On the Hardy number of Koenigs domains
Abstract
This work studies the Hardy number of hyperbolic planar domains satisfying Abel’s inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that the Hardy number of a Koenings domains whose complement is non-polar is greater than or equal to 1/2, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.
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Analysis and Mathematical Physics (2024) 14:119 https://doi.org/10.1007/s13324-024-00981-4 On the Hardy number of Koenigs domains Manuel D. Contreras1·Francisco J. Cruz-Zamorano1·Maria Kourou2· Luis Rodríguez-Piazza3 Received: 25 June 2024 / Revised: 22 September 2024 / Accepted: 4 October 2024 / Published online: 22 October 2024 © The Author(s) 2024 Abstract This work studies the Hardy number of hyperbolic planar domains satisfying Abel’s inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that the Hardy number of a Koenings domains whose complement is non-polar is greater than or equal to 1/2, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case. Keywords Hardy spaces ·Abel’s equation ·Koenigs domain ·Iteration in the unit disc ·Koenigs map M. D. Contreras, F. J. Cruz-Zamorano and L. Rodríguez-Piazza are partially supported by Ministerio de Innovación y Ciencia, Spain, project PID2022-136320NB-I00, and Junta de Andalucía, project P20_00664. F. J. Cruz-Zamorano is also partially supported by Ministerio de Universidades, Spain, through the action Ayuda del Programa de Formación de Profesorado Universitario, reference FPU21/00258. M. Kourou is partially supported by the Alexander von Humboldt Foundation. BManuel D. Contreras [email protected] Francisco J. Cruz-Zamorano [email protected] Maria Kourou [email protected] Luis Rodríguez-Piazza [email protected] 1Departamento de Matemática Aplicada II and IMUS, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n, 41092 Sevilla, Spain 2Julius-Maximilians-Universität Würzburg, Institut für Mathematik, Emil Fischer Straße 40, 97074 Würzburg, Germany 3Departmento de Análisis Matemático and IMUS, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia, s/n, 41012 Sevilla, Spain
119 Page 2 of 21 M. D. Contreras et al. Mathematics Subject Classification Primary 30D05 ·30H10 ·30C85; Secondary 39B32 ·37F99 1 Introduction Through the Hardy spaces on the unit disc, Hp(D), one can define the Hardy number of a holomorphic function f∈Hol(D,C), given by h(f):= sup {p>0:f∈Hp(D)}∪{0}∈[0,+∞], which somehow measures the “growth” of f. A similar idea was introduced by Hansen [9] in order to examine the range of holomorphic functions taking values in a domain . The so-called Hardy number of a domain is defined as h() := inf{h(f):f∈Hol(D,)}. The study of the Hardy number of a domain focuses on the case when is unbounded, since h() =+∞if is bounded. A classical problem has been to determine the range of the Hardy number of a hyperbolic planar domain (i.e., those whose boundary contains at least two points) in terms of its geometry and boundary behavior. As a matter of fact, several works have contributed to estimates on the Hardy number of certain classes of hyperbolic domains. Hansen [9,10] provided a characterization for the Hardy number of starlike and spirallike with respect to the origin domains. A few years later, Essén [7] obtained estimates of the Hardy number of a general hyperbolic domain in terms of harmonic measures and the logarithmic capacity. Thereafter, Kim and Sugawa [14] examined the Hardy number of unbounded K-quasidisks. Quite recently, the range of the Hardy number of comb domains was studied by Karafyllia [13]. The stepping stone, however, for the current work is the paper by Poggi-Corradini [16], who examined the Hardy number of hyperbolic domains satisfying Schröder’s inclusion property; i.e. λ ⊆,forsomeλ∈D. Inspired by the impact of Schröder’s and Abel’s functional equations on the discrete dynamics of the unit disc D, our main focus lies on the range of the Hardy number of hyperbolic domains satisfying Abel’s inclusion property; namely +1⊆. Due to its relevance, these domains will be called Koenigs domains. One can easily notice that all domains in the aforementioned class are unbounded. It turns out that the Hardy number of these domains strongly depends on the logarithmic capacity of their complement; see Sect. 2.1. Recall that, if ⊆Cis a domain whose complement is polar (i.e., it has zero logarithmic capacity), then the Hardy number of vanishes. This follows, for instance, from [15, Theorems 5.1.1 and 5.4.2, p. 209 and 211], where it is proved that every universal covering map p:D→has non-tangential limits almost nowhere on the unit circle Tif the logarithmic capacity of C\is zero. In contrast, our main result provides a lower bound for the Hardy number of a Koenigs domain whose complement has positive logarithmic capacity:
On the Hardy number of Koenigs domains Page 3 of 21 119 Theorem 1.1 Let ⊆Cbe a Koenigs domain. Then, h() ≥1/2if and only if the complement of is non-polar. In particular, for every Koenigs domain it holds that h() ∈{0}∪[1/2,+∞]. In Sect.5we emphasize the role of the Koenigs domains in the latter result. Namely, for any prefixed p∈(0,+∞)we build a domain ⊆Cwhose Hardy number is exactly p.Forp∈[1/2,+∞), easier examples are known (for instance, see the comments after Lemma 2.3). In the case where p∈(0,1/2), as far as we know, Theorem 5.2 is novel. The proof of Theorem 1.1 strongly depends on potential theory. More precisely, a sharp estimate of the logarithmic capacity of compact sets obtained as union of integer translations of a compact non-polar subset of C. Recall that a compact set in the complex plane is polar if its logarithmic capacity is zero. Theorem 1.2 Let E be a compact non-polar subset of D(0,1/4). Set Kn= n j=1 (E+j). Then, lim n→+∞ cap(Kn) cap([0,n])=1. Moreover, log(cap(Kn)) =log(n/4)+O(1/√n). After a preliminary introduction on logarithmic capacity, the Hardy number of domains and harmonic measure in Sect.2, we prove Theorem 1.2 in Sect.3and Theorem 1.1 in Sect.4. After this, the aforementioned family of examples appears in Sect.5. The main motivation for the latter results is that Abel’s inclusion property is directly connected to non-elliptic discrete dynamics of the unit disc, mainly through the use of Koenigs maps. To this extent, it is of interest to study any implications of Theorem 1.1 in the context of discrete iteration theory. To motivate this, several examples are presented in Sect.6to relate the Hardy number of Koenigs maps with the properties of its associated self-map. Notation 1.3 We write an=O(bn)if there exist C >0and N ∈Nsuch that |an|≤Cbnfor all n ≥N. Moreover, we write xn≥yn+O(bn)if there exists an=O(bn)such that xn≥yn+anfor all n ∈N. 2 Preliminaries 2.1 Logarithmic capacity Let us introduce some topics about the fundamentals of potential theory, which can be found in [18, Chapters 3-5].
119 Page 4 of 21 M. D. Contreras et al. Definition 2.1 Let μbe a finite positive measure on Cwith compact support. Its (logarithmic) potential is the function pμ:C→[−∞,+∞)given by pμ(z)=C log |z−w|dμ(w), z∈C. Its (logarithmic) energy I(μ) ∈[−∞,+∞)is given by I(μ) =C pμ(z)dμ(z)=CC log |z−w|dμ(w)dμ(z). From these concepts one can define the (logarithmic) capacity of a set X⊆C, that is cap(X)=sup μ exp(I(μ)) (2.1) where the supremum is taken among every probability measure μwhose support is a compact subset of X. In these definitions, we understand that I(μ) =−∞if the former integral is not convergent. Indeed, if cap(X)=0, then I(μ) =−∞for every probability measure μwith compact support in X. If this happens, Xis said to be a polar set. A property is said to be satisfied nearly everywhere in X⊂C, if it is satisfied for all points in X, except maybe for a Borel polar subset. In general, polar sets are negligible from the potential-theoretic point of view. Moreover, the following lemma holds: Lemma 2.2 [18, Corollary 3.2.5] A countable union of Borel polar sets is polar. When it comes to compacta, the theory of the potentials is richer: if X⊂Cis a compact set, then there exists a probability measure νwith support in Xattaining the supremum in (2.1). In fact, this measure is unique if Xis non-polar and in such a case νis said to be the equilibrium measure of X;see[18, Theorem 3.7.6]. The potential associated to the equilibrium measure satisfies the following property: Theorem A (Frostman’s Theorem) [18, Theorem 3.3.4] Let X ⊆Cbe a non-polar compact set, and let νbe its equilibrium measure. Then, pν(z)≥I(ν) for all z ∈C. Moreover, pν≡I(ν) nearly everywhere on X. 2.2 Hardy numbers Given 0 <p<+∞, the Hardy space Hp(D)is defined as the set of all holomorphic maps f:D→Csuch that sup 0<r<1T|f(rξ)|pdm(ξ) < +∞, where mis the normalized length measure on the boundary of the unit disc T.In case p=+∞, the Hardy space H∞(D)stands for the set of all holomorphic maps
On the Hardy number of Koenigs domains Page 5 of 21 119 f:D→Cthat are bounded, that is, sup z∈D|f(z)|<+∞. We refer to [6] for a complete introduction to this topic. One of the properties of these spaces is that they form a decreasing family, that is, Hp(D)⊇Hq(D)if 0 < p≤q≤+∞. These relations suggest the following idea: given a holomorphic map f:D→C, its Hardy number h(f)is defined as h(f)=sup({0}∪{p>0:f∈Hp(D)})∈[0,+∞]. Note that f∈Hp(D)for every 0 <p<h(f), and f/∈Hp(D)if p>h(f). In a similar manner, this idea can be translated to domains. Given a domain ⊆C, its Hardy number is defined as h() =inf{h(f):f∈Hol(D,)}. Here, Hol(D,) denotes the set of all holomorphic maps f:D→Csuch that f(D)⊆. Let us state some well-known properties of the Hardy number of a domain: Lemma 2.3 [14, Lemmas 2.1 and 2.3] Let ,⊆Cbe two domains. Then (a) h() =+∞,ifis bounded. (b) h()≤h(),if⊆. (c) h(ϕ()) =h() for a complex affine map ϕ(z)=az +b, a = 0. (d) h() =0,ifC\is bounded. (e) h() ≥1 2,if = Cis simply connected. (f) h() =h(p), where p :D→is a universal covering map of . The Hardy number of certain simply connected domains is already known. For instance, h(H)=1 for a half-plane H. In the case of a strip e.g. S(a,b)={z∈ C:a<Im z<b}, with a,b∈R, it is known that h(S(a,b)) =+∞.TheHardy number of sectors has also been examined in [9]: suppose θ∈(0,2π]and Sθ:= reiφ:r>0,|φ|<θ 2, then h(Sθ)=π θ. A characterization for the Hardy number of certain domains can be obtained through the following classical result: Theorem B [15, Theorems 5.1.1 and 5.4.2, p. 209 and 211] A universal covering map p:D→has non-tangential limits almost everywhere on Tif and only if the complement of is a non-polar. Recall that any function whose Hardy number is positive (that is, it belongs to some Hardy space) has non-tangential limits almost everywhere on T. Then, from the above result and the properties of the Hardy number of a domain, it follows that the Hardy number of every domain with polar complement is zero.
119 Page 6 of 21 M. D. Contreras et al. 2.3 Harmonic measure For a given domain ⊆Cwhose complement is non-polar, let B⊆∂ be a Borel set. The harmonic measure ω(z,B,) of Bat a point z∈is the solution of the generalized Dirichlet problem in with boundary values 1 on Band 0 on ∂\B. For a fixed Borel set B⊆∂,z→ ω(z,B,)is a harmonic and bounded function. In addition, for a fixed point z∈,themapB→ ω(z,B,)is a Borel probability measure on ∂. We refer to [18, Section 4.3] for an introduction to harmonic measure. In [7, Lemma 1], Essén proposed a relation between Hardy number and harmonic measure, which was later improved by Kim and Sugawa in the following result, where we use the notation D(z,R):= {w∈C:|w−z|<R}. Theorem C [14, Lemma 3.2] Let ⊆Cbe a domain with 0∈. Then, h() =lim inf R→+∞−log ω(0,FR,R) log R, where Ris the connected component of ∩D(0,R)containing the origin and FR=∂R∩{|z|=R}. This result will be useful in Sect.4to deduce Theorem 1.1 from Theorem 1.2, and in Sect.5to provide examples of domains satisfying h() =pfor every given p∈(0,+∞). 3 Proof of Theorem 1.2 Before moving on to the proof of Theorem 1.2, we prove two auxiliary lemmas related to the construction of the equilibrium measure of the compact sets Kn,n∈N. In the sequel we will use the equilibrium measure of a compact interval, which can easily be derived from [19, Eq. (1.7), p. 25]. For any n∈N, the equilibrium measure μof the interval [0,n]is absolutely continuous (with respect to Lebesgue’s measure m) and it is given by dμ dm(t)=χ[0,n](t) π√t(n−t).(3.1) In particular, it follows (cf. [19, Eq. (1.8), p. 25]) pμ(t)=I(μ) =log(cap([0,n])) =log(n/4), for all t∈[0,n]; (3.2) the above result is a combination of Theorem Aand [18, Theorems 4.2.2 and 4.2.4]. Using this equilibrium measure, we define the following coefficients: αj:= j j−1 dμ(t)=1 πj j−1 dt √t(n−t),j∈N,1≤j≤n.
On the Hardy number of Koenigs domains Page 7 of 21 119 Notice that αjalso depends on n. However, in the seek of clearance, this it not explicitly written in the notation. Some properties of these numbers can be derived directly from the definition. For example, it follows that αj>0 for all j=1, ..., nand that n j=1αj=1. It is also possible to notice that these numbers are endowed with some symmetry, namely αj=αn−j+1, and that αjis non-increasing for j= 1, ..., (n+1)/2, where xdenotes the integer part of the real number x. We prove the following estimations: Lemma 3.1 (a) There exists C1>0such that αj≤C1/√n, for all n ∈Nand all j=1,...,n. (b) There exists C2>0such that n j=1 j=k αj |j−k|≤C2 √n, for all n ∈Nand all k =1,...,n. Proof (a) Fix a natural number n≥2. Then α1=1 π1 0 dt √t(n−t)≤1 π√n−11 0 dt √t=O1 √n. Due to the monotonicity and symmetric properties of the coefficients αj, we have that b∗ 1≤1, b∗ 2≤1, b∗ 3≤1/2, b∗ 4≤1/2, and hence (a) holds. (b) Fix n∈Nand define S(k)= n j=1 j=k αj |j−k|,k=1,...,n. By symmetry, notice that S(k)=S(n−k+1). Thus, it is enough to work with k∈Nsuch that k≤(n+1)/2. If this is the case, notice that for any j∈Nwith 1≤j≤(n+1)/2 it is possible to check that αj=αn−j+1but |j−k|≤|n−j+1−k|. Therefore, S(k)≤2(n+1)/2 j=1 j=k αj |j−k|. Given two finite sequences {a1,a2, ..., am}and {b1,b2, ..., bm}of non-negative real numbers, the Hardy-Littlewood inequality asserts that m j=1 ajbj≤ m j=1 a∗ jb∗ j,
119 Page 8 of 21 M. D. Contreras et al. where {a∗ j}denote the sequence of elements ajarranged in decreasing order and similarly for {b∗ j}(see [11, §10.2] and also [3, p. 43]). Notice that the map {1,...,(n+ 1)/2} j→ αjis decreasing. Moreover, taking bj=1/|j−k|,for j=1, ..., (n+ 1)/2,j= k, and bk=0, we have that b∗ 1≤1,b∗ 2≤1,b∗ 3≤1/2,b∗ 4≤1/2,..., b∗ (n+1)/2=0. In particular, b∗ j≤2/j. Thus (n+1)/2 j=1 j=k αj |j−k|=(n+1)/2 j=1 αjbj≤2(n+1)/2 j=1 αj j. But now, notice that, for n≥2, (n+1)/2 j=1 αj j≤2 π(n+1)/2 0 dt (t+1)√t(n−t) ≤2 π√(n−1)/2+∞ 0 dt (t+1)√t=O1 √n, where, in the first inequality, we have used that 1/j≤2/(t+1)whenever j−1≤ t≤j. Thus, the result follows. To proceed with the proof of Theorem 1.2,letE⊆D(0,1/4)be a compact nonpolar set with 0 ∈Eand equilibrium measure ν.Fixn∈Nand set Kn= n j=1 (E+j). Let us define the positive measure σgiven by σ(A)= n j=1 αjν(A−j), where A⊆Cis a Borel set. Notice that σis a probability measure whose support is a compact set lying on Kn. Once more, observe that σdepends on n. Recalling the Definition 2.1, let us prove the following lemma: Lemma 3.2 Under the above notation, the following statements hold: (a) pσ(x)≥log(n/4)+O(1/√n)for every x ∈Kn. (b) pσ(x)=log(n/4)+O(1/√n)for nearly every x ∈Kn. (c) |pσ(x)−pσ(y)|=O(1/√n)for nearly every x,y∈Kn. (d) pσ(x)=log(n/4)+O(1/√n)for every x ∈E. (e) |pσ(x)−pσ(y)|=O(1/√n)for every x ∈E and nearly every y ∈Kn. In fact, the underlying constants do not depend on x and y.
On the Hardy number of Koenigs domains Page 9 of 21 119 Proof In this proof we will always assume that n≥2. Note that, by the definition of the probability measure σ, its potential pσcan be written as pσ(z)= n j=1 αjpν(z−j), z∈C. Let k∈{1, ..., n}. Notice that, by Theorem A,pν(x−k)≥I(ν) for all x∈E+k and pν(x−k)=I(ν) nearly everywhere in x∈E+k. Therefore pσ(x)= n j=1 αjpν(x−j)≥αkI(ν) + n j=1 j=k αjE log |x−j−y|dν(y) for all x∈E+kand the inequality turns to be an equality nearly everywhere. We claim that, given x∈E+k, αkI(ν) + n j=1 j=k αjE log |x−j−y|dν(y)=log n 4+O1 √n.(3.3) Thus, assuming this claim, we clearly conclude (a) and (b). Let us prove the claim. Take x∈E+kand write x=x−k∈E. Then log |x−j−y|=log |x−y+k−j|=log |k−j|+log 1+x−y k−j =log |k−j|+O1 |k−j|, for all y∈E, since |x−y|≤1/2. Due to Lemma 3.1.(b), we obtain αkI(ν) + n j=1 j=k αjE log |x−j−y|dν(y) =αkI(ν) + n j=1 j=k αjlog |k−j|+O1 √n.(3.4)
119 Page 16 of 21 M. D. Contreras et al. define the domain =C\∞ n=1 (CRn\n), n=z=Rneiθ:|θ|<π R2p n.(5.1) Let us also define R,N=D(0,R)\N n=1 (CRn\n),ω R,N=ω0,CR,R,N,R>RN. We can now prove the following: Theorem 5.2 For any p ∈(0,+∞)there exists a domain ⊆Cfor which h() =p. Proof Fix p>0. The domain is constructed as in (5.1), where the sequence {Rn} is inductively constructed depending on p. To do this, set R1=2. Suppose that Rnis constructed for all 1 ≤n≤N.LetR>RN, and notice that the maximum principle implies that ωR,N≤ω(0,N,D(0,RN)) =1 R2p N ,(5.2) as ω(0,·,D(0,R)) is the normalized Lebesgue measure on CR. In particular, limR→R+ NωR,N≤1/R2p N, where the limit converges because the maximum principle implies that the function R→ ωR,Nis decreasing for R>RN. We now claim that there must exists R>RNsuch that ωR,N=1/Rp.Tosee this, arguing by contradiction, using the continuity given in Lemma 5.1 and the above inequality, we have that ωR,N<1/Rpfor all R>RN. Then, Theorem Cimplies that h(C\(∪N n=1(CRn\n)) ≥p. But this contradicts Lemma 2.3.(d). Thus, it is possible to define RN+1=min{R>RN:ωR,N=1/Rp}.From(5.2), we see that RN+1≥R2 N, which yields that RN→+∞as N→+∞. It is clear that for every R>R1there exists N∈Nwith RN<R≤RN+1.From the construction, it follows ω(0,CR,∩D(0,R)) =ωR,N≤1/Rp, where equality is attained for R=RN+1. Therefore, applying Theorem C, h() =p. Remark 5.3 In the setting of Lemma 5.1, it is possible to prove that limR→+∞ ωRlog(R) >0. This fact could be used instead of Theorem Cto show the claim we made in order to prove the existence of RN+1.
On the Hardy number of Koenigs domains Page 17 of 21 119 6 Applications to discrete iteration theory Theorem 1.1 can be of interest in the non-elliptic discrete dynamics of the unit disc. To introduce this, let φbe a holomorphic self-map of D. Let us denote by (φn)n∈N, where φn=φ◦... ◦φcomposing ntimes, the sequence of iterates of φ. As usual, if φhas a fixed point in D, then φis called elliptic. In the case where φis not an elliptic automorphism of D, the Denjoy–Wolff Theorem asserts that (φn)converges locally uniformly in Dto a point τφ∈D. The point τφis called the Denjoy–Wolff point of φ. The research spectrum of the current work focuses on the case where φhas no fixed points in Dand thus, τφ∈∂D. For this case, it is known that 0 <φ (τφ)≤1(in the angular limit sense). More specifically, if φ(τφ)<1, then φis called hyperbolic, whereas, if φ(τφ)=1, φis called parabolic. Parabolic self-maps are further divided into two subcategories depending on its hyperbolic step. The parabolic self-map φ:D→Dis of zero hyperbolic step if for some –and hence, for all– z∈D, it is true that ρD(φn(z), φn+1(z)) →0, n→+∞, where ρDdenotes the pseudo-hyperbolic distance in D. Otherwise, φis parabolic of positive hyperbolic step. For a further introduction on this topic, we refer to a recent book by Abate [1, Chapter 4]. A prominent tool in examining properties of the iterates (φn)are the solutions to the so-called Abel’s equation, that is, holomorphic maps σ:D→Csatisfying σ◦φ=σ+1. In the course of the past century, the existence of solutions for this equation in the case where φis non-elliptic was shown. Indeed, this was achieved through the individual works by several mathematicians: Valiron examined the hyperbolic case [20], while the two different parabolic cases were covered by Pommerenke [2,17], the second one in collaboration with Baker. The main idea of these three works was to explicitly construct a solution σto the Abel’s equation for a given self-map φin terms of its “normalized” iterates. Some time afterwards, Cowen [5] proposed to find solutions of a general functional equation related to the self-map φ. This eventually led to the following concept: a triple (0,,) is said to be a model for φif 0⊆Cis a domain, :D→0 is a holomorphic function, and is an automorphism of 0, satisfying the following three conditions: (i) ◦φ=◦, (6.1) (ii) n∈N −n((D)) =0,
119 Page 18 of 21 M. D. Contreras et al. (iii) there exists a domain A⊆Dsuch that is injective on A,φ(A)⊆A, and for every z∈Dthere exists n∈Nwith φn(z)∈A. The domain 0is called a base space, the holomorphic function is called an intertwining map, the automorphism is called a normal form, and Ais called an absorbing set. Cowen showed that every non-elliptic self-map φadmits a model, which might be chosen so that it follows a certain canonical form. This can be formulated in the following way, where the notation H={z∈C:Im(z)>0}is used: Theorem D Let φbe a non-elliptic holomorphic self-map of D. There exists a model (0,σ,)for φ, where (z)=z+1, so that: (i) φis hyperbolic if and only if there exists λ>1such that 0=S(λ) := {z∈ C:0<Im(z)<π/log(λ)}. (ii) φis parabolic of positive hyperbolic step if and only if 0=Hor 0=−H. (iii) φis parabolic of zero hyperbolic step if and only if 0=C. Remark 6.1 This theorem is stated in [1, Theorem 4.6.8], although it has been reformulated in such a way that (6.1) always corresponds to the Abel’s equation for φ. Every non-elliptic holomorphic self-map admits an essentially unique holomorphic model (see [1, Corollary 3.5.9]). As a byproduct, the function σin Theorem Dis unique if we assume that Re(σ(0)) =0 in cases (i) and (ii), and σ(0)=0 in case (iii). From now on, such normalized solution to the Abel’s equation σwill be called the Koenigs map of φ. The image =σ(D)plays a prominent role in the properties of φ. Note that, from Abel’s equation, needs to satisfy that +1⊆. Therefore, is a Koenigs domain in the sense of the previous sections. Indeed, the conclusion in Theorem 1.1 might be strengthened if is assumed to be a Koenigs domain of a certain self-map φof D, depending on the properties of φ. For example, if φis hyperbolic, its Koenigs domain is contained in a horizontal strip. This means that this Koenigs domain and the associated Koenigs map have an infinite Hardy number, see Sect. 2.2. Therefore, in the hyperbolic case, the Koenigs map is in Hpfor all 0 <p<+∞. A little more can be obtained through the space BMOA of analytic functions with bounded mean oscillation, that is, the set of analytic functions f:D→Cwith sup w∈Dsup r∈[0,1)2π 0|fw(reiθ)|2dθ<+∞, where fw(z)=fz+w 1+wz−f(w), z∈D. We refer to [8] for a complete introduction to BMOA. It holds H∞⊆BMOA ⊆Hp, for 0 <p<+∞. The key point is that Riemann mappings from Donto any strip S(λ) are known to be in BMOA. Therefore, by a subordination argument (see [8, Corollary 10.1]), the Koenigs map σcorresponding to a hyperbolic self-map φis also in BMOA. This can also be noticed geometrically using a deeper result of Hayman and Pommerenke, see [12, Theorem 1].
On the Hardy number of Koenigs domains Page 19 of 21 119 In the parabolic case, characterizes the hyperbolic step of φ, as noticed in Theorem D. Therefore, if φis a parabolic self-map of positive hyperbolic step, the conclusion on Theorem 1.1 can be improved: its Koenigs domain is contained in a half-plane, and so the Hardy number of the Koenigs domain and the associated Koenigs map is always greater than or equal to one. Furthermore, for every p∈[1,+∞], it is possible to find a parabolic self-map φof positive hyperbolic step such that h() =h(σ) =pfor some Koenigs map σ. In the case where pis finite, this construction is achieved by means of sectors: fix some θ∈(0,π]and set the domain S(θ) := {z∈C:arg(z)∈(0,θ)}.Let σ:D→S(θ) be a Riemann mapping, and define a self-map φ:D→Dgiven by φ(z)=σ−1(σ (z)+1). By Theorem D,φis a parabolic self-map of positive hyperbolic step and, up to a translation, σis the Koenigs map for φ. Therefore, the properties of Hardy numbers show that h(σ) =h(σ(D)) =h(S(θ)) =π/θ; see Sect. 2.2.By choosing θ∈(0,π]appropriately, it is clear that for any p∈[1,+∞)one can find explicit examples with h(σ ) =p. Similarly, for the infinite value of Hardy number, we consider the domain = {z=x+iy ∈C:x>0,0<y<√x}. With the definitions given above, note that for every θ∈(0,π/2)there exists t≥0 such that +t⊆S(θ). Due to the properties of the Hardy numbers, it follows h() =h( +t)≥h(S(θ)) =π/θ. Letting θ→0+leads to h() =+∞. Once more, consider a Riemann mapping σ:D→and define a self-map φ:D→Dgiven by φ(z)=σ−1(σ (z)+1).By Theorem D,φis a parabolic self-map of Dof positive hyperbolic step and, up to a translation, σis a Koenigs map for φ. Once more, the properties of Hardy numbers lead to h(σ) =h(σ(D)) =h() =+∞. While considering parabolic self-maps of zero hyperbolic step, Koenigs domains are not included in some horizontal half-plane of C. Therefore the main difference is that now it may actually occur that C\has zero logarithmic capacity and thus, h(σ) can be zero. For example, set =C\{n∈Z:n<0}and let p:D→be a universal covering map with p(0)=0. Using [4, Theorems 8.1 and 8.2], one can see that there exists a parabolic inner function φ:D→Dof zero hyperbolic step for which pis its Koenigs map in the sense of Theorem D,buth(p)=h() =0. If the complement of the Koenigs domain is non-polar, the latter situation can not happen since Theorem 1.1 can be applied. It can also be shown that no uniform upper bound for h(σ) nor h() exists. To be more precise: for every p∈[1/2,+∞] there exists a parabolic self-map φof zero hyperbolic step such that h(σ) =h() =p for the Koenigs map for φ. Similarly as before, in the finite cases, this is done by means of the sectors S(θ) := {z∈C:|arg(z)|<θ/2}with h(S(θ)) =π/θ and a suitable Riemann mapping. The argument is also similar for the infinite value, where ={z=x+iy ∈C:x>0,|y|<√x}with h() =+∞is used. To sum up, the foregoing considerations can be abbreviated as follows: Proposition 6.2 Let φbe a holomorphic non-elliptic self-map of Dand consider its associated Koenigs map σ. (i) If φis hyperbolic, then σis in BMOA.
119 Page 20 of 21 M. D. Contreras et al. (ii) If φis parabolic of positive hyperbolic step, then h(σ ) ≥1. Indeed, for every p∈[1,+∞] there exists a parabolic self-map φof positive hyperbolic step for which h(σ) =p. (iii) If φis parabolic of zero hyperbolic step and the complement of σ(D)is nonpolar, then h(σ ) ∈[1/2,+∞]. Indeed, for every p ∈[1/2,+∞] there exists a parabolic self-map φof zero hyperbolic step for which h(σ) =p. Moreover, there exists an inner parabolic self-map of zero hyperbolic step, for which h(σ) =0. Acknowledgements The authors would like to express their gratitude to the anonymous referee for the thorough reading of the article and the helpful comments. Author Contributions Authors have been discussing and working together on the manuscript, contributing equally to the content, presentation, and reviewing the manuscript. Funding Funding for open access publishing: Universidad de Sevilla/CBUA Data Availability No datasets were generated or analysed during the current study. Declarations Conflict of interest The authors declare no competing interests. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Abate, M.: Holomorphic Dynamics on Hyperbolic Riemann Surfaces, volume 89. Walter de Gruyter GmbH & Co KG, (2022). 2. Baker, I.N., Pommerenke, C.: On the iteration of analytic functions in a halfplane II. J. Lond. Math. Soc. 20, 93–97 (1979) 3. Bennett, C., Sharpley, R.: Interpolation of Operators. Pure Appl. Math., vol. 129, Academic Press Inc. (1988) 4. Contreras, M.D., Díaz-Madrigal, S., Pommerenke, C.: Iteration in the unit disk: the parabolic zoo. In Complex and harmonic analysis, 63–91. DEStech Publ., Inc. Lancaster, PA (2007) 5. Cowen, C.C.: Iteration and the solution of functional equations for functions analytic in the unit disk. Trans. Am. Math. Soc. 265, 69–95 (1981) 6. Duren, P.L.: Theory of HpSpaces. Academic press, Boca Raton (1970) 7. Essén, M.: On analytic functions which are in Hpfor some positive p.Ark.Mat19, 43–51 (1981) 8. Girela, D.: Analytic functions of bounded mean oscillation. In Complex function spaces (Mekrijärvi, 1999), 61–170. Univ. Joensuu Dept. Math. Rep. Ser., 4(2001) 9. Hansen, L.J.: Hardy classes and ranges of functions. Michigan Math. J 17, 235–248 (1970) 10. Hansen, L.J.: The Hardy class of a spiral-like function. Michigan Math. J 18, 279–282 (1971) 11. Hardy, G. H., Littlewood, J. E., Pólya, G.: Inequalities, Cambridge Mathematical Library, (1988) 12. Hayman, W.K., Pommerenke, Ch.: On analytic functions of bounded mean oscillation. Bull. Lond. Math. Soc 10, 219–224 (1978)
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