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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 626254, 26 pages
http://dx.doi.org/10.1155/2011/626254
Research Article

Inner Functions in Lipschitz, Besov, and Sobolev Spaces

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

Received 23 February 2011; Accepted 12 April 2011

Academic Editor: WolfgangΒ Ruess

Copyright Β© 2011 Daniel Girela et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spaces 𝐻𝑝𝛼 with 1/𝑝≀𝛼<∞ or any of the Besov spaces 𝐡𝛼𝑝,π‘ž with 0<𝑝,π‘žβ‰€βˆž and 𝛼β‰₯1/𝑝, except when 𝑝=∞, 𝛼=0, and 2<π‘žβ‰€βˆž or when 0<𝑝<∞, π‘ž=∞, and 𝛼=1/𝑝 are finite Blaschke products. Our assertion for the spaces 𝐡0∞,π‘ž, 0<π‘žβ‰€2, follows from the fact that they are included in the space VMOA. We prove also that for 2<π‘ž<∞, VMOA is not contained in 𝐡0∞,π‘ž 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 {πΌβˆ’1(π‘Ž)}, |π‘Ž|<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={π‘§βˆˆβ„‚βˆΆ|𝑧|<1} the open unit disk in the complex plane β„‚, and by Hol(𝔻) the class of analytic functions on 𝔻. The classical Hardy space 𝐻𝑝, 0<π‘β‰€βˆž, consists of those functions π‘“βˆˆHol(𝔻) for whichβ€–π‘“β€–π»π‘βˆΆ=sup0<π‘Ÿ<1𝑀𝑝(π‘Ÿ,𝑓)<∞,(1.1) where π‘€π‘βˆ«(π‘Ÿ,𝑓)=((1/2πœ‹)02πœ‹|𝑓(π‘Ÿπ‘’π‘–πœƒ)|π‘π‘‘πœƒ)1/𝑝 if 0<𝑝<∞, and π‘€βˆž(π‘Ÿ,𝑓)=max|𝑧|=π‘Ÿ|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(πœ•π”»). The sequence {π‘§π‘˜} of zeros of an 𝐻𝑝 function, counting multiplicities, satisfies the so-called Blaschke condition: βˆ‘π‘˜(1βˆ’|π‘§π‘˜|)<∞. This condition characterizes the zero sequences of 𝐻𝑝 functions. By writing π‘π‘Ž(𝑧)=(|π‘Ž|/π‘Ž)((π‘Žβˆ’π‘§)/(1βˆ’π‘Žπ‘§)) when π‘Žβˆˆπ”»,π‘Žβ‰ 0, and 𝑏0(𝑧)=𝑧, the Blaschke condition implies that the product βˆπ‘˜π‘π‘§π‘˜(𝑧) converges absolutely and uniformly in each compact subset of 𝔻, hence defining a function 𝐡(𝑧), called the Blaschke product associated to the sequence {π‘§π‘˜}, which is analytic in the unit disk 𝔻, and whose exact sequence of zeros, counting multiplicities, is {π‘§π‘˜}. Also, β€–π΅β€–π»βˆžβ‰€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)=𝑒𝑖𝛾𝐡(𝑧)𝑆(𝑧), where 𝛾 is a real constant, 𝐡 is a Blaschke product (carrying all the zeros of 𝐼), and 𝑆 is what it is called a singular inner function, having the formξ‚΅βˆ’ξ€œπ‘†(𝑧)=exp02πœ‹π‘’π‘–πœƒ+π‘§π‘’π‘–πœƒξ‚Άβˆ’π‘§π‘‘πœ‡(πœƒ),(1.2) where ΞΌ is a positive Borel measure on [0,2πœ‹), 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 𝐢. Their dependence on other quantities, if specified, will appear as subindexes. In the same expression, the constant 𝐢 may change from one occurrence to the other. Two quantities or expressions, 𝐴 and 𝐡, are said to be comparable (written 𝐴≍𝐡) if there exists a positive constant 𝐢 such that πΆβˆ’1𝐡≀𝐴≀𝐢𝐡. 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 𝐴𝑝,𝛼, 0<𝑝<∞, βˆ’1<𝛼<∞ consists of those functions π‘“βˆˆHol(𝔻) such that‖𝑓‖𝐴𝑝,π›Όξ‚΅βˆΆ=𝛼+1πœ‹ξ€œπ”»||||𝑓(𝑧)𝑝1βˆ’|𝑧|2𝛼𝑑𝐴(𝑧)1/𝑝<∞,(1.3) where 𝑑𝐴(𝑧)=𝑑π‘₯𝑑𝑦 denotes the Lebesgue area measure in 𝔻. We mention the books [21, 22] as general references for the theory of Bergman spaces.

If βˆ‘π‘“(𝑧)=βˆžπ‘˜=0𝑓(π‘˜)π‘§π‘˜ is analytic in 𝔻 and 𝛼>0, the fractional derivative of order 𝛼 of 𝑓, 𝐷𝛼𝑓 is defined as𝐷𝛼𝑓(𝑧)=βˆžξ“π‘˜=0(π‘˜+1)𝛼𝑓(π‘˜)π‘§π‘˜.(1.4) (This definition also makes sense for 𝛼≀0, only that the name β€œderivative” would look a bit awkward.) For a positive integer 𝑛, it is actually an β€œπ‘›-th derivative”: 𝐷𝑛𝑓(𝑧)=((𝑑/𝑑𝑧)𝑧)𝑛𝑓(𝑧). Besides, the integral means 𝑀𝑝(π‘Ÿ,𝐷𝑛𝑓) and 𝑀𝑝(π‘Ÿ,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,1𝑓(𝑧)=ξ€œΞ“(𝛼)10𝐷𝛼𝑓(𝑠𝑧)logπ›Όβˆ’11𝑠𝑑𝑠.(1.5) This yields the estimate||||ξ€œπ‘“(𝑧)≀𝐢10||𝐷𝛼||𝑓(𝑠𝑧)(1βˆ’π‘ )π›Όβˆ’1𝑑𝑠.(1.6)

If 0<π‘β‰€βˆž and 0≀𝛼<∞, then a function π‘“βˆˆHol(𝔻) is said to belong to the Lipschitz space Λ𝑝,𝛼 if‖𝑓‖Λ𝑝,π›ΌβˆΆ=sup0<π‘Ÿ<1(1βˆ’π‘Ÿ)π‘€π‘ξ€·π‘Ÿ,𝐷1+𝛼𝑓<∞.(1.7) The subspace πœ†π‘,𝛼 consists of those π‘“βˆˆHol(𝔻) for whichlimπ‘Ÿβ†’1(1βˆ’π‘Ÿ)π‘€π‘ξ€·π‘Ÿ,𝐷1+𝛼𝑓=0.(1.8) Notice that Ξ›βˆž,0 is another name for the usual Bloch space ℬ, of functions π‘“βˆˆHol(𝔻) such that supπ‘§βˆˆπ”»(1βˆ’|𝑧|2)|π‘“ξ…ž(𝑧)|<∞. Analogously, πœ†βˆž,0 is the little Bloch space ℬ0, of functions π‘“βˆˆHol(𝔻) such that lim|𝑧|β†’1(1βˆ’|𝑧|2)|π‘“ξ…ž(𝑧)|=0.

Replacing the sup-norm with an πΏπ‘ž-norm in the above definition gives way to the Besov space 𝐡𝛼𝑝,π‘ž, 0<π‘β‰€βˆž, 0<π‘ž<∞, 0≀𝛼<∞, consisting of those functions π‘“βˆˆHol(𝔻) for which‖𝑓‖𝐡𝛼𝑝,π‘žξ‚΅ξ€œβˆΆ=10(1βˆ’π‘Ÿ)π‘žβˆ’1π‘€π‘žπ‘ξ€·π‘Ÿ,𝐷1+π›Όπ‘“ξ€Έξ‚Άπ‘‘π‘Ÿ1/π‘ž<∞.(1.9) The space obtained when π‘ž=∞ is precisely 𝐡𝛼𝑝,βˆžβ‰‘Ξ›π‘,𝛼. Other classes treated in this paper are the Hardy-Sobolev spaces 𝐻𝑝𝛼, 0<π‘β‰€βˆž, 0≀𝛼<∞, consisting of those functions π‘“βˆˆHol(𝔻) for whichβ€–π‘“β€–π»π‘π›ΌβˆΆ=‖𝐷𝛼𝑓‖𝐻𝑝<∞.(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 givesπ‘€π‘π‘ξ€œ(π‘Ÿ,𝑓)≀𝐢10(1βˆ’π‘ )π‘π›Όβˆ’1𝑀𝑝𝑝(π‘Ÿπ‘ ,𝐷𝛼𝑓)𝑑𝑠,(0<𝑝<1,0<𝛼),(1.11)π‘€π‘ξ€œ(π‘Ÿ,𝑓)≀𝐢10(1βˆ’π‘ )π›Όβˆ’1𝑀𝑝(π‘Ÿπ‘ ,𝐷𝛼𝑓)𝑑𝑠,(1β‰€π‘β‰€βˆž,0<𝛼).(1.12) The same estimate (1.6) combined with variants of Hardy's inequality givessup0<πœŒβ‰€π‘Ÿ(1βˆ’πœŒ)π›Όπ‘€π‘ξ€·πœŒ,𝐷𝛽𝑓≍sup0<πœŒβ‰€π‘Ÿ(1βˆ’πœŒ)π›Όβˆ’π›½+𝛾𝑀𝑝(𝜌,π·π›Ύξ€œπ‘“),(0<π‘β‰€βˆž,𝛼>max{0,π›½βˆ’π›Ύ}),(1.13)10(1βˆ’π‘Ÿ)π›Όβˆ’1π‘€π‘žπ‘ξ€·π‘Ÿ,π·π›½π‘“ξ€Έξ€œπ‘‘π‘Ÿβ‰10(1βˆ’π‘Ÿ)π›Όβˆ’π‘ž(π›½βˆ’π›Ύ)βˆ’1π‘€π‘žπ‘(π‘Ÿ,𝐷𝛾𝑓)π‘‘π‘Ÿ,(0<π‘ž<∞,0<π‘β‰€βˆž,𝛼>max{0,π‘ž(π›½βˆ’π›Ύ)}).(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]):π‘€π‘ž(π‘Ÿ,𝑓)≀𝐢𝑝,π‘ž(1βˆ’π‘Ÿ)1/π‘žβˆ’1/𝑝𝑀𝑝1+π‘Ÿ2,𝑓,(0<𝑝<π‘žβ‰€βˆž).(1.15)

Finally, we provide also the following estimates, due to Littlewood and Paley [25, Theorems 5 and 6], for 𝑝β‰₯1, and to Vinogradov [26, Lemma 1.4], for 0<𝑝<1.sup0β‰€π‘Ÿ<1𝑀𝑝(π‘Ÿ,𝑓)β‰€πΆπ‘ξ€œ10(1βˆ’π‘Ÿ)π‘βˆ’1π‘€π‘π‘ξ€·π‘Ÿ,𝐷1π‘“ξ€Έπ‘‘π‘Ÿ,(0<𝑝≀2),(1.16)ξ€œ10(1βˆ’π‘Ÿ)π‘βˆ’1π‘€π‘π‘ξ€·π‘Ÿ,𝐷1π‘“ξ€Έπ‘‘π‘Ÿβ‰€πΆπ‘sup0β‰€π‘Ÿ<1𝑀𝑝(π‘Ÿ,𝑓),(𝑝β‰₯2),(1.17) which can be restated as 𝐡0𝑝,π‘βŠ†π»π‘, for 0<𝑝≀2, and π»π‘βŠ†π΅0𝑝,𝑝, for 𝑝β‰₯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 π‘ž<∞, 𝐡𝛼𝑝,π‘žβŠ†πœ†π‘,𝛼: If π‘“βˆˆπ΅π›Όπ‘,π‘ž, then ∫1π‘Ÿ(1βˆ’πœŒ)π‘žβˆ’1π‘€π‘žπ‘(𝜌,𝐷1+𝛼𝑓)π‘‘πœŒβ†’0 as π‘Ÿβ†’1, so by using the increasing behavior of the means 𝑀𝑝(π‘Ÿ,𝐷1+𝛼𝑓), obtain that (1βˆ’π‘Ÿ)𝑀𝑝(π‘Ÿ,𝐷1+𝛼𝑓)β†’0, as π‘Ÿβ†’1, that is, that π‘“βˆˆπœ†π‘,𝛼.(𝑃2)If π‘ž1β‰€π‘ž2, 𝐡𝑝,π‘ž1π›ΌβŠ†π΅π‘,π‘ž2𝛼, because (1βˆ’π‘Ÿ)𝑀𝑝(π‘Ÿ,𝐷1+𝛼𝑓)β†’0, as π‘Ÿβ†’1, for all π‘“βˆˆπ΅π‘,π‘ž1𝛼.(𝑃3)If 𝑝1≀𝑝2, 𝐡𝑝2𝛼,π‘žβŠ†π΅π‘1𝛼,π‘ž, and 𝐻𝑝2π›ΌβŠ†π»π‘1𝛼, because 𝑀𝑝1(π‘Ÿ,𝑔)≀𝑀𝑝2(π‘Ÿ,𝑔) for all π‘Ÿβˆˆ(0,1) and all π‘”βˆˆHol(𝔻).(𝑃4)If 𝛼1≀𝛼2, 𝐡𝛼𝑝,π‘ž2βŠ†π΅π›Όπ‘,π‘ž1. Indeed, for π‘“βˆˆπ΅π›Όπ‘,π‘ž2, (1.14) gives β€–π‘“β€–π‘žπ΅π›Ό1𝑝,π‘žβ‰ξ€œ10(1βˆ’π‘Ÿ)π‘žβˆ’1+π‘ž(𝛼2βˆ’π›Ό1)π‘€π‘žπ‘ξ€·π‘Ÿ,𝐷1+𝛼2π‘“ξ€Έπ‘‘π‘Ÿ<β€–π‘“β€–π‘žπ΅π›Ό2𝑝,π‘ž<∞.(1.18)(𝑃5)If 𝛼1≀𝛼2, 𝐻𝑝𝛼2βŠ†π»π‘π›Ό1, because of (1.11) and (1.12). (𝑃6)If 0<𝑝1≀𝑝2, 𝐡𝑝1,π‘ž1/𝑝1βŠ†π΅π‘2,π‘ž1/𝑝2. (When π‘ž=∞, this says that Λ𝑝,1/𝑝 increases with 𝑝). A proof of this lies in an application of (1.15) and (1.14),β€–π‘“β€–π‘žπ΅π‘22,π‘ž1/π‘β‰ξ€œ10(1βˆ’π‘Ÿ)π‘žβˆ’1+π‘ž(1/𝑝1βˆ’1/𝑝2)π‘€π‘žπ‘2ξ€·π‘Ÿ,𝐷1+1/𝑝1π‘“ξ€Έπ‘‘π‘Ÿβ‰€πΆβ€–π‘“β€–π‘žπ΅π‘11,π‘ž1/𝑝.(1.19)(𝑃7)Λ𝑝,π›ΌβŠ†βˆ©{𝐡𝛽𝑝,π‘žβˆΆπ›½<𝛼,0<π‘ž}. This is again an application of (1.14), β€–π‘“β€–π‘žπ΅π›½π‘,π‘žβ‰ξ€œ10(1βˆ’π‘Ÿ)π‘ž(π›Όβˆ’π›½)βˆ’1+π‘žπ‘€π‘žπ‘ξ€·π‘Ÿ,𝐷1+π›Όπ‘“ξ€Έπ‘‘π‘Ÿβ‰€πΆβ€–π‘“β€–Ξ›π‘,𝛼,(1.20) where ∫𝐢=10(1βˆ’π‘Ÿ)π‘ž(π›Όβˆ’π›½)βˆ’1π‘‘π‘Ÿ<∞. (𝑃8)π»π‘π›ΌβŠ†βˆ©{𝐡𝛽𝑝,π‘žβˆΆπ›½<𝛼,0<π‘ž}. Again an application of (1.14). (𝑃9)If 0<𝑝≀2, 𝐡𝛼𝑝,π‘βŠ†π»π‘π›Ό. This is (1.16). (𝑃10)If 2≀𝑝, π»π‘π›ΌβŠ†π΅π›Όπ‘,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 π‘Žβˆˆπ”», its Frostman shift πΌπ‘Ž is defined asπΌπ‘Ž(𝑧)=𝐼(𝑧)βˆ’π‘Ž1βˆ’π‘ŽπΌ(𝑧),π‘§βˆˆπ”».(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 πΌπ‘Ž are Blaschke products for all π‘Žβˆˆπ”» 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 π‘Žβˆˆπ”»β§΅πΈ, with 𝐸 a set of logarithmic capacity zero, the Frostman shift πΌπ‘Ž 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,π‘žβ‰€βˆž, 0≀𝛼, 0<𝛿<1/2, and π‘ŽβˆˆπΎπ›Ώ={π‘§βˆˆπ”»βˆΆπ›Ώ<|𝑧|<1βˆ’π›Ώ}. Then β€–πΌπ‘Žβ€–π΅π›Όπ‘,π‘žβ‰β€–πΌβ€–π΅π›Όπ‘,π‘ž, with constants depending only on 𝑝,π‘ž,𝛼, and 𝛿, but not on 𝐼 or π‘Ž. In the case π‘ž=∞, the formulation is sup0β‰€πœŒ<π‘Ÿ(1βˆ’πœŒ)π‘€π‘ξ€·πœŒ,𝐷1+π›ΌπΌπ‘Žξ€Έβ‰sup0β‰€πœŒ<π‘Ÿ(1βˆ’πœŒ)π‘€π‘ξ€·πœŒ,𝐷1+𝛼𝐼.(1.22)

Proof. Put 1+𝛼=𝑛+𝛽, with 𝑛 a positive integer and π›½βˆˆ[0,1). We shall only consider the case π‘ž<∞. The procedure for π‘ž=∞ is rather similar. Also, it is enough to estimate the 𝐡𝛼𝑝,π‘ž-norm of πΌπ‘Ž in terms of that of 𝐼, for 𝐼=(πΌπ‘Ž)βˆ’π‘Ž.
Writing πœ“π‘Ž(𝑧)=π‘Ž/(1βˆ’π‘Žπ‘§), π‘§βˆˆπ”» observe that the first derivative of πΌπ‘Ž can be written as πΌξ…žπ‘Ž=((1βˆ’|π‘Ž|2)/π‘Ž2)(πœ“π‘Žβˆ˜πΌ)ξ…ž, and, in general, for a positive integer 𝑛, the 𝑛th derivative is given by πΌπ‘Ž(𝑛)=((1βˆ’|π‘Ž|2)/π‘Ž2)(πœ“π‘Žβˆ˜πΌ)(𝑛). This, together with the FaΓ  di Bruno's formula for the 𝑛th derivative of a composition, gives πΌπ‘Ž(𝑛)=1βˆ’|π‘Ž|2π‘Ž2ξ€·πœ“π‘Žξ€Έβˆ˜πΌ(𝑛)=1βˆ’|π‘Ž|2π‘Ž2𝑛!π‘˜1!β‹―π‘˜π‘›!πœ“π‘Ž(π‘˜)βˆ˜πΌπ‘›ξ‘π‘—=1𝐼(𝑗)𝑗!π‘˜π‘—,(1.23) where βˆ‘π‘˜=𝑛𝑗=1π‘˜π‘—, and the sum runs over all 𝑛-tuples βƒ—π‘˜=(π‘˜1,…,π‘˜π‘›) of nonnegative integers such that βˆ‘π‘›π‘—=1π‘—π‘˜π‘—=𝑛. Observe that if π‘ŽβˆˆπΎπ›Ώ, the quantities (1βˆ’|π‘Ž|2)/|π‘Ž|2 and |πœ“(k)π‘Žβˆ˜πΌ|=|πœ“βˆ˜πΌ|π‘˜+1 are bounded away from 0 and ∞ by constants depending only on π‘˜ and 𝛿, but not on π‘Ž.
Now, to estimate β€–πΌπ‘Žβ€–π‘žπ΅π›Όπ‘,π‘ž, we use, in order, (1.14), (1.23), twice HΓΆlder's inequality (HΓΆ) with indices {𝑛/(π‘—π‘˜π‘—)βˆΆπ‘˜π‘—β‰ 0}, again (1.14), and finally we appeal to the fact (I) that |𝐼(𝑛)(𝑧)|≀𝐢𝑛(1βˆ’|𝑧|)βˆ’π‘› (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]), β€–β€–πΌπ‘Žβ€–β€–π‘žπ΅π›Όπ‘,π‘ž=ξ€œ10(1βˆ’π‘Ÿ)π‘žβˆ’1π‘€π‘žπ‘ξ€·π‘Ÿ,𝐷𝑛+π›½πΌπ‘Žξ€Έπ‘‘π‘Ÿ(1.14)ξ€œβ‰€πΆ10(1βˆ’π‘Ÿ)π‘ž(1βˆ’π›½)βˆ’1π‘€π‘žπ‘ξ‚€π‘Ÿ,πΌπ‘Ž(𝑛)ξ‚π‘‘π‘Ÿ(1.23)ξ“β‰€πΆβƒ—π‘˜ξ€œ10(1βˆ’π‘Ÿ)π‘ž(1βˆ’π›½)βˆ’1π‘€π‘žπ‘ξƒ©π‘Ÿ,𝑛𝑗=1𝐼(𝑗)ξ€Έπ‘˜π‘—ξƒͺπ‘‘π‘Ÿ(HΜˆβ€Œo)ξ“β‰€πΆβƒ—π‘˜ξ€œ10(1βˆ’π‘Ÿ)π‘›π‘ž(1βˆ’π›½)βˆ’1𝑗=1π‘€π‘žπ‘˜π‘—π‘π‘›/π‘—ξ€·π‘Ÿ,𝐼(𝑗)ξ€Έπ‘‘π‘Ÿ(HΜˆβ€Œo)ξ“β‰€πΆβƒ—π‘˜π‘›ξ‘π‘—=1ξ‚΅ξ€œ10(1βˆ’π‘Ÿ)π‘ž(1βˆ’π›½)βˆ’1π‘€π‘žπ‘›/𝑗𝑝𝑛/π‘—ξ€·π‘Ÿ,𝐼(𝑗)ξ€Έξ‚Άπ‘‘π‘Ÿπ‘—π‘˜π‘—/𝑛(1.14)ξ“β‰€πΆβƒ—π‘˜π‘›ξ‘π‘—=1ξ‚΅ξ€œ10(1βˆ’π‘Ÿ)π‘ž(1βˆ’π›½)βˆ’1+(π‘žπ‘›(π‘›βˆ’π‘—))/π‘—π‘€π‘žπ‘›/𝑗𝑝𝑛/π‘—ξ€·π‘Ÿ,𝐼(𝑛)ξ€Έξ‚Άπ‘‘π‘Ÿπ‘—π‘˜π‘—(/𝑛I)ξ“β‰€πΆβƒ—π‘˜π‘›ξ‘π‘—=1ξ‚΅ξ€œ10(1βˆ’π‘Ÿ)π‘ž(1βˆ’π›½)βˆ’1π‘€π‘žπ‘ξ€·π‘Ÿ,𝐼(𝑛)ξ€Έξ‚Άπ‘‘π‘Ÿπ‘—π‘˜π‘—/𝑛(1.14)β‰€πΆβ€–πΌβ€–π‘žπ΅π›Όπ‘,π‘ž.(1.24)

2. Inner Functions in the Spaces 𝐡𝛼𝑝,π‘ž and 𝐻𝑝𝛼, 𝛼β‰₯1/𝑝

Ahern and JevtiΔ‡ [5] proved that a Blaschke product lies in the space 𝐡𝑝,∞1/𝑝≑Λ𝑝,1/𝑝 (0<𝑝<∞) if and only if its sequence of zeros is a finite union of exponential sequences, (see also VerbitskiΔ­ [27] for the case 1≀𝑝<∞). We refer the reader to the recent work of JevtiΔ‡ [15] on this subject where references to previous works are given. In particular, we havefor0<𝑝<∞,thespace𝐡𝑝,∞1/𝑝containsinfiniteBlaschkeproducts.(2.1) Our results in this section imply that the opposite is true for all the spaces 𝐡𝛼𝑝,π‘ž with 0<𝑝,π‘žβ‰€βˆž and 𝛼β‰₯1/𝑝, except for the mentioned case, 0<𝑝<∞, π‘ž=∞ and 𝛼=1/𝑝, and when 𝑝=∞, 𝛼=0 and 2<π‘žβ‰€βˆž.

Theorem 2.1. (a) Let 0<𝑝,π‘žβ‰€βˆž and 𝛼>1/𝑝. Then the only inner functions in 𝐡𝛼𝑝,π‘ž are finite Blaschke products.
(b) If 0<𝑝,π‘ž<∞ then the only inner functions in 𝐡𝑝,π‘ž1/𝑝 are finite Blaschke products.

Proof. To prove (a) observe that 𝐡𝛼𝑝,π‘žβŠ†π΅π›Όπ‘,βˆžβ‰‘Ξ›π‘,π›ΌβŠ†Ξ›βˆž,π›Όβˆ’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 Ξ›βˆž,𝛽, with 𝛽>0, (even if it is forced to be a constant), belongs to the disk algebra π’œ (that is, it admits a continuous extension to 𝔻). Thus if we are in the conditions of part (π‘Ž) and 𝐼 is an inner function in 𝐡𝛼𝑝,π‘ž then πΌβˆˆπ’œ. Then it follows easily that 𝐼 is a finite Blaschke product. Indeed, write 𝐼(𝑧)=𝑆(𝑧)𝐡(𝑧), where 𝐡 is a Blaschke product and 𝑆 is a singular inner function. The fact that πΌβˆˆπ’œ readily implies that 𝐡 is a finite Blaschke product and then it follows that 𝑆 also belongs to π’œ. Then 𝑆 is a function in the disk algebra without zeros and with |𝑆(πœ‰)|=1, for all πœ‰βˆˆπœ•π”». 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 Hp1/p, then B is a finite Blaschke product.Theorem B (see [5, Theorem 3.2]). For each 0<𝑝<∞ there exists πœ€π‘>0 such that if 𝐡 is a Blaschke product and limsupπ‘Ÿβ†’1(1βˆ’π‘Ÿ)π‘€π‘ξ€·π‘Ÿ,𝐷1+1/𝑝𝐡<πœ€π‘,(2.2) then 𝐡 is a finite Blaschke product. In particular, the only Blaschke products in πœ†π‘,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 𝐻𝑝1/𝑝 or πœ†π‘,1/𝑝 are the finite Blaschke products.Now Theorem 2.1(b) follows from this result and the fact that 𝐡𝑝,π‘ž1/π‘βŠ†πœ†π‘,1/𝑝.

The fact that π»π‘π›ΌβŠ†βˆ©{𝐡𝛽𝑝,π‘žβˆΆπ›½<𝛼,0<π‘ž} immediately implies the following.

Corollary 2.3. Let 0<π‘β‰€βˆž and 𝛼>1/𝑝. If π‘“βˆˆπ»π‘π›Ό then 𝑓 admits a continuous extension to 𝔻. Consequently, the only inner functions in 𝐻𝑝𝛼, with 𝛼>1/𝑝, are finite Blaschke products.

It remains to consider the case 𝑝=∞ and 𝛼=0. Of course, 𝐻∞0β‰‘π»βˆž contains the whole class of inner functions.

Let us deal now with the spaces 𝐡0∞,π‘ž. First of all, 𝐡0∞,∞ is the Bloch space, and, hence, it contains all inner functions.

Bishop [29] proved that the little Bloch space, πœ†βˆž,0, contains infinite Blaschke products. Since, for π‘ž<∞, 𝐡0∞,π‘ž is a subspace of πœ†βˆž,0, the natural question rises as whether 𝐡0∞,π‘ž contains or not infinite Blaschke products. The answer depends on the result of intersecting 𝐡0∞,π‘ž with the subspace 𝑉𝑀𝑂𝐴 of πœ†βˆž,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 π‘“βˆˆπ»1 is said to belong to 𝑉𝑀𝑂𝐴 iflim|π‘Ž|β†’1β€–β€–π‘“βˆ˜πœ‘π‘Žβ€–β€–βˆ’π‘“(π‘Ž)𝐻𝑝=0,(2.3) for some (or, equivalently, for all) finite positive 𝑝. Here, πœ‘π‘Ž(𝑧)=(π‘Žβˆ’π‘§)/(1βˆ’π‘Žπ‘§) is the typical involutive automorphism of 𝔻 interchanging the points 0 and π‘Žβˆˆπ”». 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 𝐡𝑀𝑂𝐴 and 𝑉𝑀𝑂𝐴.) In the following result, we use another characterization of 𝑉𝑀𝑂𝐴; it is the space of functions π‘“βˆˆHol(𝔻) such that1||𝐽||ξ€œπ‘†(𝐽)ξ€·1βˆ’|𝑧|2ξ€Έ||π‘“ξ…ž(||||𝐽||𝑧)𝑑𝐴(𝑧)⟢0,as⟢0,(2.4) where 𝐽 is an interval in πœ•π”», |𝐽| is its length, and 𝑆(𝐽) is the Carleson square defined by 𝑆(𝐽)={π‘Ÿπ‘’π‘–πœƒβˆΆπ‘’π‘–πœƒβˆˆπ½,1βˆ’|𝐽|β‰€π‘Ÿ<1}.

Theorem 2.4. (a), If 0<π‘žβ‰€2, then 𝐡0∞,π‘žβŠ†π‘‰π‘€π‘‚π΄. Consequently, any inner function in 𝐡0∞,π‘ž is a finite Blaschke product.
(b) There are infinite Blaschke products in ∩2<π‘ž<∞𝐡0∞,π‘ž.

Proof. To prove (a) observe that, since 𝐡∞,π‘ž10βŠ†π΅βˆž,q20 for 0<π‘ž1<π‘ž2<∞, it suffices to settle the result for π‘ž=2.
Thus, take π‘“βˆˆπ΅0∞,2 and take an interval 𝐽 in πœ•π”» with |𝐽|<1/2, then 1||𝐽||ξ€œπ‘†(𝐽)ξ€·1βˆ’|𝑧|2ξ€Έ||π‘“ξ…ž||(𝑧)21𝑑𝐴(𝑧)≀2||𝐽||ξ€œ1||𝐽||1βˆ’ξ€œ(1βˆ’π‘Ÿ)𝐽||π‘“ξ…žξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||2ξ€œπ‘‘πœƒπ‘‘π‘Ÿβ‰€21||𝐽||1βˆ’(1βˆ’π‘Ÿ)𝑀2βˆžξ€·π‘Ÿ,π‘“ξ…žξ€Έπ‘‘π‘Ÿ,(2.5) and observe that, since π‘“βˆˆπ΅0∞,2, the right hand side tends to 0 as |𝐽|β†’0.
To prove (b), observe that, by Theorem 5.2 of [33], there is a (singular) inner function 𝐼 such that ξ€·1βˆ’|𝑧|2ξ€Έ||πΌξ…ž||(𝑧)||||1βˆ’πΌ(𝑧)2≀logβˆ’1/2𝑒1βˆ’|𝑧|.(2.6) This implies that 𝐼∈𝐡0∞,π‘ž for all π‘ž>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 π‘Žβˆˆπ”» such that the Frostman shift πΌπ‘Ž is an infinite Blaschke product (actually, this is true for all π‘Žβˆˆπ”» except for those in a set of zero logarithmic capacity). Now, Lemma 1.1 shows that πΌπ‘Ž is an infinite Blaschke product in ∩2<π‘ž<∞𝐡0∞,π‘ž.

Once Theorem 2.4 is proved, it is natural to ask whether or not the inclusion π‘‰π‘€π‘‚π΄βŠ†π΅0∞,π‘ž holds for 2<π‘ž<∞. An argument based on duality shows that this is not so.

Theorem 2.5. If 2<π‘ž<∞, then the class 𝑉𝑀𝑂𝐴⧡𝐡0∞,π‘ž is nonempty.

Proof. Observe that the dual of 𝑉𝑀𝑂𝐴 is 𝐻1 under the usual pairing: βŸ¨π‘“,π‘”βŸ©=limπ‘Ÿβ†’1βˆ‘π‘˜ξπ‘“(π‘˜)̂𝑔(π‘˜)π‘Ÿπ‘˜, (see [32, 34]). Also, using the same techniques as in [4], we get that the dual of 𝐡0∞,π‘ž is 𝐡1,π‘žβ€²0, 1/π‘ž+1/π‘žξ…ž=1, under the same pairing as before. Thus, the problem reduces to show that 𝐡1,π‘žβ€²0⧡𝐻1 is nonempty.
It is shown in Theorem 3 of [35] that the function 𝑓(𝑧)=((1βˆ’π‘§)log(2𝑒/(1βˆ’π‘§)))βˆ’1, π‘§βˆˆπ”» is univalent in 𝔻 and π‘“βˆ‰π»1. Also, an argument given in [24, page 61] shows that there exist 𝑐>0 and π‘Ÿ0∈(0,1) such that𝑀1ξ€·π‘Ÿ,π‘“ξ…žξ€Έβ‰€π‘(1βˆ’π‘Ÿ)(log(2𝑒/(1βˆ’π‘Ÿ))),π‘Ÿ0<π‘Ÿ<1.(2.7) It then follows that π‘“βˆˆπ΅1,π‘žβ€²0, whenever 1<π‘žβ€²<∞. This finishes the proof.

3. The Case Max {0,1/π‘βˆ’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 {πΌβˆ’1(π‘Ž)}, π‘Žβˆˆπ”». We start introducing certain counting functions.

If 𝐼 is an inner function and π‘Žβˆˆπ”», denote by {π‘§π‘˜(π‘Ž)} the exact sequence of zeros, multiplicities included, of πΌπ‘Ž, placed in increasing modulus as the subindex π‘˜ increases (in other words, {π‘§π‘˜(π‘Ž)} is the ordered sequence of preimages of π‘Ž). Writing π‘‘π‘˜(π‘Ž)=1βˆ’|π‘§π‘˜(π‘Ž)|, the distribution of zeros in each annulus may be studied with the sequences {π‘˜π‘›(π‘Ž)}βˆžπ‘›=0 and {πœˆπ‘›(π‘Ž)}βˆžπ‘›=0:π‘˜π‘›ξ€½(π‘Ž)=Cardπ‘˜βˆΆ2βˆ’π‘›<π‘‘π‘˜ξ€Ύξ€½(π‘Ž)=maxπ‘˜βˆΆ2βˆ’π‘›<π‘‘π‘˜ξ€Ύ,𝜈(π‘Ž)𝑛(π‘Ž)=Cardπ‘˜βˆΆ2βˆ’π‘›βˆ’1<π‘‘π‘˜(π‘Ž)≀2βˆ’π‘›ξ€Ύ=π‘˜π‘›+1(π‘Ž)βˆ’π‘˜π‘›(π‘Ž).(3.1) Observe that π‘˜0(π‘Ž)=0 always. When π‘Ž=0, just write {π‘§π‘˜}, {π‘‘π‘˜}, {π‘˜π‘›}, and {πœˆπ‘›}. 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){2βˆ’π‘›π›Όπœˆπ›½π‘›(π‘Ž)}βˆˆβ„“βˆž if and only {π‘‘π›Όπ‘˜(π‘Ž)π‘˜π›½}βˆˆβ„“βˆž, and, in either case, their β„“βˆž-norms are comparable.(b)βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπœˆπ›½π‘›βˆ‘(π‘Ž)β‰π‘˜β‰₯1π‘‘π›Όπ‘˜(π‘Ž)π‘˜π›½βˆ’1.

Proof. In order to keep up with readability, it is better to omit the letter value π‘Ž in what follows, that is, assume π‘Ž=0.
To prove (a), assume first that π‘‘π›Όπ‘˜π‘˜π›½β‰€πΆ for all π‘˜, then for each 𝑛=0,1…, 2βˆ’π‘›π›Όπœˆπ›½π‘›=2βˆ’π‘›π›Όξ€·π‘˜π‘›+1βˆ’π‘˜π‘›ξ€Έπ›½β‰€2π›Όπ‘‘π›Όπ‘˜π‘›+1π‘˜π›½π‘›+1≀𝐢.(3.2) In the other direction, assume that 2βˆ’π‘›π›Όπœˆπ›½π‘›β‰€πΆ for all 𝑛. Given π‘˜ find the unique 𝑛=𝑛(π‘˜) such that 2βˆ’π‘›βˆ’1<π‘‘π‘˜β‰€2βˆ’π‘›. This implies that π‘˜β‰€π‘˜π‘›+1, and thus, π‘‘π›Όπ‘˜π‘˜π›½β‰€2βˆ’π‘›π›Όπ‘˜π›½π‘›+1=𝑛𝑗=02βˆ’π›Όπ‘—/π›½πœˆπ‘—2βˆ’π›Ό(π‘›βˆ’π‘—)/𝛽ξƒͺ𝛽,≀𝐢𝑛𝑗=02βˆ’π›Ό(π‘›βˆ’π‘—)/𝛽ξƒͺ𝛽≀𝐢1βˆ’2βˆ’π›Ό/π›½ξ€Έβˆ’π›½.(3.3)
To prove (b), assume first that βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπœˆπ›½π‘›<∞. In the case 𝛽≀1, use an easy integral estimate and the fact that π‘˜π›½π‘›+1βˆ’π‘˜π›½π‘›β‰€(π‘˜π‘›+1βˆ’π‘˜π‘›)𝛽, to obtain the desired result, ξ“π‘˜β‰₯1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1=ξ“π‘˜π‘›β‰₯0𝑛+1ξ“π‘˜=π‘˜π‘›+1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1≀𝑛β‰₯02π‘˜βˆ’π‘›π›Όπ‘›+1ξ“π‘˜=π‘˜π‘›+1π‘˜π›½βˆ’1≀𝑛β‰₯02βˆ’π‘›π›Όξ€œπ‘˜π‘›+1π‘˜π‘›π‘₯π›½βˆ’11𝑑π‘₯=𝛽𝑛β‰₯02βˆ’π‘›π›Όξ‚€π‘˜π›½π‘›+1βˆ’π‘˜π›½π‘›ξ‚β‰€1𝛽𝑛β‰₯02βˆ’π‘›π›Όξ€·π‘˜π‘›+1βˆ’π‘˜π‘›ξ€Έπ›½=1𝛽𝑛β‰₯02βˆ’π‘›π›Όπœˆπ›½π‘›.(3.4)
In the case 𝛽>1, it is easy to arrive at βˆ‘π‘˜β‰₯1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1β‰€βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπ‘˜π›½π‘›+1. Now we imitate the proof of Hardy's inequality given in [36, Theorem 326 in page 239]. Write β„Žπ‘›=2βˆ’π‘›π›Ό/π›½π‘˜π‘›+1=2βˆ’π‘›π›Ό/π›½βˆ‘π‘›π‘—=0πœˆπ‘—, for 𝑛β‰₯0, and β„Žβˆ’1=0. Then πœˆπ‘›=2𝑛𝛼/𝛽(β„Žπ‘›βˆ’2βˆ’π›Ό/π›½β„Žπ‘›βˆ’1), for 𝑛β‰₯0, and by the inequality between the geometric and arithmetic means [36, Theorem 9 in page 17], β„Žπ›½π‘›βˆ’11βˆ’2βˆ’π›Ό/π›½β„Žπ‘›π›½βˆ’12βˆ’π‘›π›Ό/π›½πœˆπ‘›=12𝛼/π›½ξ‚€β„Žβˆ’1π‘›π›½βˆ’1β„Žnβˆ’1βˆ’β„Žπ›½π‘›ξ‚β‰€12𝛼/π›½ξƒ©βˆ’1(π›½βˆ’1)β„Žπ›½π‘›+β„Žπ›½π‘›βˆ’1π›½βˆ’β„Žπ›½π‘›ξƒͺ=1𝛽2𝛼/π›½ξ€Έξ‚€β„Žβˆ’1π›½π‘›βˆ’1βˆ’β„Žπ›½π‘›ξ‚.(3.5) So the sum on 𝑛 of the left hand side is negative, and, therefore, using HΓΆlder's inequality with exponents 𝛽/(π›½βˆ’1) and 𝛽, 𝑛β‰₯02βˆ’π‘›π›Όπ‘˜π›½π‘›+1=𝑛β‰₯0β„Žπ›½π‘›β‰€11βˆ’2βˆ’π›Ό/𝛽𝑛β‰₯0β„Žπ‘›π›½βˆ’12βˆ’π‘›π›Ό/π›½πœˆπ‘›β‰€11βˆ’2βˆ’π›Ό/𝛽𝑛β‰₯0β„Žπ›½π‘›ξƒͺπ›½βˆ’1/𝛽𝑛β‰₯02βˆ’π‘›π›Όπœˆπ›½π‘›ξƒͺ1/𝛽,(3.6) giving as a result that βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπ‘˜π›½π‘›+1≀(1βˆ’2βˆ’π›Ό/𝛽)βˆ’π›½βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπœˆπ›½π‘›, as desired.
In the other direction, assume that βˆ‘π‘˜β‰₯1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1<∞. In the case 𝛽<1, it is easily verified that βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπœˆπ›½π‘›β‰€βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπ‘˜π›½π‘›+1. To continue, use that π‘˜0=0 and that the function π‘₯↦π‘₯π›½βˆ’1 is decreasing in (0,∞), 𝑛β‰₯02βˆ’π‘›π›Όπ‘˜π›½π‘›+1=𝑛β‰₯02βˆ’π‘›π›Όπ‘˜π‘˜π›½βˆ’1𝑛+1𝑛+1ξ“π‘˜=11≀𝑛β‰₯02π‘˜βˆ’π‘›π›Όπ‘›+1ξ“π‘˜=π‘˜π‘›+1π‘˜π›½βˆ’1+𝑛β‰₯12βˆ’π‘›π›Όπ‘˜π›½π‘›β‰€2π›Όξ“π‘˜π‘›β‰₯0𝑛+1ξ“π‘˜=π‘˜π‘›+1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1+2βˆ’π›Όξ“π‘›β‰₯02βˆ’π‘›π›Όπ‘˜π›½π‘›+1,(3.7) from where it follows that βˆ‘π‘›β‰₯02βˆ’π‘›π›Όπ‘˜π›½π‘›+1≀2𝛼(1βˆ’2βˆ’π›Ό)βˆ’1βˆ‘π‘˜β‰₯1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1<∞.
It remains to deal with the case 𝛽β‰₯1 under the assumption βˆ‘π‘˜β‰₯1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1<∞. Here, we use that, when 0β‰€π‘Žβ‰€π‘, (π‘βˆ’π‘Ž)π›½β‰€π‘π›½βˆ’π‘Žπ›½ (because ((π‘βˆ’π‘Ž)𝛽+π‘Žπ›½)1/𝛽≀𝑏), and use also the Mean Value Theorem, 𝑛β‰₯02βˆ’π‘›π›Όπœˆπ›½π‘›=𝑛β‰₯02βˆ’π‘›π›Όξ€·π‘˜π‘›+1βˆ’π‘˜π‘›ξ€Έπ›½β‰€ξ“π‘›β‰₯02βˆ’π‘›π›Όξ‚€π‘˜π›½π‘›+1βˆ’π‘˜π›½π‘›ξ‚=2𝛼𝑛β‰₯02π‘˜βˆ’(𝑛+1)𝛼𝑛+1ξ“π‘˜=π‘˜π‘›+1ξ€·π‘˜π›½βˆ’(π‘˜βˆ’1)𝛽≀2π›Όπ›½ξ“π‘˜π‘›β‰₯0𝑛+1ξ“π‘˜=π‘˜π‘›+1π‘‘π›Όπ‘˜π‘˜π›½βˆ’1=2π›Όπ›½ξ“π‘˜β‰₯1π‘‘π›Όπ‘˜π‘˜π›½βˆ’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,π‘ž<∞, that 0<𝛼<1, and that 𝐼 is an inner function. Then the following quantities are comparable, β€–πΌβ€–π‘žπ΅π›Όπ‘,π‘ž=ξ€œ10(1βˆ’π‘Ÿ)π‘žβˆ’1π‘€π‘žπ‘ξ€·π‘Ÿ,𝐷1+π›ΌπΌξ€Έπ‘‘π‘Ÿ,(3.9)ξ€œ10(1βˆ’π‘Ÿ)(1βˆ’π›Ό)π‘žβˆ’1π‘€π‘žπ‘ξ€·π‘Ÿ,πΌξ…žξ€Έπ‘‘π‘Ÿ,(3.10)ξ€œ10(1βˆ’π‘Ÿ)βˆ’π›Όπ‘žβˆ’1ξ‚΅1ξ€œ2πœ‹02πœ‹ξ€·||𝐼1βˆ’π‘Ÿπ‘’π‘–πœƒξ€Έ||ξ€Έπ‘ξ‚Άπ‘‘πœƒπ‘ž/π‘π‘‘π‘Ÿ,(3.11)ξ€œ10(1βˆ’π‘Ÿ)βˆ’π›Όπ‘žβˆ’1ξ‚΅1ξ€œ2πœ‹02πœ‹||πΌξ€·π‘’π‘–πœƒξ€Έξ€·βˆ’πΌπ‘Ÿπ‘’π‘–πœƒξ€Έ||π‘ξ‚Άπ‘‘πœƒπ‘ž/π‘π‘‘π‘Ÿ.(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 π‘ž=∞ 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‖Λ𝑝,𝛼=sup0β‰€π‘Ÿ<1(1βˆ’π‘Ÿ)π‘€π‘ξ€·π‘Ÿ,𝐷1+𝛼𝐼,(3.13)sup0β‰€π‘Ÿ<1(1βˆ’π‘Ÿ)1βˆ’π›Όπ‘€π‘ξ€·π‘Ÿ,πΌξ…žξ€Έ,(3.14)sup0β‰€π‘Ÿ<1(1βˆ’π‘Ÿ)βˆ’π›Όξ‚΅1ξ€œ2πœ‹02πœ‹ξ€·||𝐼1βˆ’π‘Ÿπ‘’π‘–πœƒξ€Έ||ξ€Έπ‘ξ‚Άπ‘‘πœƒ1/𝑝,(3.15)sup0β‰€π‘Ÿ<1(1βˆ’π‘Ÿ)βˆ’π›Όξ‚΅1ξ€œ2πœ‹02πœ‹||πΌξ€·π‘’π‘–πœƒξ€Έξ€·βˆ’πΌπ‘Ÿπ‘’π‘–πœƒξ€Έ||π‘ξ‚Άπ‘‘πœƒ1/𝑝.(3.16)

For Blaschke products, the third author [15, 38] gave sufficient (and in special cases, necessary) conditions for their membership in 𝐡𝛼𝑝,π‘ž in terms of the distribution of their zeros. Recall that a Carleson-Newman sequence is a finite union of interpolating sequences, and a sequence {π‘§π‘˜} in the unit disk is called interpolating if it is uniformly separated, that is, infπ‘›ξ‘ξ€½π‘˜βˆΆπ‘§π‘˜β‰ π‘§π‘›ξ€Ύ||||𝑧kβˆ’π‘§π‘›1βˆ’π‘§π‘˜π‘§π‘›||||>0.(3.17)

Theorem E (see [15, 38]). Let 0<𝑝,𝛼<∞ be such that max{0,1/π‘βˆ’1}<𝛼<1/𝑝. Assume that 0<π‘žβ‰€βˆž, and that 𝐡 is a Blaschke product. If {(2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›)1/𝑝}βˆˆβ„“π‘ž, then π΅βˆˆπ΅π›Όπ‘,π‘ž and ‖𝐡‖𝐡𝛼𝑝,π‘žβ€–β€–ξ‚†ξ€·2β‰€πΆβˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›ξ€Έ1/π‘ξ‚‡β€–β€–β„“π‘ž.(3.18) On the other hand, if the zero sequence {π‘§π‘˜} of 𝐡 is Carleson-Newman and π΅βˆˆπ΅π›Όπ‘,π‘ž, then {(2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›)1/𝑝}βˆˆβ„“π‘ž 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/π‘βˆ’1}<𝛼<1/𝑝, 0<π‘žβ‰€βˆž, and 𝐼 is an inner function satisfying ξ‚΅ξ€œπΎπ›Ώβ€–β€–ξ‚†ξ€·2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›ξ€Έ(π‘Ž)1/π‘ξ‚‡β€–β€–π‘β„“π‘žξ‚Άπ‘‘π΄(π‘Ž)1/𝑝<∞,(3.19) for 𝐾𝛿={π‘Žβˆˆπ”»βˆΆπ›Ώβ‰€|π‘Ž|≀1βˆ’π›Ώ}, and 0<𝛿<1/2, then πΌβˆˆπ΅π›Όπ‘,π‘ž, and the norm ‖𝐼‖𝐡𝛼𝑝,π‘ž 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 𝐡𝛼𝑝,π‘ž.

Theorem 3.3. Let 0<𝑝,𝛼<∞ be such that max{0,1/π‘βˆ’1}<𝛼<1/𝑝. Assume that 0<π‘žβ‰€βˆž, and that 𝐼 is an inner function. Then πΌβˆˆπ΅π›Όπ‘,π‘ž if and only if (3.19) holds for some π›Ώβˆˆ(0,1/2). In that case, both quantities, ‖𝐼‖𝐡𝛼𝑝,π‘ž 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 π‘ž=∞.

Lemma 3.4. If 0<𝑝<∞, 0<π‘žβ‰€βˆž, 0≀𝛼<∞, and 1≀𝑑<∞, then Ξ›βˆž,0βˆ©π΅π›Όπ‘,π‘žβŠ†π΅π‘π‘‘,π‘žπ‘‘π›Ό/𝑑, that is, Bloch functions in 𝐡𝛼𝑝,π‘ž are also in 𝐡𝑝𝑑,π‘žπ‘‘π›Ό/𝑑 for any 𝑑β‰₯1. Furthermore, the following relation holds: ‖𝑓‖𝐡𝑝𝑑,π‘žπ‘‘π›Ό/𝑑≀𝐢‖𝑓‖Λ1βˆ’1/π‘‘βˆž,0‖𝑓‖𝐡1/𝑑𝛼𝑝,π‘ž.(3.20)

Proof of Lemma 3.4. The case π‘ž=∞ will not be treated due to its similarity with the other cases. Take 0<𝑝,π‘ž<∞, 0≀𝛼<∞, 1≀𝑑<∞, and π‘“βˆˆΞ›βˆž,0βˆ©π΅π›Όπ‘,π‘ž. We need to show that π‘“βˆˆπ΅π‘π‘‘,π‘žπ‘‘π›Ό/t. For that, use (1.14) to find an equivalent quantity to ‖𝑓‖𝐡𝑝𝑑,π‘žπ‘‘π›Ό/𝑑, and then separate it into two factors, the first will be controlled by β€–π‘“β€–Ξ›βˆž,0, and the second by ‖𝑓‖𝐡𝛼𝑝,π‘ž. ‖𝑓‖𝐡𝑝𝑑,π‘žπ‘‘π›Ό/𝑑=ξ‚΅ξ€œ10(1βˆ’π‘Ÿ)π‘žπ‘‘βˆ’1π‘€π‘žπ‘‘π‘π‘‘ξ€·π‘Ÿ,𝐷1+𝛼/π‘‘π‘“ξ€Έξ‚Άπ‘‘π‘Ÿ1/π‘žπ‘‘(1.14)ξ‚΅ξ€œβ‰€πΆ10(1βˆ’π‘Ÿ)π‘žπ‘‘βˆ’1βˆ’π‘žπ›Ό+π‘žπ›Όπ‘‘π‘€π‘žπ‘‘π‘π‘‘ξ€·π‘Ÿ,𝐷1+π›Όπ‘“ξ€Έξ‚Άπ‘‘π‘Ÿ1/π‘žπ‘‘ξƒ©ξ€œ=𝐢10(1βˆ’π‘Ÿ)π‘ž(π‘‘βˆ’1)(1+𝛼)+(π‘žβˆ’1)ξ‚΅ξ€œ02πœ‹||𝐷1+π›Όπ‘“ξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||π‘π‘‘ξ‚Άπ‘‘πœƒπ‘ž/𝑝ξƒͺπ‘‘π‘Ÿ1/π‘žπ‘‘β‰€πΆsup0β‰€π‘Ÿ<1ξ€·(1βˆ’π‘Ÿ)1+π›Όπ‘€βˆžξ€·π‘Ÿ,𝐷1+𝛼𝑓1βˆ’1/π‘‘ξ‚΅ξ€œ10(1βˆ’π‘Ÿ)π‘žβˆ’1π‘€π‘žπ‘ξ€·π‘Ÿ,𝐷1+π›Όπ‘“ξ€Έξ‚Άπ‘‘π‘Ÿ1/π‘žπ‘‘β‰€πΆβ€–π‘“β€–Ξ›1βˆ’1/π‘‘βˆž,0‖𝑓‖𝐡1/𝑑𝛼𝑝,π‘ž.(3.21)

Two more lemmas are needed.

Lemma F (see [15, Corollary 4.5]). If 𝑝β‰₯1, 𝐼 is an inner function, and if 1βˆ’2βˆ’π‘›<π‘Ÿβ‰€1βˆ’2βˆ’(𝑛+1), then 2βˆ’π‘›πœˆπ‘›β‰€πΆπ‘ξ€œ02πœ‹log𝑝1||πΌξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||π‘‘πœƒ.(3.22)

Lemma G (see [15, Corollary 4.7]). If 0<𝑝<∞, 0<𝛿<1/2, 𝐼 is an inner function, and if π‘§βˆˆπ”», then ξ€œπΎπ›Ώlog𝑝1||πΌπ‘Ž||(𝑧)𝑑𝐴(π‘Ž)≀𝐢𝑝,𝛿||||ξ€Έ1βˆ’πΌ(𝑧)𝑝.(3.23)

Proof of Theorem 3.3. Again, we deal only with the case π‘ž<∞. The sufficiency of condition (3.19) has already been established. To prove its necessity, assume that πΌβˆˆπ΅π›Όπ‘,π‘ž and, rather than imposing the whole restriction max{0,1/π‘βˆ’1}<𝛼<1/𝑝, just assume 0<𝛼,𝑝,π‘ž<∞. Observe that the integral in (3.19) (without the power 1/𝑝) remains unchanged if we replace 𝑝,π‘ž,𝛼 with 𝑝𝑑,π‘žπ‘‘,𝛼/𝑑. Now choose 𝑑β‰₯1 such that 𝛼/𝑑<1, 𝑝𝑑>1 and π‘žπ‘‘>1. If the result holds in this situation, then, by the homogeneity property of Lemma 3.4, we have ξ‚΅ξ€œπΎπ›Ώβ€–β€–ξ‚†ξ€·2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›ξ€Έ(π‘Ž)1/π‘ξ‚‡β€–β€–π‘β„“π‘žξ‚Άπ‘‘π΄(π‘Ž)1/𝑝=ξ‚΅ξ€œπΎπ›Ώβ€–β€–ξ‚†ξ€·2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›ξ€Έ(π‘Ž)1/π‘π‘‘ξ‚‡β€–β€–β„“π‘π‘‘π‘žπ‘‘ξ‚Άπ‘‘π΄(π‘Ž)1/𝑝≀𝐢‖𝐼‖𝑑𝐡𝑝𝑑,π‘žπ‘‘π›Ό/π‘‘β‰€πΆβ€–πΌβ€–Ξ›π‘‘βˆ’1∞,0‖𝐼‖𝐡𝛼𝑝,π‘ž.(3.24) So it suffices to prove the result for 0<𝛼<1 and 1<𝑝,π‘ž<∞. In what follows π‘Ÿπ‘›=1βˆ’2βˆ’π‘›. First assume that 𝑝>π‘ž. Then use, in order, Minkowski's inequality for 𝑝/π‘ž>1, the fact that βˆ«π‘Ÿπ‘›+1π‘Ÿπ‘›π‘Ÿβˆ’π›Όπ‘žβˆ’1π‘‘π‘Ÿβ‰2π‘›π›Όπ‘ž, and finally Lemmas F and G together with Theorem C to arrive at the desired estimate for (3.19), ξ‚΅ξ€œπΎπ›Ώβ€–β€–ξ‚†ξ€·2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›ξ€Έ(π‘Ž)1/π‘ξ‚‡β€–β€–π‘β„“π‘žξ‚Άπ‘‘π΄(π‘Ž)1/𝑝=βŽ›βŽœβŽœβŽξ€œπΎπ›Ώξƒ©ξ“π‘›β‰₯02π‘›π›Όπ‘žξ€·2βˆ’π‘›πœˆπ‘›ξ€Έ(π‘Ž)π‘ž/𝑝ξƒͺ𝑝/π‘žβŽžβŽŸβŽŸβŽ π‘‘π΄(π‘Ž)(π‘ž/𝑝)(1/π‘ž)≀𝑛β‰₯02π‘›π›Όπ‘žξ‚΅ξ€œπΎπ›Ώ2βˆ’π‘›πœˆπ‘›ξ‚Ά(π‘Ž)𝑑𝐴(π‘Ž)π‘ž/𝑝ξƒͺ1/π‘žβŽ›βŽœβŽœβŽξ“β‰€πΆπ‘›β‰₯0ξ€œπ‘Ÿπ‘›+1r𝑛(1βˆ’π‘Ÿ)βˆ’π›Όπ‘žβˆ’1ξƒ©ξ€œπΎπ›Ώξ€œ02πœ‹log𝑝1||πΌπ‘Žξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||ξƒͺπ‘‘πœƒπ‘‘π΄(π‘Ž)π‘ž/π‘βŽžβŽŸβŽŸβŽ π‘‘π‘Ÿ1/π‘žβŽ›βŽœβŽœβŽξ€œβ‰€πΆ10(1βˆ’π‘Ÿ)βˆ’π›Όπ‘žβˆ’1ξƒ©ξ€œ02πœ‹ξ€œπΎπ›Ώlog𝑝1||πΌπ‘Žξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||ξƒͺ𝑑𝐴(π‘Ž)π‘‘πœƒπ‘ž/π‘βŽžβŽŸβŽŸβŽ π‘‘π‘Ÿ1/π‘žξƒ©ξ€œβ‰€πΆ10(1βˆ’π‘Ÿ)βˆ’π›Όπ‘žβˆ’1ξ‚΅ξ€œ02πœ‹ξ€·||𝐼1βˆ’π‘Ÿπ‘’π‘–πœƒξ€Έ||ξ€Έπ‘ξ‚Άπ‘‘πœƒπ‘ž/𝑝ξƒͺπ‘‘π‘Ÿ1/π‘žβ‰€πΆβ€–πΌβ€–π΅π›Όπ‘,π‘ž.(3.25)
The case 𝑝<π‘ž follows the same procedure, only that instead of using Minkowski's inequality, we use HΓΆlder's with exponents π‘ž/(π‘žβˆ’π‘) and π‘ž/𝑝, and then, after applying Lemma F and before Lemma G, use again Minkowski's inequality with π‘ž/𝑝>1,ξ‚΅ξ€œπΎπ›Ώβ€–β€–ξ‚†ξ€·2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›ξ€Έ(π‘Ž)1/π‘ξ‚‡β€–β€–π‘β„“π‘žξ‚Άπ‘‘π΄(π‘Ž)1/𝑝=βŽ›βŽœβŽœβŽξ€œπΎπ›Ώξƒ©ξ“π‘›β‰₯02π‘›π›Όπ‘žξ€·2βˆ’π‘›πœˆπ‘›ξ€Έ(π‘Ž)π‘ž/𝑝ξƒͺ𝑝/π‘žβŽžβŽŸβŽŸβŽ π‘‘π΄(π‘Ž)1/π‘β‰€ξ‚΅ξ€œπΎπ›Ώξ‚Άπ‘‘π΄(π‘Ž)1/π‘βˆ’1/π‘žξƒ©ξ€œπΎπ›Ώξ“π‘›β‰₯02π‘›π›Όπ‘žξ€·2βˆ’π‘›πœˆπ‘›ξ€Έ(π‘Ž)π‘ž/𝑝ξƒͺ𝑑𝐴(π‘Ž)1/π‘žξƒ©ξ“=𝐢𝑛β‰₯02π‘›π›Όπ‘žξ€œπΎπ›Ώξ€·2βˆ’π‘›πœˆπ‘›ξ€Έ(π‘Ž)π‘ž/𝑝ξƒͺ𝑑𝐴(π‘Ž)1/π‘žβŽ›βŽœβŽœβŽξ“β‰€πΆπ‘›β‰₯0ξ€œπ‘Ÿπ‘›+1π‘Ÿπ‘›(1βˆ’π‘Ÿ)βˆ’π›Όπ‘žβˆ’1βŽ›βŽœβŽœβŽξ€œπΎπ›Ώξƒ©ξ€œ02πœ‹log𝑝1||πΌπ‘Žξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||ξƒͺπ‘‘πœƒπ‘ž/π‘βŽžβŽŸβŽŸβŽ π‘‘π΄(π‘Ž)(𝑝/π‘ž)(π‘ž/𝑝)βŽžβŽŸβŽŸβŽ π‘‘π‘Ÿ1/π‘žβŽ›βŽœβŽœβŽξ€œβ‰€πΆ10(1βˆ’π‘Ÿ)βˆ’π›Όπ‘žβˆ’1βŽ›βŽœβŽœβŽξ€œ02πœ‹ξƒ©ξ€œπΎπ›Ώlogπ‘ž1||πΌπ‘Žξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||ξƒͺ𝑑𝐴(π‘Ž)𝑝/π‘žβŽžβŽŸβŽŸβŽ π‘‘πœƒπ‘ž/π‘π‘‘βŽžβŽŸβŽŸβŽ 1/π‘žξƒ©ξ€œβ‰€πΆ10π‘Ÿβˆ’π›Όπ‘žβˆ’1ξ‚΅ξ€œ02πœ‹ξ€·||𝐼1βˆ’π‘Ÿπ‘’π‘–πœƒξ€Έ||ξ€Έπ‘ξ‚Άπ‘‘πœƒπ‘ž/𝑝ξƒͺπ‘‘π‘Ÿ1/π‘žβ‰€πΆβ€–πΌβ€–π΅π›Όπ‘,π‘ž.(3.26)

Remark 3.5. Along the proof of this theorem, we have actually proved that if 𝐼 is an inner function in 𝐡𝛼𝑝,π‘ž, with 0<𝛼,𝑝<∞ and 0<π‘žβ‰€βˆž, then (3.19) holds and ξ‚΅ξ€œπΎπ›Ώβ€–β€–ξ‚†ξ€·2βˆ’π‘›(1βˆ’π›Όπ‘)πœˆπ‘›ξ€Έ(π‘Ž)1/π‘ξ‚‡β€–β€–π‘β„“π‘žξ‚Άπ‘‘π΄(π‘Ž)1/𝑝≀𝐢‖𝐼‖𝐡𝛼𝑝,π‘ž.(3.27)
Also, as we observed before, the integral in (3.19), or (3.27), is unchanged if 𝑝,π‘ž,𝛼 is replaced with 𝑝𝑑,π‘žπ‘‘,𝛼/𝑑, (𝑑>0). This allows us to extend the homogeneity property of Lemma 3.4 to other values of 𝑑, provided that we can apply Theorem 3.3, that is, that we work with inner functions and that max{0,1/(𝑝𝑑)βˆ’1}<𝛼/𝑑<1/(𝑝𝑑).

Corollary 3.6. Let 0<𝑝,𝛼<∞ be such that 0<𝛼<1/𝑝. Assume that 0<π‘žβ‰€βˆž, and that 𝐼 is an inner function in 𝐡𝛼𝑝,π‘ž. Then πΌβˆˆπ΅π‘π‘‘,π‘žπ‘‘π›Ό/𝑑 for all 𝑑>1/π‘βˆ’π›Ό.

Remark 3.7. Of course, when 𝛼>1/𝑝, the class of inner functions in 𝐡𝛼𝑝,π‘ž coincides with that of 𝐡𝑝𝑑,π‘žπ‘‘π›Ό/𝑑 for any 𝑑>0 (or any 𝐡𝑝1,π‘ž1𝛼1 with 𝛼1>1/𝑝1, whatsoever), because they only contain the finite Blaschke products. The same reasoning applies when 𝛼=1/𝑝 and 0<π‘ž<∞. Is it the same for π‘ž=∞? 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 of 𝑑>0. The answer is negative. Ahern and Clark [3, Lemma 2] have constructed a Blaschke product 𝐡 in 𝐡1,11/2 but not in 𝐻11/2. By property (𝑃9), we deduce that π΅βˆ‰π΅11/2,1/2, and this is the space that would be obtained from 𝐡1,11/2 by taking 𝑑=1/2, which coincides with 1/π‘βˆ’π›Ό with the usual notation.

Proof of Corollary 3.6. Notice first that Remark 3.5 implies that (3.27) holds for 𝑝,π‘ž,𝛼, and thus the integral on the left hand side is unchanged for 𝑝𝑑,π‘žπ‘‘,𝛼/𝑑. Now, take 𝑑>1/π‘βˆ’π›Ό>0. Then 1/(𝑝𝑑)βˆ’1<𝛼/𝑑<1/(𝑝𝑑). If now 𝑑≀1/𝑝, then 0≀1/(𝑝𝑑)βˆ’1, and so max{0,1/(𝑝𝑑)βˆ’1}<𝛼/𝑑<1/(𝑝𝑑), proving that πΌβˆˆπ΅π‘π‘‘,π‘žπ‘‘π›Ό/𝑑 by Theorem 3.3. If, on the contrary, 𝑑>1/𝑝, then 1/(𝑝𝑑)βˆ’1<0, and we still have max{0,1/(𝑝𝑑)βˆ’1}=0<𝛼/𝑑<1/(𝑝𝑑), and again, πΌβˆˆπ΅π‘π‘‘,π‘žπ‘‘π›Ό/𝑑 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 𝐡 is a Blaschke product satisfying βˆ‘π‘˜π‘‘π‘˜2βˆ’π‘<∞ for some π‘βˆˆ(1,2). By Lemma 3.1(b), this condition is equivalent to βˆ‘π‘›β‰₯02βˆ’π‘›(2βˆ’π‘)πœˆπ‘›<∞, which implies that π΅βˆˆπ΅π‘,𝑝1βˆ’1/𝑝 by Theorem E. (Notice that this is equivalent to π΅β€²βˆˆπ΄π‘ by (1.14).) Observe now that we can apply Corollary 3.6 with 𝑑=1/𝑝, obtaining that 𝐡∈𝐡1,1π‘βˆ’1, which is the aforementioned result by Protas, with our notation. On the other hand, notice also that if π΅ξ…žβˆˆπ΄π‘, for some 1<𝑝<2, and the zero sequence of 𝐡 is Carleson-Newman then, by Theorem E, βˆ‘π‘˜π‘‘π‘˜2βˆ’π‘<∞.
Next we will turn to study the membership of inner functions in the spaces 𝐻𝑝𝛼. Properties (𝑃9) and (𝑃10), that is, the Littlewood-Paley inequalities, relate quite well 𝐻𝑝𝛼 with 𝐡𝛼𝑝,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<π‘β‰€βˆž and 𝛼>max{0,1/π‘βˆ’1}. Then the class of inner functions in 𝐻𝑝𝛼 coincides with that of inner functions in 𝐡𝛼𝑝,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 𝐻𝑝𝛼. Then π‘“βˆˆπ»π‘π‘‘π›Ό/𝑑 for all 𝑑β‰₯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 𝐢=𝐢𝛼,𝛽 such that if π‘“βˆˆΞ›βˆž,0 then ||𝐷𝛼||𝑓(𝑧)≀𝐢max0≀𝑑≀1||𝐷𝛽||𝑓(𝑑𝑧)𝛼/𝛽.(3.28)

The original proof of this lemma runs with 𝐻∞ functions instead of Bloch (Ξ›βˆž,0) functions. However, the lemma can be proved for Bloch functions by just noticing the validity of the estimate |𝐷𝛽𝑓(𝑧)|≀𝐢(1βˆ’|𝑧|)βˆ’π›½ for Bloch functions.

Proof of Theorem 3.10. First notice that when 𝛼β‰₯1/𝑝, the only inner functions in 𝐻𝑝𝛼 and 𝐡𝛼𝑝,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 𝐡𝛼𝑝,π‘βŠ†π»π‘π›Ό by (𝑃9). To go in the other direction, take an inner function 𝐼 in 𝐻𝑝𝛼. Then, by Proposition 3.11, πΌβˆˆπ»π‘π‘‘π›Ό/𝑑 for all 𝑑β‰₯1. For 𝑑β‰₯2/𝑝β‰₯1, we have 𝑝𝑑β‰₯2, and hence 𝐻𝑝𝑑𝛼/π‘‘βŠ†π΅π‘π‘‘,𝑝𝑑𝛼/𝑑 by (𝑃10). Using now that max{0,1/π‘βˆ’1}<𝛼<1/𝑝, we get 1/𝑑>1/(𝑝𝑑)βˆ’π›Ό/𝑑, and we conclude that πΌβˆˆπ΅π›Όπ‘,𝑝 by Corollary 3.6.
Next consider the case 2≀𝑝<∞. (Then max{0,1/π‘βˆ’1}=0.) By (𝑃10), π»π‘π›ΌβŠ†π΅π›Όπ‘,𝑝. To go in the other direction, take an inner function 𝐼 in 𝐡𝛼𝑝,𝑝. Since 0<𝛼<1/𝑝 then, by Corollary 3.6, πΌβˆˆπ΅π‘π‘‘,𝑝𝑑𝛼/𝑑 for all 𝑑>1/π‘βˆ’π›Ό. Choose π‘‘βˆˆ(1/π‘βˆ’π›Ό,2/𝑝]. Thus, as 𝑝𝑑≀2, (𝑃9) gives 𝐡𝑝𝑑,𝑝𝑑𝛼/π‘‘βŠ†π»π‘π‘‘π›Ό/𝑑. Finally, as 1/𝑑β‰₯𝑝/2β‰₯1, Proposition 3.11 gives πΌβˆˆπ»π‘π›Ό.

Remark 3.12. A careful reading of this proof shows that, in fact, any inner function 𝐼 in 𝐡𝛼𝑝,𝑝 is also in 𝐻𝑝𝛼, whenever 0<π‘β‰€βˆž 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 𝐻𝑝𝛼. Then πΌβˆˆπ»π‘π‘‘π›Ό/𝑑 for all 𝑑>1/π‘βˆ’π›Ό.

Proof. Take 𝑑>1/π‘βˆ’π›Ό. If 𝑑β‰₯1 then πΌβˆˆπ»π‘π‘‘π›Ό/𝑑 by Proposition 3.11. Otherwise we have 1/π‘βˆ’π›Ό<𝑑<1, which implies 1/π‘βˆ’1<𝛼, and, together with 0<𝛼<1/𝑝, it gives max{0,1/π‘βˆ’1}<𝛼<1/𝑝. So, by Theorem 3.10, πΌβˆˆπ΅π›Όπ‘,𝑝 and, by Corollary 3.6, πΌβˆˆπ΅π‘π‘‘,𝑝𝑑𝛼/𝑑 and, finally, by Remark 3.12, πΌβˆˆπ»π‘π‘‘π›Ό/𝑑.

Remark 3.14. Again, since it only contains finite Blaschke products, the class of inner functions in 𝐻𝑝𝛼, with 𝛼β‰₯1/𝑝, coincides with that of 𝐻𝑝𝑑𝛼/𝑑 for all 𝑑>0, or for the case, with the class of inner functions of any 𝐻𝑝1𝛼1, with 𝛼1β‰₯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 𝑑>0, the same example given in Remark 3.8 shows that it is impossible. The Blaschke product 𝐡 constructed in [3, Lemma 2] is in 𝐡1,11/2 but not in 𝐻11/2, which is the space that would be obtained from 𝐻11/2 by taking 𝑑=1/2(=1/π‘βˆ’π›Ό with the usual notation). By Theorem 3.10, 𝐡∈𝐡1,11/2 implies that 𝐡∈𝐻11/2.

Remark 3.15. As an application, we again regain a known result of Protas [9, Theorem 2], namely, if 𝐡 is a Blaschke product such that βˆ‘π‘˜π‘‘π‘˜1βˆ’π‘<∞, for some 1/2<𝑝<1, then π΅βˆˆπ»π‘1. Indeed, by Lemma 3.1(b), βˆ‘π‘˜π‘‘π‘˜1βˆ’π‘β‰βˆ‘π‘›β‰₯02βˆ’π‘›(1βˆ’π‘)πœˆπ‘›, and so 𝐡∈𝐡1𝑝,𝑝 by Theorem E, and we are done with π΅βˆˆπ»π‘1 by Theorem 3.10. On the other hand, Theorem E also says that if the zero sequence of 𝐡 is Carleson-Newman then the condition βˆ‘π‘˜π‘‘π‘˜1βˆ’π‘<∞ is also necessary for π΅βˆˆπ»π‘1.

4. The Case 1/(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]). If 𝐼 is an inner function and π‘€π‘π‘ξ€·π‘Ÿ,𝐷1/(2𝑝)𝐼1=olog1βˆ’π‘Ÿ,asπ‘ŸβŸΆ1,(4.1) for some 0<𝑝<∞, then 𝐼 is a Blaschke product.

Now, all functions in 𝐻𝑝1/(2𝑝) satisfy condition (4.1) and, by (𝑃5), the same is true for all 𝐻𝑝𝛼-functions with 𝛼β‰₯1/(2𝑝). So the following is immediate:

Corollary 4.1. Let 0<𝑝,𝛼<∞ with 𝛼β‰₯1/(2𝑝). Then the only inner functions in 𝐻𝑝𝛼 are Blaschke products, finite ones if 𝛼β‰₯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,π‘ž,𝛼<∞ with 𝛼β‰₯1/(2𝑝). Then the only inner functions in 𝐡𝛼𝑝,π‘ž are Blaschke products, finite ones if 𝛼β‰₯1/𝑝.

Remark 4.2. These results are again accurate, for the β€œatomic” singular inner function 𝑆(𝑧)=exp(βˆ’(1+𝑧)/(1βˆ’π‘§)) is in 𝐡𝛼𝑝,π‘žβˆ©π»π‘π›Ό for all 0<𝑝,π‘ž,𝛼<∞ with 𝛼<1/(2𝑝). Indeed, in [39], it is shown that, for any positive integer 𝑛, 𝑀1/(2𝑛)ξ€·π‘Ÿ,𝑆(𝑛)ξ€Έ1≍log1βˆ’π‘Ÿ.(4.2) This implies that π‘†βˆˆπ»π›½1/(2𝑛) for all 𝛽<𝑛, because by (1.11), 𝑀1/(2𝑛)1/(2𝑛)ξ€·π‘Ÿ,π·π›½π‘†ξ€Έξ€œβ‰€πΆ10(1βˆ’π‘ )(π‘›βˆ’π›½)/(2𝑛)βˆ’1𝑀1/(2𝑛)1/(2𝑛)(π‘Ÿπ‘ ,π·π‘›ξ€œπ‘†)𝑑𝑠≀𝐢10(1βˆ’π‘ )(π‘›βˆ’π›½)/(2𝑛)βˆ’1log1/(2𝑛)ξ‚€11βˆ’π‘Ÿπ‘ π‘‘π‘ <∞,(4.3) and also π‘†βˆˆπ΅1/(2𝑛),Μƒπ‘žπ›½ for all 𝛽<𝑛 and all 0<Μƒπ‘ž<∞, because by (1.14), ξ€œ10(1βˆ’π‘Ÿ)Μƒπ‘žβˆ’1π‘€Μƒπ‘ž1/(2𝑛)ξ€·π‘Ÿ,𝐷1+π›½π‘†ξ€Έξ€œπ‘‘π‘Ÿβ‰€πΆ10(1βˆ’π‘Ÿ)Μƒπ‘ž(π‘›βˆ’π›½)βˆ’1π‘€Μƒπ‘ž1/(2𝑛)(π‘Ÿ,π·π‘›ξ€œπ‘†)π‘‘π‘Ÿβ‰€πΆ10(1βˆ’π‘Ÿ)Μƒπ‘ž(π‘›βˆ’π›½)βˆ’1ξ‚€1logΜƒπ‘žξ‚1βˆ’π‘Ÿπ‘‘π‘Ÿ<∞.(4.4) Hence, given 0<𝑝,π‘ž,𝛼<∞ with 𝛼<1/(2𝑝), take the smallest integer 𝑛 such that 1/(2𝑝)≀𝑛 and then take 𝛽=2𝑛𝑝𝛼<𝑛 and Μƒπ‘ž=π‘ž/(2𝑛𝑝). In this way, π‘†βˆˆπ»π›½1/