# Composition Operator, Boundedness, Compactness, Hyperbolic Bloch-Type Spaceβμ∗, Hyperbolic-Type Space

Authors: Shuan Tang; Pengcheng Wu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we obtain some characterizations of composition operatorsCφ, which are induced by an analytic self-map φ of the unit disk Δ, from hyperbolic Bloch type space βμ∗ into hyperbolic type space QK,p,q∗.


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## Body

## 1. Introduction and Results

LetΔ=z:∣z∣<1 be the unit disk in the complex plane ℂ, Δ∗=ℂ\Δ¯ and HΔ denote the space of all analytic functions on Δ. Let BΔ denote the subspace of HΔ consisting of those analytic self-maps φ for which φΔ⊂Δ. For every φ∈BΔ, the linear composition operator is defined as
(1)Cφ=f∘φ:HΔ→HΔ.An important problem is to characterize the boundedness and compactness ofCφ in analytic function spaces by means of the geometric and analytic properties of φ. The reader is invited to the monographs [1, 2] for an excellent overview of this subject. Composition operators from the Bloch-type space into some other analytic function spaces have been much investigated in recent years, see [3–19] and references therein. Also, there are many works concerning composition operators from hyperbolic Bloch type spaces into some other hyperbolic classes, see [20–23]. This work is intended to characterize the boundedness and compactness of composition operators Cφ from hyperbolic Bloch type spaces βμ∗ into hyperbolic type spaces QK,p,q∗. We start with some notations and definitions.A positive continuous functionμ on the interval 0,1 is called normal if there exist constants 0≤δ<1 and 0<a<b such that
(i)
μr/1−ra is decreasing on [δ, 1) and limr⟶1μr/1−ra=0;(ii)
μr/1−rb is increasing on [δ, 1) and limr⟶1μr/1−rb=∞.It is clear that such a normal weight functionμ is decreasing in the neighborhood of the point 1 and satisfies
(2)limr⟶1μr=0.It should be pointed out that the normal weight functionμ has been usually used to define the mixed norm spaces [6]. Let μ be normal, the Bloch type space βμ is defined as the set of all analytic functions f so that
(3)fβμ=f0+supz∈Δμzf′z<∞.The little Bloch-type spaceβμ,0 consists of all f∈βμ such that
(4)lim∣z∣→1μzf′z=0.Letμz=1−z2α, α>0. Then, the Bloch-type space βμ becomes the so-called α-Bloch space Bα, which for α=1 reduces to the classical Bloch space B. If μz=1−z2αloge/1−z2s, where α>0 and s≤0, then, we obtain the so-called logarithmic Bloch-type space βlogsα which consists of all analytic functions f∈HΔ so that
(5)fβlogsα=f0+supz∈Δ1−z2αloge1−z2sf′z<∞.The little logarithmic Bloch-type spaceβlogs,0α consists of all analytic functions f∈βlogsα and
(6)lim∣z∣⟶11−z2αloge1−z2sf′z=0.For more results onβμ space, see, e.g., [24–26] and related references therein.Hyperbolic function classes are subset ofBΔ and are defined by using the hyperbolic derivative
(7)f∗z=f′z1−fz2.It is clear that the hyperbolic derivative of the compositionf∘φ satisfies the chain rule
(8)f∘φ∗z=f∗φzφ′z.The hyperbolic Bloch-type spaceβμ∗ consists of all analytic functions f∈BΔ with
(9)fβμ∗=f0+supz∈Δμzf∗z<∞.The little hyperbolic Bloch-type spaceβμ,0∗ is the set of all analytic functions f∈BΔ for which f∈βμ∗ and
(10)lim∣z∣⟶1μzf∗z=0.Replacing the derivativef′ by hyperbolic derivative f∗, we can define the corresponding hyperbolic α-Bloch space Bα,∗ and hyperbolic logarithmic Bloch-type space βlogsα,∗ and βlogs,0α,∗.For anya∈Δ, the function
(11)φaz=a−z1−a¯z,z∈Δ,is a Möbius map of the unit disk and interchanges the points a to 0. The Green function gz,a of Δ is defined as
(12)gz,a=log1φaz.Letp>0,q>−2 and K:0,∞⟶0,∞ be right continuous and nondecreasing. The space QK,p,q consists of all analytic functions f in Δ such that
(13)fQK,p,qp=supa∈Δ∬Δf′zp1−z2qKgz,adAz<∞,where dA is Lebesgue measure on Δ, normalized so that the unit disk has Lebesgue measure equals to 1. The definition was introduced in [23] and was motivated by the theory of Qp, Fp,q,s, and QK, see [16, 27–31] and references therein. We note that the case when p=2 and q=0 is the classic QK space, which was introduced by Wulan and Wu in [28] and has been much investigated in recent years. We also observe that the case when Kt=ts is the space Fp,q,s introduced by Zhao in [31]. For α=q+2/p, the spaces QK,p,q are subsets of space Bα. Throughout this paper, we assume that
(14)∫011−r2qKlog1rrdr<∞.Otherwise,QK,p,q contains constant functions only (see [32]).The corresponding hyperbolic spaceQK,p,q∗ is defined as the set of all analytic functions f in Δ such that
(15)fQK,p,q∗p=supa∈Δ∬Δf∗zp1−z2qKgz,adAz<∞,where K:0,∞⟶0,∞ is defined as before and satisfies the condition (14). Replacing the derivative f′ by hyperbolic derivative f∗, we can define the corresponding hyperbolic space Qp∗.Yang, Xu, and Kotilainen [30] obtained some characterizations of the boundedness and compactness of the composition operatorsCφ:βμ⟶QK,p,q. For an r∈0,1 and φ∈BΔ, set Ωr=z∈Δ:φz>r and Δr=z∈Δ:z>r. We use 1E to denote the characteristic function of a set E⊂Δ.Theorem 1 (see [18]).
Assume thatμ is a normal function and φ∈BΔ. Let p>0, q>−2 and K be nonnegative and nondecreasing in 0,∞. Then
(i)
Cφ:βμ⟶QK,p,q is bounded if and only if(16)supa∈Δ∬Δφ′zp1−z2qμpφzKgz,adAz<∞.(ii)
Cφ:βμ⟶QK,p,q is compact if and only if(17)limr⟶1supa∈Δ∬Δφ′zp1−z2qμpφzKgz,a1ΩrdAz=0.In [11], the authors considered bounded composition operators from hyperbolicα-Bloch spaces Bα into hyperbolic space Qp∗. They proved the following result.Theorem 2.
Assume thatφ∈BΔ. Let 0<α≤1, 0≤p≤1. Then the following statements are equivalent:
(i)
Cφ:Bα⟶Qp is bounded;(ii)
Cφ:Bα∗⟶Qp∗ is bounded;(iii)
Cφ:Bα,0∗⟶Qp∗ is bounded.In this paper, we extend Theorem1 to the corresponding hyperbolic classes. For a real increasing function hr defined on 0,1, its growth order is defined as
(18)ρh≔limsupr⟶1log+hrlog1/1−r.We obtain the following result.Theorem 3.
Suppose thatμ is a normal function defined on 0,1 and the growth order of function 1/μ is strictly less than 1. Let φ∈BΔ, p>0, q>−2, and K be nonnegative and nondecreasing in 0,∞. Then, the following statements are equivalent:
(1)
Cφ:βμ⟶QK,p,q is bounded;(2)
Cφ:βμ∗⟶QK,p,q∗ is bounded;(3)
Cφ:βμ,0∗⟶QK,p,q∗ is bounded.It should be pointed out that there are many normal functions which satisfy the condition of Theorem3. For example, if μr=1−r2α with 0<α<1, then the growth order of 1/μ is α. As a corollary, we obtain the following result by taking μr=1−r2αloge/1−r2s, where 1>α>0 and s≤0.Corollary 4.
Let1>α>0, s≤0, and φ∈BΔ. Let p>0, q>−2, and K be nonnegative and nondecreasing in 0,∞. Then, the following statements are equivalent:
(1)
Cφ:βlogsα⟶QK,p,q is bounded;(2)
Cφ:βlogsα,∗⟶QK,p,q∗ is bounded;(3)
Cφ:βlogs,0α,∗⟶QK,p,q∗ is bounded.We now turn to the compactness of composition operatorCφ:βμ∗⟶QK,p,q∗. It is clear that βμ∗ is not a linear space since the sum of two functions in BΔ does not necessarily belong to BΔ. It is also known that there is a natural metric in βμ∗, which is defined by
(19)df,g,βμ∗≔f−gβμ∗+f−gβμ+f0−g0.The classesβμ∗ equipped with the metric d·,·,βμ∗ is a complete metric space, and βμ,0∗ is a closed subspace of βμ∗ (see [5, 9]). Also, there is a natural metric in QK,p,q∗ defined as
(20)df,g,QK,p,q∗≔f−gQK,p,q∗+f−gQK,p,q+f0−g0.A similar argument shows that the classesQK,p,q∗ equipped with the metric d·,·,QK,p,q∗ is a complete metric space (see [21, 22]).Recall that an operatorCφ:βμ∗⟶QK,p,q∗ is compact if it maps any ball in βμ∗ onto a related relatively compact set in QK,p,q∗. This is equivalent to say that the operator Cφ:βμ∗⟶QK,p,q∗ is completely continuous. We obtain the following result.Theorem 5.
Suppose thatμ is a normal function defined on 0,1 and the growth order of function 1/μ is strictly less than 1. Let φ∈BΔ, p>0, q>−2, and K be nonnegative and nondecreasing in 0,∞. Then, the following statements are equivalent:
(1)
Cφ:βμ∗⟶QK,p,q∗ is compact(21)limr⟶1supa∈Δ∬Δφ′zp1−z2qμpφzKgz,a1ΩrdAz=0.Similarly, if we takeμr=1−r2αloge/1−r2s, where 1>α>0 and s≤0, then we haveCorollary 6.
Let1>α>0, s≤0, and φ∈BΔ. Let p>0, q>−2, and K be nonnegative and nondecreasing in 0,∞. Then, the following statements are equivalent:
(1)
Cφ:βlogsα,∗⟶QK,p,q∗ is compact(22)limr⟶1supa∈Δ∬Δφ′zp1−z2q1−φz2pαloge/1−φz2psKgz,a1ΩrdAz=0.The paper is organized as follows. We shall prove Theorem3 in section 2 and prove Theorem 5 in section 3. Throughout this paper, we use the notation a≲b to denote that there is a constant C>0 such that a≤Cb, and the notation a≈b to indicate that a≲b≲a.


## 2. Proof of Theorem 1.1

To prove Theorem3, we need the following Lemma [30], which gives some lower estimate of the βμ norm of some functions in βμ.Lemma 7 (see [18]).
Suppose thatμ is a normal function defined on 0,1. Then, there exist two functions f1,f2∈βμ such that for all z∈Δ,
(23)f′1z+f′2z≥Cμzfor some positive constant C>0.We shall prove a hyperbolic version of Lemma7.Lemma 8.
Suppose thatμ is a normal function defined on 0,1 and the growth order of function 1/μ is strictly less than 1. Then, there exist two functions g1,g2∈βμ∗ such that for all z∈Δ,
(24)g1∗z+g2∗z≥Cμz
for some positive constantC>0.Proof.
By Lemma7, there are two functions f1,f2∈βμ so that
(25)f1′z+f2′z≈1/μ∣z∣.
Consequently, we get(26)fiz≤∫0zf′iwdw≤fiβμ∫011μrdr,i=1,2.
Lethr=1/μr and note that the growth order of hr is less than 1, we deduce that for positive number 1+ρh/2>0, there exists r0 with 1>r0>0, such that for all r>r0,
(27)hr<11−r1+ρh/2.
On the other hand, sinceμr is a positive continuous function, there exists positive number M>0 such that for all 0≤r≤r0, hr≤M. Thus, combining (26) and (27) yields that
(28)fiz≤fiβμMr0+11−ρh/2,i=1,2,hold for all z∈Δ. This implies that the functions fi are bounded.
LetCi=fiβμMr0+1/1−ρh/2 and gi=fi/2Ci,i=1,2. Then, giH∞≤1/2 and gi∗z≈gi′z,i=1,2. This gives the following
(29)g1∗z+g2∗z≈1/μz.
The proof follows.We now start our proof of Theorem1.1.Proof of Theorem 1.1.
Sinceβμ,0∗ is a closed subset of βμ∗, it is obvious that 2 implies 3. Thus, it is enough to show that 1⟹2 and 3⟹1.
We first show that1⟹2. Assume that (1) holds, that is Cφ:βμ⟶QK,p,q is bounded. Then, for any f∈βμ∗, it follows from Theorem 1 that
(30)∬Δf∘φ∗zp1−z2qKgz,adAz≤fβμ∗p∬Δφ′zp1−z2qμpφzKgz,adAz<∞,uniformly for all a∈Δ. This implies that Cφ:βμ∗⟶QK,p,q∗ is bounded, and we complete the proof of 1⟹2.
We next show that3⟹1. By Lemma 8, there exist two functions g1,g2∈βμ∗ such that for all z∈Δ,
(31)g1∗z+g2∗z≥Cμzfor some positive constant C>0. Consequently, by the assumption that Cφ:βμ,0∗⟶QK,p,q∗ is bounded, we have
(32)∬Δφ′zp1−z2qμpφzKgz,adAz≈∬Δg1∗φz+g2∗φzpφ′zp1−z2qKgz,adAz≤2p∬Δg1∘φ∗zp+g2∘φ∗zp1−z2qKgz,adAz≲g1βμ∗p+g2βμ∗p<∞.
The proof of Theorem1.1 is completed.


## 3. Proof of Theorem 1.3

In this section, we shall prove Theorem5. First, we find some test functions in βμ. Let μ be a normal weight function defined in 0,1, q∈ℕ be large and bi⊂1/2,1, consider a family analytic functions defined by
(33)fi,m,θz=1bi∑j=1∞bieiθqjμ1−1/qjzqj+qmqj+qm,where 0≤θ≤2π, i,m∈ℕ, and bi⟶1 as i⟶∞.It is proved in [21] that if we take the nature numberq large enough, then, the sequence fi,m,θzm=1∞ weakly converges to zero in βμ .Lemma 9 (see [33]).
Letfi,m,θz be the functions defined in (33). Then, there are natural number q∈ℕ such that fi,m,θz∈βμ,0 and positive constant M>0 such that fi,m,θβμ≤M for all i,m and θ∈0,2π. Moreover, set gi,m,θz=fi,m,θz/M, then
(34)limm→∞¯supi,θCφgi,m,θQK,p,qp≳limr⟶1supa∈Δ∬∣φz∣>rφ′zpμφzpdAq,K,az,where dAq,K,az=1−z2qKgz,adAz.Now, we are in a position to prove Theorem5.Proof of Theorem 1.3.
We first show that1⟹2. Let fi,m,θz be the analytic functions defined as in Lemma 9. Therefore, there is a positive constant M>0 such that
(35)f′i,m,θz≤Mμz,z∈Δ,for all i,m, and θ∈0,2π. Observe that the growth order of function 1/μ is less than 1. Arguing as we have done in Lemma 8, we can find a constant C>0 such that fi,m,θH∞<C for all i,m and θ∈0,2π. Take gi,m,θ=fi,m,θ/2C, where i,m∈ℕ, and θ∈0,2π. Then, gi,m,θH∞≤1/2 and gi,m,θ∗z≈gi,m,θ′z,i=1,2.
On the other hand, by the definition ofgi,m,θ, it is clear that the analytic function sequence gi,m,θm=1∞ converges uniformly on compact subset of Δ to 0. Consequently, by Lemma 9, the sequence gi,m,θm=1∞ belongs to βμ,0∗ and tends to 0 weakly as m⟶∞.
SinceCφ:βμ∗⟶QK,p,q∗ is compact, we have
(36)limm→∞¯supi,θCφgi,m,θQK,p,q∗=0.
Thus, it follows from Lemma9 that
(37)limr⟶1supa∈Δ∬∣φz∣>rφ′zpμφzp1−z2qKgz,adAz≲limm→∞¯supi,θCφgi,m,θQK,p,qp≈limm→∞¯supi,θCφgi,m,θQK,p,q∗p=0.
This completes the proof of1⟹2.We now prove that2⟹1. Suppose that (2) holds. Let B≔f∈βμ∗:df,g,βμ∗≤ρ, where g∈βμ∗ and ρ>0, and fnn=1∞⊂B be any sequence. By the definition of compactness of Cφ, it is sufficient to show that its image Cφfnn=1∞ has a convergent subsequence in QK,p,q∗.Sincefnn=1∞⊂BΔ is a normal family, there is a subsequence fnjj=1∞ which converges uniformly on compact subset of Δ to an analytic function f. It is clear that f′njj=1∞ also converges uniformly on compact subset of Δ to f′. Consequently, the sequences fnj∘φj=1∞ and f′nj∘φj=1∞ converge uniformly on compact subsets of Δ to f∘φ and f′∘φ.We claim thatf∈B⊂βμ∗. Indeed, for any fixed R with 0<R<1, we have
(38)supz<Rf′z1−fz2−g′z1−gz2μz+sup∣z∣<Rf′z−g′zμ∣z∣+f0−g0=limj→∞sup∣z∣<Rf′njz1−fnjz2−g′z1−gz2μ∣z∣+limj→∞sup∣z∣<Rf′njz−g′zμ∣z∣+fnj0−g0≤ρ.LettingR⟶1 gives the desired result.For anyε>0, by the uniform convergence, there is N0∈ℕ such that for all j≥N0,
(39)∣fnj∘φ0−f∘φ0∣<ε.By Theorem1, the condition (4) implies that the composition operator Cφ:βμ⟶QK,p,q is compact. Therefore, by passing once more to a subsequence and adjusting the notations, we can assume that
(40)fnj∘φ−f∘φQK,p,q<ε,for all j≥N1, for some N1∈ℕ.On the other hand, since (2) holds, we may findr with 0<r<1 so that
(41)supa∈Δ∬Δφ′zp1−z2qμp∣φz∣Kgz,a1ΩrdAz<ε.Observe thatfnjj=1∞⊂B and f∈B, it follows that
(42)dfnj,f,βμ∗≤dfnj,g,βμ∗+df,g,βμ∗≤2ρ.Consequently, we have(43)supa∈Δ∬Δf′njφz1−fnjφz2−f′φz1−fφz2pφz′p1ΩrdAq,K,az≤2ρpsupa∈Δ∬Δφ′zp1−z2qμp∣φz∣Kgz,a1ΩrdAz<2ρpε.Finally, by the uniform convergence on compact subset ofΔ, we can find N2∈ℕ so that for all j≥N2 and for all z with ∣φz∣≤r,
(44)f′njφz1−fnjφz2−f′φz1−fφz2<ε.Note again that the condition (4) implies that the composition operator Cφ:βμ⟶QK,p,q is compact. Thus, Cφ:βμ⟶QK,p,q is also bounded, and we get from Theorem 1 that
(45)C≔supa∈Δ∬Δφ′zp1−z2qμp∣φz∣Kgz,adAz<∞.Therefore, we have for allj≥N2,
(46)supa∈Δ∬Δf′njφz1−fnjφz2−f′φz1−fφz2pφz′p1−1ΩrdAq,K,az≤εpsupa∈Δ∬Δφz′p1−1ΩrdAq,K,az≲εpsupa∈Δ∬Δφ′zp1−z2qμp∣φz∣Kgz,a1−1ΩrdAz≲εp.Combining (39), (40), (43), and (46) yields
(47)limj⟶∞dfnj,f,QK,p,q∗=0.The proof of Theorem1.3 is completed.



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