝,𝑞}.

Using Theorem 9 of [2], we obtain the following result for Hardy-Sobolev spaces.

Theorem 5.5. Let 0<𝑝,𝛼<∞ with max{1/𝑝−1,1/(2𝑝)}<𝛼<1/𝑝. Assume that 𝐵 is a Blaschke product whose sequence of zeros {𝑧_𝑘} lies in a fixed Stolz angle, and satisfies ∑_𝑘𝑑^{𝑘(1−𝛼𝑝)/(𝛼𝑝)}<∞. Then 𝐵∈𝐻𝑝_𝛼̃, for all 0<𝑝̃<𝑝.

Proof. Observe that the condition max{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∈𝐻𝑝_𝛼̃, for all max{1/(2𝛼),1/(1+𝛼)}<𝑝̃<𝑝.
For such 𝑝̃, 1/2<𝛼𝑝̃<𝛼𝑝 and, also, (1−𝛼𝑝)/(𝛼𝑝)<(1−𝛼𝑝)̃/(𝛼𝑝)̃. So, by Theorem 9 of [2], 𝐵′∈𝐻^𝛼𝑝̃, that is, 𝐵∈𝐻^𝛼₁𝑝̃. Now, the fact that 1/(1+𝛼)<𝑝̃ implies that 1/𝛼>1/(𝛼𝑝)̃−1. So, by Corollary 3.13, we conclude that 𝐵∈𝐻𝑝_𝛼̃.

Acknowledgments

Daniel Girela and Cristóbal are partially supported by grants from “el Ministerio de Ciencia e Innovación, Spain" (MTM2007-60854 and “Ingenio Mathematica (i-MATH)” No. CSD2006-00032) and from “La Junta de AndalucÍa", Spain, (FQM210 and P09-FQM4468). Daniel Girela is also supported by the European Networking Programme “HCAA" of the European Science Foundation. Miroljub Jevtić is supported by a grant from MN ON144010, Serbia.