A Characterization of One-component Inner Functions

We present a characterization of one-component inner functions in terms of the location of their zeros and their associated singular measure. As consequence we answer several questions posed by J. Cima and R. Mortini. In particular we prove that for any inner function $\Theta$ whose singular set has measure zero, one can find a Blaschke product $B$ such that $\Theta B$ is one-component. We also obtain a characterization of one-component singular inner functions which is used to produce examples of discrete and continuous one-component singular inner functions.


Introduction and main results
Let D be the open unit disc of the complex plane and let BD be the unit circle. An inner function is a bounded analytic function in D having unimodular radial limits almost everywhere on BD. It is a classical result that any inner function can be factorized as the product of a Blaschke product, a singular inner function and unimodular constant ( [14]). Recall that, for a given sequence tz n u Ă D satisfying ř n p1´|z n |q ă 8, the Blaschke product with zeros tz n u is defined by Bpzq " ź n |z n | z n z n´z 1´z n z , z P D.
Here each zero is repeated according to its multiplicity and the convention |z n |{z n " 1 is used when z n " 0. A singular inner function is an inner function of the form where σ is a positive measure on BD, singular with respect to the Lebesgue measure. The singular set of an inner function Θ, which will be denoted by sing Θ, consists of all points on BD in which Θ does not have an analytic continuation. If Θ factors as Θ " λBS, where |λ| " 1, B is a Blaschke product and S is the singular inner function associated to the singular measure σ, then sing Θ is precisely the union of the accumulation points of zeros of Θ and the closed support of the measure σ. See Chapter II of [14]. We focus on so-called one-component inner functions introduced by B. Cohn in [12], which are inner functions Θ whose level set tz P D : |Θpzq| ă εu is connected for some 0 ă ε ă 1. For simplicity, we denote by I c the set of all one-component inner functions. The main motivation to study I c comes from the theory of model spaces K p Θ " H p X zΘH p , 1 ă p ă 8, generated by the inner function Θ. For instance, B. Cohn characterized Carleson measures for K 2 Θ when Θ P I c in terms of their action on reproducing kernels ( [12]), and then S. Treil and A. Volberg generalized Cohn's result to all p P p1, 8q ( [25]). It is also worth mentioning that N. Nazarov and A. Volberg proved that Cohn's result does not hold for arbitrary inner functions. See [20]. The class I c also appears naturally in several recent results in the context of operator theory in K 2 Θ [5,6,7,8]. A. Aleksandrov obtained a series of nice descriptions of inner functions in I c in terms of the behaviour of their derivatives ( [3]). As a byproduct he proved also a strong form of the Schwarz-Pick lemma for inner functions in I c . Using Aleksandrov's descriptions, J. Cima, R. Mortini and the second author constructed some concrete examples of one-component inner functions [10,11,23]. In particular singular inner functions associated to a finite sum of weighted Dirac masses are one-component and thin Blaschke products are not in I c . A Blaschke product B whose zeros tz n u 8 n"1 (ordered by non-decreasing moduli) lie in a Stolz angle is one-component if The main motivation of our work is to give descriptions and examples of one-component inner functions in terms of the location of their zeros and their associated singular measures. As it will be explained, our results provide answers to several questions posed by J. Cima and R. Mortini. Let Θ be an inner function which factors as Θ " λBS, where |λ| " 1, B is a Blaschke product with zeros tz n u and S is the singular inner function associated to the singular measure σ. Consider the measure µpΘq on D defined as µpΘq " ÿ n p1´|z n |qδ zn`σ .
Here δ z denotes the Dirac point measure at the point z. We will describe one-component inner functions Θ in terms of the mass given by µpΘq to Carleson squares defined as Qpzq " tw P D : | arg z´arg w| ď p1´|z|q{2, |w| ě |z|u, z P D.
This idea originates in [9] where C. Bishop described inner functions Θ in the little Bloch space in terms of the behaviour of the corresponding measure µpΘq. Similar ideas have been used in [21] and [19]. This result can be proved applying [3, Theorem 1.2] by A. Aleksandrov but in Section 2, we present a self-contained proof relying on Hall's lemma and a stopping time argument. Let us say that an inner function is a finite-component inner function if it has a finitely connected level set. Applying the proof of Theorem 1, we show in Section 2 that all finite-component inner functions belong to I c .
Corollary 2. Let Θ be an inner function. If there exists a constant C with 0 ă C ă 1 such that tz P D : |Θpzq| ă Cu is finitely connected, then Θ P I c .
For 1 ă α ă 8, let Γ α pe iθ q " z P D : |z´e iθ | ă αp1´|z|q ( denote the Stolz angle with vertex at e iθ P BD. As another consequence of Theorem 1, in Section 2, we characterize one-component Blaschke products whose zeros are contained in a Stolz angle. A celebrated result by L. Carleson says that interpolating sequences for the algebra of bounded analytic functions in D are precisely the uniformly separated sequences. A Blaschke product with uniformly separated zeros is called an interpolating Blaschke product. Given a measurable set E Ă BD let |E| denote its Lebesgue measure. One-component inner functions have many special properties. For instance the singular set of an inner function in I c has Lebesgue measure zero. See [4]. Theorem 4 below gives an affirmative answer to the following question posed in [10] by J. Cima and R. Mortini: Can every inner function Θ with | sing Θ| " 0 be multiplied by a one-component inner function B into I c ? In addition, we show that B can be chosen to be an interpolating Blaschke product.
Theorem 4. Let Θ be an inner function whose singular set has Lebesgue measure zero. Then there exists an interpolating Blaschke product B P I c such that BΘ P I c .
Theorem 4 implies also a negative answer to the following question posed in [11] by J. Cima and R. Mortini: Is the singular set of any inner function in I c necessarily countable?
The main effort in the proof of Theorem 4 is to find a suitable Blaschke product B. Once the zeros of B are well located, the assertion can be proved quite easily by using Theorem 1. Roughly speaking, B is chosen such that sing B " sing Θ, zeros of B are close enough to BD (depending on Θ) and form a chain where the pseudohyperbolic distance of adjacent points is fixed. The detailed proof is presented in Section 3.
Recall that an analytic function f in D belongs to the Nevanlinna class N if where log`0 " 0 and log`x " maxt0, log xu for 0 ă x ă 8. Deep results on inner functions whose derivative is in the Nevanlinna class have been recently obtained by O. Ivrii. See [16] and [17]. Applying Theorem 4, we deduce that some one-component inner functions might be bad-behaving in several ways. As an example, we show in Section 3 that I c is not contained in tf : f 1 P N u.
In Section 4 we study one-component singular inner functions. As another application of Theorem 1, we present the following characterization of one-component singular inner functions.
Note that the set Ω in the statement is a sawtooth region, as the following figure shows. Figure 1. The region DzΩ, painted in grey, consists of a union of tents whose bottoms are the complementary arcs of supp σ in BD. Each tent has the same shape, but size depends on the length of the complementary arc.
Using Theorem 6, we show that any singular inner function associated to a Cantor measure in a symmetric Cantor set is one-component. On the other hand, also as an application of Theorem 6, we construct discrete measures whose associated singular inner functions are not one-component. Theorem 6 also implies the following result which provides an affirmative answer to another question of J. Cima and R. Mortini (Question 4.4 i) in [11]). Corollary 7. Let E be a closed countable set of the unit circle. Let σ be a positive singular measure supported on E such that σptξuq ą 0 for any ξ P E. Then the singular inner function associated to σ is one-component.
We finish the paper in Section 4 with an open question.

Proofs of Theorem 1, Corollaries 2 and 3
We first fix some notation. Consider the pseudohyperbolic distance ρpz, wq between the points z, w P D given by ρpz, wq " |pz´wq{p1´zwq|. A sequence of points tz n u Ă D is called separated if there exists δ " δptz n uq P p0, 1q such that ρpz j , z k q ą δ for all distinct j and k. Moreover, we write f À g if there exists an absolute constant C ą 0 such that f ď Cg, while f Á g is understood analogously. If g À f À g, then the notation fg is used.
Proof of Theorem 1. Let us first prove the necessity. Assume contrary that there exists a sequence of points tz n u 8 n"1 Ă D such that µpΘqpQpz n qq ą 0 and |Θpz n q| Ñ 1´as n Ñ 8. Since Θ P I c , we find C P p0, 1q such that the level set Ω " tz P D : |Θpzq| ď Cu is connected. Given a domain A Ă C and a subset E Ă BA, let ωpz, E, Aq be the harmonic measure of E in the domain A, that is, the harmonic function in A whose boundary values are identically 1 on E and identically zero on BAzE. Since Θ is inner, the subharmonicity of log |Θpzq| gives that log |Θpzq| ď plog Cq ωpz, BΩ, DzΩq for any z P DzΩ. Then by Hall's lemma (see [13,Chap. 12,Lemma 4]), there exists an absolute constant C 1 ą 0 such that where pBΩq˚" tz{|z| : z P BΩu is the radial projection of BΩ. Since µpΘqpQpz n qq ą 0 and the diameter of the connected set Ω depends only on Θ and C, we find N P N such that |pBΩq˚X 2Qpz n q| ě p1´|z n |q{2 for n ě N . Consequently, Combining estimates (2.1) and (2.2), we deduce that there exists a positive constant D ă 1 such that |Θpz n q| ď D for all n ě N . This contradiction finishes the proof of the necessity. Next we prove the sufficiency. Pick a constant C 1 P p0, 1q such that p1´C 1 q{p1´Cq is very small. In particular, we assume C 1 ą pC`9{10q{p1`9C{10q. Consider the decomposition of D into dyadic Carleson squares where n ě 2 and 0 ď k ă 2 n . Let T pQ n,k q " tz P Q n,k : |z| ď 1´π2´n´1u denote the top half of Q n,k . Let G " tQ j u be the collection of maximal dyadic Carleson squares such that sup zPT pQ j q |Θpzq| ě C 1 . If w P D satisfies ρpw, T pQ j qq ď 9{10 for some j, then |Θpwq| ě C. This is easy to deduce from the estimate where the first inequality is due to the Schwarz-Pick lemma. Thus our hypothesis implies µpΘqpQpwqq " 0 when w is as above. In particular, µpΘqp2Q j q " 0 for all j.
Let z j be the center of T pQ j q. By the Schwarz-Pick lemma, we find C 2 " C 2 pC 1 q P p0, 1q with C 2 " C 2 pC 1 q Ñ 1´as C 1 Ñ 1´, such that |Θpz j q| ě C 2 for all j. Next we show that there exists a universal constant C 3 ą 0 such that |Θpzq| ě |Θpz j q| C 3 ě C C 3 2 , z P Q j .
Set Ω 1 " tz P D : |Θpzq| ă C 1 u. We show that Ω 1 is connected arguing by contradiction. Let G be the union of all Carleson squares in the family G. First we note that DzG Ă Ω 1 by the construction of G. Since DzG is connected, we find a connected component Ω 2 of Ω 1 with DzG Ă Ω 2 . Assume that Ω 3 is a connected component of Ω 1 satisfying Ω 2 X Ω 3 " H. In particular, Ω 3 Ă G. Hence estimate (2.3) gives By the maximum principle Ω 3 is simply connected and we can consider a conformal mapping ϕ : D Ñ Ω 3 . Then g " C´1 1 Θ˝ϕ is an inner function. This fact was noted in the proof of [ which is a contradiction. Thus Ω 1 is connected and the proof is complete. l Next we prove Corollaries 2 and 3.
Proof of Corollary 2. Assume contrary that Θ is not a one-component inner function. By Theorem 1, there exists a sequence tz j u 8 j"1 Ă D such that µpΘqpQpz n qq ą 0 for every n and |Θpz j q| Ñ 1´as j Ñ 8. By the assumption we have tz P D : |Θpzq| ă Cu " Ť N n"1 Ω n where Ω n Ă D are connected sets. Then Qpz j q X Ω k ‰ H for some k P t1, . . . , N u and all j P N. Applying Hall's lemma, we obtain log |Θpzq| ď log C ωpz, BΩ, DzΩ k q ď log C ωpz, pBΩ k q˚, Dq, z P DzΩ k , where the notation is same as in (2.1). Again arguing as in the proof of Theorem 1, one can find a constant D ă 1 such that |Θpz j q| ď D for all j sufficiently large. Since this is a contradiction, the assertion is proved. l Proof of Corollary 3. We can assume e iθ " 1. Let 0 ă β ă π{2 and let Γ β be a cone in D with aperture 2β and vertex at 1. Assume without loss of generality that the zeros of B are contained in Γ β . Let us first prove the necessity. Assume contrary that exists a sequence tr n u 8 n"1 Ă p0, 1q such that |Bpr n q| Ñ 1´as n Ñ 8. Since µpBqpQpr n qq ą 0 for any n P N, Theorem 1 implies B R I c . This is a contradiction and the necessity is proved.
Next we prove the sufficiency. Let us define Γ γ in a similar way as Γ β , and choose γ " γpβq, β ă γ ă π{2, such that Qpzq is contained in DzΓ β when z P DzΓ γ . Using the Schwarz-Pick lemma together with the assumption lim sup rÑ1´| Bprq| ă 1, we find a constant C " Cpγq ă 1 such that lim sup zPΓγ ,zÑ1´| Bpzq| ď C. By Theorem 1 it suffices to show that µpBqpQpzqq " 0 for z P DzΓ γ . Since this follows from the choice of γ, the proof is complete. l

Proofs of Theorem 4 and Corollary 5
We go directly to the proofs.
Proof of Theorem 4. We first consider a Whitney type decomposition of the open set BDz sing Θ " Ť 8 n"1 I n , where each I n is a closed arc on BD satisfying |I n | -distpI n , sing Θq. Next we construct the Blaschke product B. Fix a sequence tε n u Ă p0, 1q with lim nÑ8 ε n " 0. Let us choose r n P p0, 1q such that, if |z| ě r n and e i arg z P I n , then |Θpzq| ě 1´ε n . Let Γ be the curve containing Ť 8 n"1 tr n ξ : ξ P I n u and the radial segments connecting arcs tr n ξ : ξ P I n u in the natural way. Now locate the zeros ZpBq " tz j u 8 j"1 of B on the curve Γ such that, for each z j , there exist distinct zeros z m , z l satisfying ρpz j , z m q " ρpz j , z l q " 1{10, while other zeros are further away from z j . In other words, place the zeros of B in the curve Γ at each 1{10 pseudohyperbolic units. Let us recall that a sequence tz j u is uniformly separated if and only if tz j u is separated and ř z j PQpzq p1´|z j |q À 1´|z| for all z P D (see [14, Chap. VII, Theorem 1.1]). Using this result together with the fact that the pseudohyperbolic distance of adjacent points in ZpBq is 1{10, one checks that B is an interpolating Blaschke product.
It is easy to check that w satisfies 2|z´w| ą 1´|z| if ρpw, zq ą 1{2. This implies that µpBqpQpzqq " 0 when |Bpzq| ą 12{21. Hence B belongs to I c by Theorem 1. By the choice of Γ, it is obvious that possible zeros of Θ are in Ω. Hence the previous argument also applies to BΘ and again by Theorem 1, BΘ is a one-component inner function. This completes the proof. l Proof of Corollary 5. Let Θ be a Blaschke product whose singular set has measure zero such that Θ 1 R N . See [22,Theorem 4]. By our Theorem 4, we can find a Blaschke product B such that | sing B| " 0 and BΘ P I c . Now it suffices to deduce pBΘq 1 R N by applying the following consequence of [2, Theorem 2 and Corollary 4]: The derivative of a Blaschke product φ belongs to N if and only if the non-tangential limit of φ 1 exists almost everywhere on BD and ż 2π 0 log`˜ÿ n 1´|z n | 2 |e iθ´z n | 2¸d θ ă 8, where tz n u is the zero-sequence of φ. l For 0 ă p ă 8 and´1 ă α ă 8, the Bergman space A p α consists of those analytic functions in D such that where dmpzq is the Lebesgue area measure on D. Note that [23,Theorem 10] Hence using Corollary 5 one can construct one-component inner functions whose derivative does not belong to certain Bergman spaces.

One-component singular inner functions
We begin with another consequence of Theorem 1, which is a slight more general version of Theorem 6 stated in the Introduction. Corollary 8. Let Θ " BS, where B is a Blaschke product with zeros tz n u and S is a singular inner function associated with a non-trivial singular measure σ. Assume that tz n u Ă Ω :" tz P D : 1´|z| ě 2 distpz{|z|, supp σqu. Then Θ P I c if and only if lim sup zPΩ,|z|Ñ1´| Θpzq| ă 1.
Proof. The necessity follows from Theorem 1 since Qpzq X supp σ is non-empty when z P Ω. Hence we only need to prove the sufficiency. By the assumption, there exists a constant C ă 1 such that |Θpzq| ď C for z P Ω. Set K " tz P D : distpz, Ωq ď p1´|z|q{2u. Applying the Schwarz-Pick lemma, we find a constant D ă 1 such that |Θpzq| ď D for z P K. Next we show that Θ is one-component applying Theorem 1. Assume that |Θpzq| ě D. Then z P DzK, and it follows that Qpzq X Ω " H, which implies µpΘqpQpzqq " 0. Consequently, Θ is a one-component inner function.
Note that the necessity in Corollary 3 or 8 follows also from [4, Lemma 6.1] by A. Aleksandrov, and this Aleksandrov's result originates from the proof of [25, Theorem 3] by S. Treil and A. Volberg.
Proof of Corollary 7. Let E " tξ n : n " 1, 2, . . .u. Then σ " ř 8 n"1 α n δ ξn , where a n ą 0. Corollary 8 and [14, Chap. II, Theorem 6.2] show that the singular inner function associated to σ is one-component. l The following example shows that even quite basic singular inner functions may lie out of I c .
Next we will show that the singular inner function associated to the Cantor measure on a symmetric Cantor set, is one-component. Let us recall the construction of symmetric Cantor measures σ associated with a sequence tδ n u, which were studied for instance in [1] by P. Ahern.
‚ Let tδ n u 8 n"0 be a strictly decreasing sequence such that δ 0 " 2π and lim nÑ8 δ n " 0. ‚ Set E 0 " r0, 2πs and n P N. Define E n inductively as follows: E n consists of 2 n pairwise disjoint intervals each of length 2´nδ n and E n`1 is obtained (from E n ) by removing a segment from each interval of E n . Write E " Ş 8 n"0 E n . ‚ Define the non-decreasing function ϕ : r0, 2πs Ñ r0, 1s as follows: ϕp0q " 0, ϕp2πq " 1, ϕ is a constant on each interval of r0, 2πszE and ϕ increases by an amount of 2´n on each intervals of E n . ‚ For 0 ď a ď b ď 2π, define the measure σ by σppa, bqq " ϕpbq´ϕpaq.
For instance E is the Cantor middle third set if δ n " 2πp2{3q n for all n P N.
Corollary 10. If S is a singular inner function associated with a symmetric Cantor measure, then S P I c .
Proof of Corollary 10. Let tδ n u be the sequence associated with the symmetric Cantor measure σ that induces S, and let Ω be as in Corollary 8. Assume that z P Ω and 2´nδ n ď 1| z| ď 2´p n´1q δ n´1 for some n P Nzt1, 2u. Then 4Qpzq contains an interval of E n from the construction of σ. Hence P rσspzq " Consequently, we obtain lim zPΩ,|z|Ñ1´| Spzq| " lim zPΩ,|z|Ñ1´e xp p´P rσspzqq " 0, and the assertion follows from Corollary 8. l Next we present another consequence of Corollary 8.
Corollary 11. Let σ be a positive singular measure in the unit circle and let S be the corresponding singular inner function. Assume that there exists a constant δ ą 0 such that for any point ξ in the closed support of σ we have lim inf hÑ0`σ ptψ P BD : |ψ´ξ| ă huq h ą δ.
Then S is a one-component inner function.
Proof of Corollary 11. Given a point z P Dzt0u let Ipzq be the arc centered at z{|z| of length 1´|z|. Note that there exists an absolute constant C ą 0 such that for any z P Dzt0u, we have ż 2π 0 1´|z| 2 |z´e iθ | 2 dσpθq ě C σpIpzqq 1´|z| .
Since the previous integral is´log |Sprξq|, Corollary 8 implies that S is one-component. l We finish the paper with an open question. Level sets of bounded analytic functions are related to the original proof of the Corona Theorem. It is well known that there exists a bounded analytic function having all level sets of infinite length. More concretely P. Jones constructed an analytic function Θ from the unit disc into itself such that for any 0 ă c ă 1, the level set tz P D : |Θpzq| " cu has infinite length. See [18]. We say that an inner function Θ has property pAq if there exist an inner function B and a constant 0 ă c ă 1 such that arc length on the level set tz P D : |ΘpzqBpzq| " cu is a Carleson measure. Roughly speaking, an inner function has property pAq if by adding more zeros, one can produce a nice level set.
Question. Does property pAq hold for any inner function?
B. Cohn proved that for any one-component inner function Θ, there exists 0 ă c ă 1 such that the arc length of the level set tz P D : |Θpzq| " cu is a Carleson measure ( [12]). Hence our Theorem 4 gives that any inner function whose singular set has measure zero, has property pAq. It is likely that using the techniques in [15] one could factor any inner function Θ into a finite number of inner functions Θ i having property pAq.