Characterizations for inner functions in certain function spaces
aa r X i v : . [ m a t h . C V ] N ov CHARACTERIZATIONS FOR INNER FUNCTIONS IN CERTAINFUNCTION SPACES
ATTE REIJONEN AND TOSHIYUKI SUGAWA
Abstract.
For ă p ă 8 , 0 ă q ă 8 and a certain two-sided doubling weight ω , wecharacterize those inner functions Θ for which } Θ } qA p,qω “ ż ˆż π | Θ p re iθ q| p dθ ˙ q { p ω p r q dr ă 8 . Then we show a modified version of this result for p ě q . Moreover, two additional char-acterizations for inner functions whose derivative belongs to the Bergman space A p,pω aregiven. Introduction and main results
Let H p D q be the space of all analytic functions in the open unit disc D of the complex plane C . For 0 ă p ă 8 , the Hardy space H p consists of those f P H p D q such that M p p r, f q “ ˆ π ż π | f p re iθ q| p dθ ˙ { p , ď r ă , is bounded. A function ω : D Ñ r , is called a (radial) weight if it is integrable over D and ω p z q “ ω p| z |q for all z P D . For 0 ă p, q ă 8 and a weight ω , the weighted mixed norm space A p,qω consists of those f P H p D q such that } f } qA p,qω “ ż M qp p r, f q ω p r q dr ă 8 . For a weight ω , we set p ω p z q “ p ω p| z |q “ ż | z | ω p s q ds, z P D . A weight ω belongs to p D if there exists C “ C p ω q ě p ω p r q ď C p ω p ` r q for all0 ď r ă ω P q D if there exist K “ K p ω q ą C “ C p ω q ą p ω p r q ě C p ω ˆ ´ ´ rK ˙ , ď r ă . Class D of so-called two-sided doubling weights is the intersection of p D and q D [21]. Analternative characterization can be given as follows: A weight ω belongs to D if and only ifthere exist C “ C p ω q ě α “ α p ω q ą β “ β p ω q ě α such that C ´ ˆ ´ r ´ s ˙ α p ω p s q ď p ω p r q ď C ˆ ´ r ´ s ˙ β p ω p s q , ď r ď s ă . (1.1)In many cases this characterization is more practical than the original definition.An inner function is a bounded analytic function having unimodular radial limits almosteverywhere on the boundary T “ t z P C : | z | “ u [6, 17]. An important subclass of Mathematics Subject Classification.
Primary: 30J05; Secondary: 30H10, 30H20 and 30H25.
Key words and phrases.
Bergman space, Blaschke product, Frostman shift, Hardy space, inner function,mixed norm space.This research was supported in part by Finnish Cultural Foundation and JSPS Postdoctoral Fellowship forNorth American and European Researchers. inner functions consists of Blaschke products [5]. For a given sequence t z n u Ă D satisfying ř n p ´ | z n |q ă 8 and a real constant λ, the Blaschke product with zeros t z n u is defined by B p z q “ e iλ ź n | z n | z n z n ´ z ´ z n z , z P D . For z n “
0, the interpretation | z n |{ z n “ ´ A p,qω satisfying certain regularityconditions. This characterization is taking advantage of the fact that the Frostman shift Θ a of an arbitrary inner function Θ, defined byΘ a p z q “ Θ p z q ´ a ´ a Θ p z q , z P D , is a Blaschke product for almost every a P D . More precisely, Frostman’s result states thatthe exceptional set E Θ where Θ a is not a Blaschke product has logarithmic capacity zero; seefor instance [8, Chapter 2, Theorem 6.4].Before our first main result, we define r j “ ´ ´ j for j P N Yt u and D δ “ t z P C : | z | ď δ u for 0 ă δ ă
1. Moreover, recall that f À g if there exists a constant C “ C p¨q ą f ď Cg , while f Á g is understood in an analogous manner. If f À g and f Á g , then wewrite f — g . Here the letter C “ C p¨q is a positive constant whose value depends only on theparameters indicated in the parentheses, and may change from one occurrence to another. Theorem 1.
Let ă p ă 8 , ă q ă 8 , ă δ ă and ω P D . If Θ is an inner functionand either (a) ă p ď and ω satisfies the right-hand inequality of (1.1) for some β ă q ´ qp , or (b) 1 ă p ă 8 , ω satisfies the right-hand inequality of (1.1) for some β ă q and theleft-hand inequality for some α ą q ´ qp ,then } Θ } qA p,qω — ÿ n p ω p r n qp ´ r n q q ´ q { p ż D δ υ n p a q q { p dA p a q , (1.2) where t z n p a qu is the zero-sequence of Θ a and υ n p a q “ t j : r n ď | z j p a q| ă r n ` u . Here and hereafter, dA p z q will stand for the 2-dimensional Lebesgue measure dxdy. Theargument of Theorem 1 utilizes the recent results regarding the derivative of inner functionsin A p,qω [24, 25], and certain estimates for Θ a originated to [14]. Using Theorem 1 togetherwith a connection between the mixed norm and Besov spaces [7], we can prove a streamlinedversion of [9, Theorem 3.3]; see Corollary 7 in Section 4. Hence it does not come as a surprisethat the proofs of Theorem 1 and [9, Theorem 3.3] have some similarities.Modifying the argument of Theorem 1 for p ě q , we may remove the integral over D δ inthe statement by assuming a P D z E Θ ; see Theorem 2 below. More precisely, we verify asan auxiliary result that the Lebesgue integral over D δ can be replaced by a certain integralwith respect to a probability measure supported in a compact subset of D ; see Lemma 8 inSection 5. Using this observation, we can prove the desired result for p “ Theorem 2.
Let ă p ă 8 and ă q ď p . Assume that ω , Θ and υ n p a q are as in Theorem 1.Then the following statements are equivalent: (i) Θ P A p,qω . (ii) There exists a set E Θ Ă D of logarithmic capacity zero such that ÿ n p ω p r n q υ n p a q q { p p ´ r n q q ´ q { p ă 8 (1.3) for every a P D z E Θ . HARACTERIZATIONS FOR INNER FUNCTIONS 3 (iii)
There exists a P D z E Θ such that (1.3) holds. For instance, ω p z q “ p ´ | z |q α and ω p z q “ p ´ | z |q α ´ log e ´| z | ¯ β satisfy the hypothesesof ω in Theorems 1 and 2 if max t´ , q ´ qp ´ u ă α ă min t q ´ qp ´ , q ´ u and β P R . Usingthis observation for ω together with [9, Lemma 3.1], it is easy to check that [27, Theorem 3]for q ď p follows from Theorem 2. Our result contains also the case ă p ă
1, unlike theoriginal theorem. Moreover, it is worth mentioning that [27, Theorem 3] was stated withoutany proof. By studying A p,qα “ A p,qω , we can also show that hypothesis (b) in Theorems 1 and 2is natural in a certain sense. More precisely, the only inner functions whose derivative belongsto A p,qα for α ď q ´ qp ´ α ą q ´
1, then the derivativeof every inner function belongs to A p,qα by the Schwarz-pick lemma.Next we turn our attention to the case of the weighted Bergman space A pω , which is themixed norm space with p “ q . If ω p z q “ p ´ | z |q α for some ´ ă α ă 8 , then the notation A pα is used for A pω . Our first characterizations for inner functions Θ whose derivative belongsto A pω are straightforward consequences of Theorem 2. In addition, we give a generalization ofthe equivalence p a q ô p b q in [11, Theorem 1]. The proof of this result is based on the existenceof approximating Blaschke products [4] and some estimates for } Θ } A pω [22, 23]. The argumentused here is essentially different from that used in [11]. Applying the above-mentioned tools, wecan also prove a characterization which utilizes the so-called Carleson curve Γ ε “ Γ ε p Θ q (in thesense of W. S. Cohn) associated with 0 ă ε ă ε are recalled also inSection 6. Theorem 3.
Let ă p ă 8 , ω P D , Θ be an inner function and t z n p a qu the zero-sequence of Θ a . Moreover, assume either (a) ă p ď and ω satisfies the right-hand inequality of (1.1) for some β ă p ´ , or (b) 1 ă p ă 8 , ω satisfies the right-hand inequality of (1.1) for some β ă p and theleft-hand inequality for some α ą p ´ .Then the following statements are equivalent: (i) Θ P A pω . (ii) There exists a set E Θ Ă D of logarithmic capacity zero such that ÿ n p ω p z n p a qqp ´ | z n p a q|q p ´ ă 8 (1.4) for every a P D z E Θ . (iii) There exists a P D z E Θ such that (1.4) holds. (iv) There exists ă C ă such that ż t z P D : | Θ p z q|ă C u p ω p z qp ´ | z |q p ` dA p z q ă 8 . (v) There exists ă ε ă such that ż Γ ε p ω p z qp ´ | z |q p | dz | ă 8 . As mentioned above, Theorem 3 implies a part of [11, Theorem 1]. In addition, the essentialcontents of classical results [1, Theorem 6.2] and [3, Theorem 3] are consequences of Theorem 3.All of these results are contained in the following corollary.
Corollary 4.
Let ă p ă , Θ be an inner function and t z n p a qu the zero-sequence of Θ a .Then the following statements are equivalent: (i) Θ P H p . ATTE REIJONEN AND TOSHIYUKI SUGAWA (ii) Θ P A p ` α ` α for every ´ ă α ă 8 . (iii) Θ P A p ` α ` α for some ´ ă α ă 8 . (iv) There exists a set E Θ Ă D of logarithmic capacity zero such that ÿ n p ´ | z n p a q|q ´ p ă 8 (1.5) for every a P D z E Θ . (v) There exists a P D z E Θ such that (1.5) holds. (vi) There exists ă C ă such that ż t z P D : | Θ p z q|ă C u dA p z qp ´ | z |q p ` ă 8 . (vii) There exists ă ε ă such that ż Γ ε | dz |p ´ | z |q p ă 8 . The remainder of this note is organized as follows. Some auxiliary results are stated inSection 2. Theorems 1, 2 and 3 are proved in Sections 3, 5 and 6, respectively. Consequencesof Theorem 1 are stated in Section 4, and the proof of Corollary 4 can be found in Section 7.In addition, the last section contains an example and some remarks.2.
Auxiliary results
We begin by stating a sufficient condition for the derivative of a Blaschke product B to bein A p,qω [24]. In addition, it is mentioned that the condition is necessary if the zero-sequenceof B is a finite union of separated sequences. Before this result we recall that a sequence t z n u Ă D is called separated if there exists δ “ δ pt z n uq ą ˇˇˇˇ z n ´ z k ´ z n z k ˇˇˇˇ ą δ, n ‰ k. Lemma A.
Let ă p ă 8 , ă q ă 8 , ω P D and B be the Blaschke product with zeros t z j u . If either (a) ă p ď and ω satisfies the right-hand inequality of (1.1) for some β ă q ´ qp , or (b) 1 ă p ă 8 , ω satisfies the right-hand inequality of (1.1) for some β ă q and theleft-hand inequality for some α ą q ´ qp ,then } B } qA p,qω À ÿ n p ω p r n q υ q { pn p ´ r n q q ´ q { p , where υ n “ t j : r n ď | z j | ă r n ` u . If, in addition to (a) or (b) , t z j u is a finite union ofseparated sequences, then } B } qA p,qω — ÿ n p ω p r n q υ q { pn p ´ r n q q ´ q { p . We say that a weight ω belongs to p D p for 0 ă p ă 8 ifsup ă r ă p ´ r q p p ω p r q ż r ω p s qp ´ s q p ds ă 8 , and ω P q D p if sup ă r ă p ´ r q p p ω p r q ż r ω p s qp ´ s q p ds ă 8 . It is worth noting that (a) and (b) in Lemma A for p ě q can be replaced by the followingconditions respectively:(A) ă p ď ω P p D q ´ q { p , HARACTERIZATIONS FOR INNER FUNCTIONS 5 (B) 1 ă p ă 8 and ω P p D q X q D q ´ q { p .This observation is relevant because conditions (a) and (b) imply (A) and (B), respectively.More precisely, if the right-hand inequality of (1.1) is satisfied for some β “ β p ω q ă p , then ω P p D p . Similarly, if the left-hand inequality is satisfied for some α “ α p ω q ą p , then ω P q D p .The validity of these implications can be checked by straightforward calculations based onintegration by parts; see [24] for details. In addition, we recall that ω P p D if and only if ω P p D p for some p [20].The next auxiliary result shows that, for ω P D X p D q and an inner function Θ, we may usethe Schwarz-Pick lemma inside the norm } Θ } A p,qω without losing any essential information [25].In addition, we give some modified asymptotic estimates for } Θ } A p,qω , which are consequencesof the following fact [25]: For 0 ă p, q ă 8 and ω P D , } f } qA p,qω — ż M qp p r, f q p ω p r q ´ r dr for all f P H p D q . Lemma B.
Let ă p, q ă 8 , ω P D X p D q and Θ be an inner function. Then } Θ } qA p,qω — ż ˆż π ˆ ´ | Θ p re iθ q| ´ r ˙ p dθ ˙ q { p ω p r q dr — ż ˆż π ˆ ´ | Θ p re iθ q| ´ r ˙ p dθ ˙ q { p p ω p r q ´ r dr — ż ˆż π | Θ p re iθ q| p dθ ˙ q { p p ω p r q ´ r dr. We close this section by recalling that the counterparts of Lemmas A and B for A pω wereoriginally proved in [22, 23]. In addition, we note that Lemma B and the first part of Lemma Afor p “ q are valid also if the hypothesis ω P D is replaced by ω P p D .3. Proof of Theorem 1
Let us begin by proving a modification of [14, Lemma 4.6].
Lemma 5.
Let ă p ă 8 and ă δ ă . Then there exists C “ C p p, δ q ą such that ż D δ ˆ log ˇˇˇˇ ´ azz ´ a ˇˇˇˇ˙ p dA p a q ď C p ´ | z |q p , z P D . Proof. If | z | ă p ` δ q{
2, then the assertion follows by observing that there exists a constant M “ M p p q ą ż D δ ˆ log ˇˇˇˇ ´ azz ´ a ˇˇˇˇ˙ p dA p a q ă M ă 8 . Hence we may assume | z | ě p ` δ q{
2. Since ˇˇˇˇ ´ azz ´ a ˇˇˇˇ “ p ´ | z | qp ´ | a | q| z ´ a | ` , we have log ˇˇˇˇ ´ azz ´ a ˇˇˇˇ ď log ` p ´ δ q ´ p ´ | z |q ` ˘ ď p ´ δ q ´ p ´ | z |q , a P D δ . Consequently, the assertion follows. (cid:3)
ATTE REIJONEN AND TOSHIYUKI SUGAWA
For x P R and a weight ω , we set ω x p z q “ ω p z qp ´ | z |q x for all z P D . If 0 ă x ă 8 and ω P D , then x ω x p z q — p ω p z qp ´ | z |q x , z P D ; (3.1)see the proof of [23, Corollary 7]. It follows that ˆ ´ r ´ s ˙ α ` x x ω x p s q À x ω x p r q À ˆ ´ r ´ s ˙ β ` x x ω x p s q , ď r ď s ă , (3.2)where α and β are from (1.1). With these preparations we are ready to prove Theorem 1. Proof of Theorem 1.
Since Θ a p z q “ Θ p z q ´ | a | p ´ a Θ p z qq , (3.3)we obtain | Θ p z q| — | Θ a p z q| for z P D and a P D δ . Hence Lemma A yields } Θ } qA p,qω — ż D δ } Θ a } qA p,qω dA p a q À ÿ n p ω p r n qp ´ r n q q ´ q { p ż D δ υ n p a q q { p dA p a q ;and consequently, the upper bound for } Θ } A p,qω is proved.Let 1 ď p ă 8 and 0 ă q ă 8 . Sincelog 1 | Θ a p z q| ě ÿ n p ´ | z | qp ´ | z n p a q| q| ´ z n p a q z | , z P D , by [8, Chapter 7, Lemma 1.2], the super-additivity of x p for 0 ă x ď υ n p a qp ´ r n q À ż π ˆ log 1 | Θ a p re iθ q| ˙ p dθ, r n ď r ă r n ` , (3.4)as observed in [14, Corollary 4.5]. Using (3.4) together with the hypothesis ω P p D , we obtain S : “ ÿ n p ω p r n qp ´ r n q q ´ q { p ż D δ υ n p a q q { p dA p a q— ÿ n ż r n ` r n p ω p r qp ´ r q q ` ż D δ υ n p a q q { p p ´ r n q q { p dA p a q dr À ÿ n ż r n ` r n p ω p r qp ´ r q q ` ż D δ ˆż π ˆ log 1 | Θ a p re iθ q| ˙ p dθ ˙ q { p dA p a q dr. (3.5)If p ă q , then (3.5), Minkowski’s inequality [12, Theorem 202] and Lemma 5 for z “ Θ p re iθ q yield S À ÿ n ż r n ` r n p ω p r qp ´ r q q ` ˜ż π ˆż D δ ˆ log 1 | Θ a p re iθ q| ˙ q dA p a q ˙ p { q dθ ¸ q { p dr À ż ˆż π ˆ ´ | Θ p re iθ q| ´ r ˙ p dθ ˙ q { p p ω p r q ´ r dr. For p ě q , we use (3.5), H¨older’s inequality and Lemma 5 to obtain S À ÿ n ż r n ` r n p ω p r qp ´ r q q ` ˆż π ż D δ ˆ log 1 | Θ a p re iθ q| ˙ p dA p a q dθ ˙ q { p dr À ż ˆż π ˆ ´ | Θ p re iθ q| ´ r ˙ p dθ ˙ q { p p ω p r q ´ r dr. Finally the lower bound of } Θ } A p,qω for 1 ď p ă 8 follows from these inequalities and Lemma B.Thus we have shown (1.2) when p ě HARACTERIZATIONS FOR INNER FUNCTIONS 7
Let ă p ă ă q ă 8 . Put x “ q { p ´ q and assume, by the hypotheses ω P D and(a), α ` x ą β ` x ă q { p . Finally asymptotic equation (3.1), (1.2) with p and q beingreplaced by 1 and q { p, respectively, and the Schwarz-Pick lemma yield S — ÿ n x ω x p r n q ż D δ υ n p a q q { p dA p a q — } Θ } q { pA ,q { pωx ď } Θ } qA p,qω . This completes the proof. l Note that the proof of the lower bound } Θ } qA p,qω Á ÿ n p ω p r n qp ´ r n q q ´ q { p ż D δ υ n p a q q { p dA p a q relies on Lemma B, not Lemma A. In particular, this means that for the lower bound it sufficesto assume only the hypotheses of Lemma B.4. Consequences of Theorem 1
The first consequence of Theorem 1 asserts that the derivative of an inner function Θ belongsto A p,qω if and only if Θ P A p ` xp { q,q ` xω x for every/some 0 ă x ă 8 . Note that this result wasoriginally proved in [24]. The argument there relies on the existence of approximating Blaschkeproducts [4], unlike the proof here. Corollary 6.
Let ă p ă 8 and ă q, x ă 8 . Assume that ω and Θ are as in Theorem 1.Then } Θ } qA p,qω — } Θ } q ` xA p ` xp { q,q ` xωx . Proof.
Let p “ p ` xp { q and q “ q ` x . Then, by (3.1), we have ÿ n p ω p r n qp ´ r n q q ´ q { p ż D δ υ n p a q q { p dA p a q — ÿ n x ω x p r n qp ´ r n q q ´ q { p ż D δ υ n p a q q { p dA p a q , where υ n p a q is as in Theorem 1. Hence the assertion follows from Theorem 1 by showing thatone of the following conditions holds:(i) ă p ď ω x satisfies the right-hand inequality of (3.2) for some β ` x ă q ´ q p and the left-hand inequality for some α ` x ą ă p ă 8 , ω x satisfies the right-hand inequality of (3.2) for some β ` x ă q and theleft-hand inequality for some α ` x ą q ´ q p .The validity of (i) or (ii) can be checked by considering the cases ă p ď
1, 1 ă p ď ` xp { q and p ą ` xp { q separately; see [24] for details. Consequently, the proof is complete. (cid:3) Next we turn our attention to the Besov space. For 0 ă α ă 8 and an analytic function f p z q “ ř n a n z n , the fractional derivative of order α is defined by D α f p z q “ ÿ n p n ` q α a n z n , z P D . Note that, for f P H p D q , n P N and 0 ă p ă 8 , we have M p p r, f p n q q — M p p r, D n f q withcomparison constants independent of r [7]. For 0 ă p, q ă 8 and 0 ď α ă 8 , the Besov space B p,qα consists of those f P H p D q such that } f } qB p,qα “ ż M qp p r, D ` α f qp ´ r q q ´ dr ă 8 . Corollary 7.
Let Θ be an inner function, ă δ ă and ă p, q, α ă 8 be such that max t , p ´ u ă α ă p . Then Θ P B p,qα if and only if ÿ n p ´ r n q q { p ´ αq ż D δ υ n p a q q { p dA p a q ă 8 , (4.1) where υ n p a q as in Theorem 1. ATTE REIJONEN AND TOSHIYUKI SUGAWA
Set K δ “ t z P C : δ ď | z | ď ´ δ u for 0 ă δ ă , and recall that [9, Theorem 3.3] is acorresponding result where (4.1) is replaced by the condition ż K δ ˜ÿ n p ´ r n q q { p ´ αq υ n p a q q { p ¸ p { q dA p a q ă 8 . One could say that Corollary 7 is a streamlined version of [9, Theorem 3.3], or a generalizationof the main result of [15]. Before the proof we underline that our argument for α ě Proof of Corollary 7.
Let 0 ă α ă
1. Then Theorem 1 together with [7, Theorem 6] yields } Θ } qB p,qα — ż M qp p r, D Θ qp ´ r q p ´ α q q ´ dr — } Θ } qA p,q p ´ α q q ´ — ÿ n p ´ r n q q { p ´ αq ż D δ υ n p a q q { p dA p a q . Note that for the last asymptotic equation it suffices to check that ω p z q “ p ´ | z |q p ´ α q q ´ satisfies the hypotheses of Theorem 1. This gives the assertion for 0 ă α ă ď α ă 8 and α ă t ă 8 . By [9, Lemma 3.4], we know B p,qα Ă B pt,qtα { t . Moreover, [9,Corollary 3.6] for t “ { t implies Θ P B p,qα if Θ P B pt,qtα { t for α ą p ´
1. Applying these factstogether with the previous case, it is easy to verify the assertion for α ě
1. This completesthe proof. l Proof of Theorem 2
Let us begin by stating an auxiliary result, which can be proved in a similar manner asLemma 5.
Lemma 8.
Let ă δ ă and σ be a probability measure supported in D δ and satisfying sup z P D ż D δ log ˇˇˇˇ ´ azz ´ a ˇˇˇˇ dσ p a q “ M ă 8 . (5.1) Then there exists C “ C p δ, M q ą such that ż D δ log ˇˇˇˇ ´ azz ´ a ˇˇˇˇ dσ p a q ď C p ´ | z |q , z P D . Before the proof of Theorem 2, we recall that a compact set K Ă D has a positive logarithmic(inner) capacity if there exits a non-zero probability measure σ supported in K and satisfying(5.1). For details, see Section 12 as well as Section 2 in [26, Chapter III]. Proof of Theorem 2.
If (1.3) holds for some a P D z E Θ , then Θ P A p,qω by Lemma A and(3.3). Consequently, condition (iii) implies (i). Moreover, since p D z E Θ q X p D z E Θ q ‰ H , theimplication p ii q ñ p iii q is clear. Hence it suffices to show p i q ñ p ii q .Assume Θ P A p,qω for some p ě
1, let 0 ă δ ă σ be a probability measure supportedin D δ and satisfying (5.1). For condition (ii) it suffices to prove I : “ ż D δ ÿ n p ω p r n q υ n p a q q { p p ´ r n q q ´ q { p dσ p a q ă 8 (5.2)because then σ ˜ a P D δ : ÿ n p ω p r n q υ n p a q q { p p ´ r n q q ´ q { p “ 8 +¸ “ . HARACTERIZATIONS FOR INNER FUNCTIONS 9
Using (3.4) for p “
1, H¨older’s inequality together with the hypothesis q ď p and Lemma 8for z “ Θ p re iθ q , we obtain I — ÿ n ż r n ` r n p ω p r qp ´ r q q ` ż D δ υ n p a q q { p p ´ r n q q { p dσ p a q dr À ÿ n ż r n ` r n p ω p r qp ´ r q q ` ż D δ ˆż π log ˆ | Θ a p re iθ q| ˙ dθ ˙ q { p dσ p a q dr ď ż p ω p r qp ´ r q q ` ˆż π ż D δ log ˆ | Θ a p re iθ q| ˙ dσ p a q dθ ˙ q { p dr À ż ˆż π ´ | Θ p re iθ q| ´ r dθ ˙ q { p p ω p r qp ´ r q q ` ´ q { p dr ď ż ˆż π ´ | Θ p re iθ q| ´ r dθ ˙ q { p x ω x p r q ´ r dr, (5.3)where x “ q { p ´ q .Next we verify some properties for ω x . By the second part of hypothesis (b) and its conse-quence ω P q D ´ x , we find α “ α p ω q ą ´ x such that x ω x p s qp ´ s q α ` x À p ω p s qp ´ s q α À p ω p r qp ´ r q α ď x ω x p r qp ´ r q α ` x , ď r ď s ă , and p ω p t qp ´ t q x À α p ω ˆ ˙ p ´ t q α ` x ÝÑ ` , t Ñ ´ . Consequently, an integration by parts together with hypothesis (b) gives ˆ ´ r ´ s ˙ α ` x x ω x p s q À x ω x p r q “ p ω p r qp ´ r q x ´ x ż r p ω p t qp ´ t q α p ´ t q α ` x ´ dt À p ω p r qp ´ r q x À ˆ ´ r ´ s ˙ β ` x x ω x p s q , ď r ď s ă , for some α “ α p ω q ą ´ x and β “ β p ω q ă q .Finally (5.3), Lemma B and Corollary 6 for p “ q “ q { p and x “ ´ x yield I À } Θ } q { pA ,q { pωx — } Θ } qA p,qω ă 8 . Hence estimate (5.2) is satisfied for p ě
1. Since the case ă p ă l It is worth noting that we can slightly weaken the hypotheses for ω in Theorem 2: Condition(a) can be replaced by the hypothesis ω P p D q ´ q { p , and the first part of (b) by ω P p D q . This isdue to the alternative version of Lemma A for q ď p , mentioned in Section 2. More precisely,we have to first prove a modification of Corollary 6 for q ď p , and then apply this result inthe proof. 6. Proof of Theorem 3
Recall that a sequence t z n u Ă D is said to be uniformly separated ifinf n P N ź k ‰ n ˇˇˇˇ z k ´ z n ´ z k z n ˇˇˇˇ ą . By [4, Theorem 2.1], for every inner function Θ, there exists a Blaschke product B Θ withuniformly separated zeros t z n u such that 1 ´ | Θ p z q| — ´ | B Θ p z q| for all z P D . B Θ is calledan approximating Blaschke product of Θ. Using the existence of approximating Blaschke products together with our auxiliary results, we prove the following proposition which impliesthe equivalence p i q ô p iv q in Theorem 3. Proposition 9.
Let ă p ă 8 , ω P D and Θ be an inner function. Moreover, assumeeither ă p ď and ω P p D p ´ , or ă p ă 8 and ω P p D p X q D p ´ . Then there exists C “ C p Θ q P p , q such that } Θ } pA pω — I C : “ ż t z P D : | Θ p z q|ă C u p ω p z qp ´ | z |q p ` dA p z q , where the comparison constants may depend on p , ω , Θ and C .Proof. For any 0 ă C ă
1, Lemma B yields } Θ } pA pω — ż D ˆ ´ | Θ p z q| ´ | z | ˙ p p ω p z q ´ | z | dA p z q ě p ´ C q p I C . Hence the lower bound for } Θ } A pω is proved.Let B Θ be an approximating Blaschke product of Θ with zeros t z n u . Since t z n u is (uni-formly) separated, we find 0 ă δ ă p z n q “ t z : | z n ´ z | ă δ p ´ | z n |qu arepairwise disjoint. Hence, using Lemma B and [23, Theorem 1] together with the hypothesesfor ω , we obtain } Θ } pA pω — ż D ˆ ´ | Θ p z q| ´ | z | ˙ p ω p z q dA p z q — ż D ˆ ´ | B Θ p z q| ´ | z | ˙ p ω p z q dA p z q— } B Θ } pA pω — ÿ n p ω p z n qp ´ | z n |q p ´ — ÿ n ż ∆ p z n q dA p z q p ω p z n qp ´ | z n |q p ` — ÿ n ż ∆ p z n q p ω p z qp ´ | z |q p ` dA p z q “ ż Ť n ∆ p z n q p ω p z qp ´ | z |q p ` dA p z q . Consequently, it suffices to find constants C and D such that 0 ă C, D ă ď n ∆ p z n q Ă t z P D : | B Θ p z q| ă D u Ă t z P D : | Θ p z q| ă C u . Since | B Θ p z q| ď | z n ´ z || ´ z n z | ď | z n ´ z | ´ | z n | ă δ, z P ∆ p z n q , the first inclusion is valid for D “ δ . If | B Θ p z q| ă D for some 0 ă D ă
1, then we find M “ M p Θ q ă ´ D such that | Θ p z q| ď ´ M p ´ | B Θ p z q|q ă ´ M p ´ D q , z P D . Thus the second inclusion is proved and the assertion follows. (cid:3)
Recall that the Carleson curve Γ ε Ă D associated with 0 ă ε ă ε “ ε p ε q P p , ε q such that ε ă | Θ p z q| ă ε for z P Γ ε X D .(2) Γ ε X D is a countable union of arcs I n with pairwise disjoint interiors such that ‚ each I n is either a radial segment or part of a circle; ‚ the end points a n and b n of I n satisfy δ ď ˇˇˇˇ a n ´ b n ´ a n b n ˇˇˇˇ ď δ for all n and some fixed δ , δ P p , q .(3) If t z n u is the sequence of the middle points of I n , then the Blaschke product B Θ withzeros t z n u is an approximating Blaschke product of Θ. HARACTERIZATIONS FOR INNER FUNCTIONS 11
Now we are ready to prove Theorem 3.
Proof of Theorem 3.
By Proposition 9, the equivalence p i q ô p iv q is valid. Assume withoutloss of generality that t z n p a qu is ordered by increasing moduli, and enumerate it such that,for all k , r j ď | z jk p a q| ă r j ` , j “ , , . . . , and t z jk p a qu is ordered by increasing moduli with k . Then the hypothesis ω P p D yields ÿ j p ω p r j q υ j p a qp ´ r j q p ´ — ÿ j ÿ k p ω p z jk p a qqp ´ | z jk p a q|q p ´ “ ÿ n p ω p z n p a qqp ´ | z n p a q|q p ´ , a P D , where υ j p a q is as in Theorem 1. Consequently, p i q ô p ii q ô p iii q by Theorem 2. Hence itsuffices to prove p i q ô p v q .By the hypotheses of ω , we know that p ω p r q{p ´ r q p is essentially increasing with r . Usingthis fact and condition (2) of Γ ε , we obtain ż Γ ε p ω p z qp ´ | z |q p | dz | “ ÿ n ż I n p ω p z qp ´ | z |q p | dz | À ÿ n | I n | p ω p ξ n qp ´ | ξ n |q p , where ξ n is the supremum of I n in the sense of absolute value. Since, for all n , | I n | — | a n ´ b n | — ´ | z n | and p ω p ξ n q — p ω p z n q by condition (2) and the hypothesis ω P p D , we obtain ż Γ ε p ω p z qp ´ | z |q p | dz | À ÿ n p ω p z n qp ´ | z n |q p ´ . In a similar manner, one can also verify the asymptotic equation Á . Consequently, condition(3) together with Lemmas A and B yields ż Γ ε p ω p z qp ´ | z |q p | dz | — ÿ n p ω p z n qp ´ | z n |q p ´ — } B Θ } pA pω — } Θ } pA pω . This means that p i q ô p v q and the proof is complete. l Proof of Corollary 4, example and remarks
Let us begin with the proof of Corollary 4.
Proof of Corollary 4.
Let B Θ be an approximation Blaschke product of Θ with zeros t z n u .Using this fact together with [22, Theorem 2], [3, Theorem 1], Lemmas A and B, we obtain } Θ } pH p — sup ď r ă ż π ˆ ´ | Θ p re iθ q| ´ r ˙ p dθ — sup ď r ă ż π ˆ ´ | B Θ p re iθ q| ´ r ˙ p dθ — } B Θ } pH p — ÿ n p ´ | z n |q ´ p — } B Θ } p ` α ` A p ` α ` α — } Θ } p ` α ` A p ` α ` α for every/some ´ ă α ă 8 . Hence the equivalences p i q ô p ii q ô p iii q are valid. Moreover,Theorem 3 gives p iii q ô p iv q ô p v q ô p vi q ô p vii q . This completes the proof.Next we give a concrete example in which we use Theorem 3. Example 10.
Let us consider the atomic singular inner function S p z q “ exp ˆ z ` z ´ ˙ , z P D . Let t z n p a qu be the zero-sequence of the Frostman shift S a of S , assume a P D zt u and set ´ π ă arg a ď π . Solving the equation S p z q “ exp ˆ z ` z ´ ˙ “ a, we can present zeros z n p a q in the form z n p a q “ c n ` c n ´ , where c n “ log | a | ` i p πn ` arg a q , for n P Z . It follows that1 ´ | z n p a q| “ | c n ´ | ´ | c n ` | | c n ´ | “ ´ c n | c n ´ | “ ´ | a || c n ´ | — | n | ´ , | n | Ñ 8 . In particular, for α P R , ÿ n p ´ | z n p a q|q α ă 8 if and only if α ą . (7.1)Hence, as a consequence of Theorem 3 and the nesting property A pα Ă A pα for ´ ă α ď α ă 8 , we obtain the following result: For ă p ă 8 and ´ ă α ă 8 , the derivative of S belongs to A pα if and only if α ą p ´ . This result originates to [16]; see also [18]. However,the argument used here is essentially different from that used in these references.By [10, Example 2], the Frostman shift S a for any a P D zt u is a Blaschke product withuniformly separated zeros. Applying this fact together with (7.1), the above-mentioned resultfollows also from Lemma A.We close this note with the following remarks, which indicate two open questions.(I) A modification of Corollary 7 for p ě q can be obtained in a similar manner as thecurrent version using Theorem 2 instead of Theorem 1. More precisely, the counterpartof (4.1) takes the form ÿ n p ´ r n q q { p ´ αq υ n p a q q { p ă 8 , (7.2)where a P D z E Θ . In addition, if Θ belongs to B p,qα with the given restrictions, thenthere exists a set E Θ Ă D of logarithmic capacity zero such that (7.2) holds for every a P D z E Θ .Applying the above-mentioned result together with Corollary 4, one can show that,for ă p ă 8 , the derivative of an inner function Θ belongs to H p if and onlyif Θ P A pp ´ . Note that for p ě ă p ď is an open question. However, since t f : f P A pp ´ u Ă H p , ă p ď , by [28, Lemma 1.4], another implication is trivially valid also for 0 ă p ď .(II) Corollary 4 contains several ways to characterize those inner functions Θ whose de-rivative belongs to H p for some ă p ă
1. Nevertheless, it does not contain animportant characterization given in [11, Theorem 1]: For ă p ă ă η ă 8 ,the derivative of an inner function Θ belongs to H p if and only if Θ is a Blaschkeproduct whose zero-sequence t z n u satisfies the condition ż π ¨˝ ÿ z n P Γ η p e iθ q ´ | z n | ˛‚ p dθ ă 8 , where Γ η p e iθ q “ t z P D : | z ´ e iθ | ď η p ´ | z |qu . For instance using Corollary 4, we may replace H p in the above-mentioned resultby A p ` α ` α , where ´ ă α ă 8 . Even so any corresponding result for general A pα has HARACTERIZATIONS FOR INNER FUNCTIONS 13 not been verified, and proving such result seems to be laborious. A reason for this isthe fact that the argument of [11, Theorem 1] utilizes the well-known identity } B } pH p “ π ż π ˜ÿ n ´ | z n | | z n ´ e iθ | ¸ p dθ, ă p ă 8 , where B is the Blaschke product with zeros t z n u [2]; and we do not have a similarresult for A pα . Acknowledgements.
The authors thank Janne Gr¨ohn for valuable comments, and thereferees for careful reading of the manuscript.
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