# Q-Type Spaces of Harmonic Mappings

Authors: Alaa Kamal
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this paper, we attempt to investigate a new harmonic mapping class denoted by the Banach spaceQHp in the unit disk. Several characteristics of the class Qp are examined. We additionally describe some findings for the little QHp harmonic space, which really is a closed subspace of QHp. We also talk about the boundedness and compactness of the composition operator in the QHp space.


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## Body

## 1. Introduction

A complex-valued functionh=u+iv of ξ=x+iy is said to be a harmonic mapping of an open subset Λ of the complex plane ℂ provided that its related Laplace equation is satisfied on Λ. That is,(1)Δh=4hξξ¯=0onΛ,where hξξ¯ stands for the mixed complex second partial derivative of h.A harmonic mappingh in the unit disk admits the canonical decomposition h=f+g where f and g are analytic functions, and g0=0. Analytic functions are harmonic mappings because their real and imaginary components are harmonic according to the Cauchy-Riemann equations. We shall assume that all the functions under consideration are defined on D=ξ∈ℂ:ξ<1. In the complex plane ℂ, T be its boundary. Let A denote the normalized Lebesgue area measure on D, and dAξ indicate the normalized Lebesgue area measure on D. ℍD signifies the family of all holomorphic functions on D. ℋarD signifies the family of all harmonic functions on D. Let σνξ be the Möbius transformations σν:D⟶D be defined by for each ν∈D.(2)σνξ=ν−ξ1−ν¯ξ,ξ∈D.It is worth noting thatσνσνξ=ξ, and therefore φν−1=σν. It has the following useful property:(3)1−σνξ2=1−ξ21−ν21−ν¯ξ2=1−ν2σν′ξ.We look at the linear structure of the harmonic equivalents of theα-Bloch spaces and the Qp space in particular. We also apply to the harmonic setting and certain well-known features that apply to the analytic case.A bounded linear operatorT:X⟶Y is said to be bounded below if there exists C>0, such that TxY≥CxX for all x∈X.Remark 1.
A closed range operator is a linear operator that is bounded below (see Proposition 6.4 in [1]). If the operator is injective, the opposite is true and then the converse also holds (see Theorem 5.17.2 in [2]).TheCϕ operator is defined on the space of complex-valued functions f with domain Λ, given an analytic self-map ϕ of D, and the composition operator induced by ϕ is defined as the operator:(4)Cϕf=f°ϕ,forf∈HD.It is obvious that such an operator preserves harmonic mappings immediately. Assuming that analytic functions are clearly harmonic, the idea of how to extend the harmonic mappings is interesting. The norm on the larger space fits with the Banach space structures of known spaces of analytic functionsX. When confined to the elements of X, the norm is X.The organization of this work will be as follows: we dedicate Section2 to study some basic facts about the harmonic spaces and the concept of QHp-spaces. In Section 3, we performed characterization of boundedness and compactness of a composition operator of Cϕ for each of these spaces.We will use the notationU≈V to indicate that the quantities are comparable; that is, there exist two positive constants C1,C2 that satisfy C1V≤U≤C2V. Similarly, if only the first or second inequality holds, we say U≽V or U≺¯V.


## 2. Harmonic Spaces

In this section, we present some useful properties and auxiliary results to discussQHp.Forf analytic in D, the Bloch constant of f is(5)Bf=supξ∈D1−ξ2f′ξ<∞.We remember that an analytic functionf belongs to ℬα, if and only if Bαf=supξ∈D1−ξ2αf′ξ<∞ (see [3]) with norm(6)fℬα=f0+Bαf<∞.Colonna (see [4]) shows that the Bloch constant of h=f+g¯ can be expressed in terms of the moduli of the derivatives of f and g as follows:(7)bh:=supξ∈D1−ξ2hξξ+hξ¯ξ.Colonna (see [4]) presented the following theorem.Theorem 1.
With h as above,h∈ℬh, if and only if f and g are in ℬ. Furthermore,(8)maxBf,Bg≤ℬh≤Bf+Bg.Forα>0, the harmonic α-Bloch space ℬHα is the collection of all h∈ℋarD such that bhα<∞, where(9)bhα:=supξ∈D1−ξ2αhξξ+hξ¯ξ.The harmonic littleα-Bloch space ℬH,0α is defined as the subspace of ℬHα consisting of the mappings h∈ℋarD such that(10)limξ⟶11−|ξ|2αhξξ+hξ¯ξ=0.This space is a continuation of Zhu’s harmonic mappings of the (analytic)α-Bloch space ℬα. Thus, representing h∈ℋarD as f+g¯ with f,g∈ℍD and g0=0, we can see that hξ=f′ and hξ¯=g′¯. Hence, bhα=supξ∈D1−ξ2αf′ξ+g′ξ, and(11)12Bαf+Bαg≤maxBαf,Bαg≤Bαf+Bαg.Consequently, a harmonic mappingh is associated with ℬHα, if and only if the unique functions f and g are analytic on D and h=f+g¯ with g0=0 and are in the ℬα space (see [2, 5] for additional details on the spaces ℬHα). The space ℬα is the classical Bloch space ℬ for α=1, and the appropriate harmonic extension will be represented by ℬHα. Automorphisms of the unit disk are of the form(12)σνξ=λν−ξ1−ν¯ξwithλ=1andν<1.In particular, ifλ=1, then we obtain the involution automorphism that exchanges 0 and ν.The Green function ofD with pole at w∈D is denoted by gξ,ν=−logσνξ. The pseudohyperbolic disk with a (pseudohyperbolic) center ν∈D and a (pseudohyperbolic) radius r∈0,1 is represented by Dν,r=ξ∈D:σνξ<r.Now, we introduce the concept ofQHp space which generalizes the concept of Qp space as follows.Definition 1.
Forp∈0,∞, a function h∈ℋarD is said to be in the class QHp if(13)Qph=supν∈D∫Dhξξ+hξ¯ξ2gpξ,νdAz1/2<∞.
The norm ofQHp is identified as(14)hQHp=h0+Qph<∞.
It is obvious that the correspondence is ongoingh↣hQHp is a norm on QHp, which we refer to as the harmonic QHp.Because the Green function is conformally invariant, it can be used in a variety of applicationsgσξ,σν=gξ,ν for all σ∈AutD and ξ,ν∈D, and each class is almost self-evident. QHp is conformally invariant in the following sense: if h∈QHp, then Qph°σ=Qph for all σ∈AutD.Remark 2.
hz reduces to h′ when h is an analytic function, and hξ¯=0. Thus, the analytic Qp space is the set of analytic functions in the harmonic Qp space, and the respective Qp norms coincide. However, due to the presence of the mixed product resulting from squaring the quantity hξξ+hξ¯ξ, this choice of the norm does not allow a modification that may lead to a (complex) inner product in this situation. As a result, we will propose an alternative norm on QHp that solves the problem while also extending the analytic Qp norm. We will see that the above-mentioned standard is, in fact, valid. Both the old and new norms are Möbius invariant.Notice that, for the proof of the main results, we need the following lemma.Lemma 1.
Letp∈0,∞, and h∈ℋarD. Then, h∈QHp, if and only if(15)supν∈D∫Dhξξ+hξ¯ξ21−σνξ2pdAξ1/2<∞.proof 1.
Proof. By the inequalities−2logt≥1−t2,t∈0,1; as well as(16)−logt≤41−t2,t∈14,1,it suffices to show(17)Qph2≺¯supν∈D∫Dhξξ+hξ¯ξ21−σνξ2pdAξ.
Because∫T|hξξ|+hξ¯ξ2dζ is a nondecreasing function of r∈0,1, one has(18)∫T|hξξ|+hξ¯ξ2dζ≺¯∫D/D0,1/4|hξξ|+hξ¯ξ21−ξ2dAξ,which implies(19)∫D/D0,1/4|hξξ|+hξ¯ξ21−ξ2dAξ≺¯Qph2.
This, together with equation (17), applies to h°σν, and hence(20)Qph≈supν∈D∫Dhξξ+hξ¯ξ21−σνξ2pdAξ1/2.
We use the following lemma due to Songxiao [6].Lemma 2.
LetX=Qp. Then,(1)
On compact sets, every bounded sequencefn in X is uniformly bounded(2)
For any sequencefn in X such that fn⟶0,fn−fn0⟶0 uniformly on compact setsWe state and prove the following corollary.Corollary 1.
Ifh is the real part and imaginary part of an analytic function f, then(21)Qph=Qpf.proof 1.
Letf=Reh, then h can be written as h=1/2f+f¯. Thus,(22)Qph=supν∈D∫D12f′ξ+12f′ξ21−φνξ2pdAξ1/2=supν∈D∫Df′ξ21−φνξ2pdAξ1/2=Qpf.
Likewise, iff=Imh, then h can be written as h=1/2if−1/2if¯. So(23)Qph=supν∈D∫D12f′ξ+12f′ξ21−φνξ2pdAξ1/2=supν∈D∫Df′ξ21−φνξ2pdAξ1/2=Qpf.Theorem 2.
: Let0<p<∞ and f,g∈HD. Then, f,g∈Qp if and only if f+g¯∈QHp. Moreover, if g0=0, then(24)12fQp+gQp≤f+g¯QHp≤2fQp+gQp.
The above estimate holds even without the assumptiong0=0, as shown in the proof.proof 1.
Assumef,g∈Qp and let h=f+g¯. Then, f′=hξ,g′=h¯ξ¯. So, using the inequality(25)|hξξ|+hξ¯ξ2<4hξξ2+hξ¯ξ2.
This derives from the fact that fora,b≥0,(26)a+b22≤maxa,b2=maxa2,b2≤a2+b2,we have(27)Qph2=supν∈D∫Dhξξ+hξ¯ξ21−φνξ2pdAξ≤4supν∈D∫Df′ξ21−φνξ2pdAξ+supν∈D∫Dg′ξ21−φνξ2pdAξ<∞.
Therefore,h∈QHp and(28)Qpf+g¯2≤4Qpf2+Qpg2.
Using the 2-th root, we have(29)Qpf+g¯≤2Qpf2+Qpg21/2<2Qpf+Qpg.
Then, we derive the upper estimate using the inequalityf0+g¯0≤f0+g0, which holds even without the assumption g0=0.
Conversely, assumef+g¯∈QHp. Then, observing that(30)f′ξ2+g′ξ2≤|f′ξ|+g′ξ2,we obtain(31)supν∈D∫Df′ξ21−φνξ2pdAξ+supν∈D∫Dg′ξ21−φνξ2pdAξ≤supν∈D∫Dhξξ+hξ¯ξ21−φνξ2pdAξ,which is finite. Thus, f,g∈Qp and(32)Qpf2+Qpg2≤Qpf+g¯2≤QHp.
Hence, noting by (17) that(33)12Qpf+Qpg≤Qpf2+Qpg21/2.
When we combine these two inequalities, we get(34)12Qpf+Qpg≤Qpf+g¯.
Sinceg0=0,1/2f0≤|f0=f0+g¯0. We deduce that(35)12fQp+gQp≤f+g¯QHp,and then we get the desired result.
The relationship between theQp seminorm of a harmonic mapping and the Qp seminorms of the corresponding real and imaginary parts is then examined.Proposition 1.
Leth be harmonic on D and let u=Reh and v=Imh. Then, h∈QHp, if and only if u,v∈Qp. Moreover,(36)14uQHp+vQHp≤hQHp≤uQHp+vQHp.proof 1.
Assume thatβ,γ∈QHp are true. The upper estimate arises immediately from the triangle inequality of the norm due to linearity. Assume h∈QHp and notice that(37)Jβ,γ=βxγy−γxβy,gives us(38)2Jβ,γ≤∇β2+∇γ2,where ∇β=βx,βy, and ∇γ=γx,γy.
As a result, we have(39)∇β2+∇γ2+2Jβ,γ1/2+∇β2+∇γ2−2Jβ,γ1/2≥2∇β2+∇γ21/2.
Indeed, when equation (39) is squared, the left-hand side yields(40)∇β2+∇γ2+2Jβ,γ+∇β2+∇γ2−2Jβ,γ+2∇β2+∇γ22−4Jβ,γ21/2.
As a result, we get by canceling opposite terms, ignoring the last word, and merging like terms.2∇β2+∇γ2, which is the square of the right-hand side of the equation (39).
By first expressing the partials with respect toξ and ξ¯ in terms of the partials with respect to x and y, then computing the modulus, and lastly applying the inequality equation (39), we can now compute hξ+hξ¯ in terms of β and γ.(41)hξ+hξ¯=βξ+iγξ+βξ¯+iγξ¯=12βx+γy+iβx−iγy+12βx−γy+iγx+βy=12βx+γy2+γx−βy21/2+12βx−γy2+γx+βy21/2=12∇β2+∇γ2+2Jβ,γ1/2+12∇β2+∇γ2−2Jβ,γ1/2≥12∇β2+∇γ21/2≥12∇β2+∇γ2,where we applied the inequality in the last step(42)ξ1,ξ2≥ξ1+ξ22forξj∈ℂ,j=1,2.
Multiplying by1−ξ2 and taking the supremum overall ξ∈D, as well as by a real-valued harmonic Qp function h, we get(43)Qph2≥12supξ∈D∫D∇βξ+∇γξ21−φνξ2pdAξ≥12maxQβp,Qγp≥14Qβp+Qγp.
Using inequality equation (42) once more, we have(44)h0≥12β0+γ0.
When equations (43) and (44) are combined, the result is(45)hQHp≥14βQp+γQp,showing that β and γ are in QHp, and that the lower estimate is correct.Theorem 3.
QHp is a Banach space under the harmonic norm.proof 1.
We simply need to prove completeness now that we have established thatQHp is a normed linear space. Assume that hn in QHp is a Cauchy sequence. The analytic functions are defined by the lower norm estimate in Theorem 2 for each n∈N, fn,gn are in Qp such that gn0=0 and hn=fn+g¯n, and the sequences fn and gn are Cauchy. Since Qp is complete, these sequences converge in the Qp norm to some analytic functions f and g, respectively.
Defineh=f+g¯. Then, by the upper estimate in Theorem 2, h∈QHp and(46)hn−hQHp≤2fn−fQp+gn−gQp⟶0,as n⟶∞. Thus, hn⟶h in QHp, proving the completeness of QHp.Theorem 4.
QHp⊆ℬh, for 0<p<∞. In addition, there is a constant C>0 that is purely dependent on p, such that for h∈QHp,(47)hℬh≤ChQHp.proof 1.
Leth∈QHp be a harmonic on D, and f and g be analytic on D, so that g0=0 and h=f+g¯. Then, according to Theorem 2, f,g∈Qp. As a result of Lemma 1.1 [7] f,g∈ℬ and(48)fℬ+gℬ≤C0fQHp+gQHp,for some positive constant C0 only dependent on p. Using Theorem 2, we see that f+g¯∈ℬh. Finally, using again Theorem 2, we see that(49)hℬh≤fℬ+gℬ≤c0fQp+gQp≤2c0hQHp.
The Möbius invariance of the analyticQp spaces is then extended to their harmonic counterparts.Theorem 5.
QHp is Möbius invariant Banach space for 0<p<∞.proof 1.
Rotations clearly have no effect on the seminorm. As a result, we just need to show that for each
h∈QHp and ν∈D,(50)Qph°φν=Qph.
Given thatφν−1 is the inverse of φν, the identities are(51)1−ξ2φ′ξ|=1−φνξ2,and(52)φν′φνξ=1φν′ξ,hold at each ξ∈D, and making the change of variables w=φνξ, we have(53)Qph°φν2=supν∈D∫D1−φνξ2ph°φνξξ+h°φνξ¯ξ2dAξ=supν∈D∫D1−φνξ2pφν′ξ2hξφνξ+hξ¯φνξ2dAξ=supν∈D∫D1−w2pφν′φνw2hξw+hξ¯w2φν′w2dAξ=supν∈D∫D1−w2p1φν′w2hξw+hξ¯w2φν′w2dAξ=supν∈D∫D1−w2phξw+hξ¯w2dAξ=Qph2,as desired.Theorem 6.
For0<p<∞, there is a constant C>0 such that for all h∈QHp and all z∈D,(54)hξ≤C1+12log1+ξ1−ξ1/2hQHp.proof 1.
Leth∈QHp and z∈D be fixed. Assume ξ≠0 because the result is plainly valid for ξ=0. The triangle inequality and by Proposition 1 in [8] then show that there exists a positive constant c0 such that(55)hξ≤fξ+gξ≤c01+12log1+ξ1−ξ1/2fQp+gQp≤2c01+12log1+ξ1−ξ1/2hQHp,where Theorem 2 was used to get the last inequality. Taking C=2c0 as an example, the following is the outcome.
In a similar way as Lemma 2.9 in [7], we state and prove the following theorem in harmonic Qp space.Theorem 7.
Let0<p<∞. Then,(a)
Every bounded sequencehn in QHp is uniformly bounded on compact subsets of D(b)
For any sequencehn in QHp such that hnQHp⟶0, the sequence hn−hn0 converges to 0 uniformly on compact subsets of Dproof 1.
To provea, set M=supn∈NhnQHp. Let C is compact in D and let r0=maxξ∈Cξ∈0,1. Then, by Theorem 5,(56)supξ∈Dhnξ≤2c01+12log1+r01−r01/2M.
Therefore,hn in QHp is uniformly bounded on C.
Next, to proveb, let C be compact in D and let r0 be as above. Since hnQHp⟶0, by the definition of norm in QHp, it follows that hn0⟶0 as n⟶∞.
By Theorem6, and the triangle inequality, we get(57)hnξ−hn0≤2c01+12log1+r01−r01/2hnξ−hn0QHp≤2c01+12log1+r01−r01/2hnzQHp+hn0.
Therefore,hn−hn0⟶0 uniformly on C.


## 3. Boundedness and Compactness Operator Norm

The following characterization of boundedness and compactness of a composition operator ofCϕ sending one subclass to another is obtained by connecting the Qp norm of a harmonic mapping h=f+g¯ (such that g0=0) with the Qp norms of the associated analytic functions f and g.Corollary 2.
Letp∈0,∞ and f∈H satisfy(58)Nf=supb∈ℂ∫ζ−b≤1nf,ζdAζ<∞.
Then,f∈Qp if and only if f∈ℬ.We need the following lemmas in the sequel.Lemma 3 (see [9]).
Letα∈0,∞ and suppose that fξ=∑j=1∞νjznj belongs to the Hadamard gap class. Then, f∈ℬα, if and only if(59)supj∈ℕνjnj1−α<∞,whereℕ=1,2,3,….Lemma 4.
There exist two functionsh1,h2∈ℬα such that(60)h1ξξ+h1ξ¯ξ+h2ξξ+h2ξ¯ξ≽1−ξ2−α,ξ∈D.proof 1.
For a large numberμ∈ℕ, choose a gap series(61)h1ξ=∑j=0∞ξμj,z∈D.
Then, apply Lemma3 to infer that 1−ξαh1ξξ≤λ holds for all ξ∈D, where λ is a constant. Furthermore, let us verify(62)1−ξh1ξξ≥λ,1−μ−k≤z≤1−μ−k+1/2,k∈ℕ.
Observe that for anyξ∈D,(63)h1ξξ≥μkzμk−∑j=0k−1μjzμj−∑k+1∞μjzμj=θ1−θ2−θ3.
And then, fix aξ with ξ∈1−μ−k,1−μ−k+1/2,k∈ℕ, and put x=ξμk.
Thus,(64)1−μ−kμk≤x≤1−μ−k+1/2μk+1/2μ−1/2.
Ifμ is large enough, then for k≥1 one has(65)13≤x≤12μ−1/2,and hence θ1≥μk/3. Since it is easy to establish(66)θ2≤∑j=0k−1μj≤μkμ−1,it remains to deal with the third term θ3. Noting that(67)ξμnμ−1≤ξμk+1μ−1,n≥k+1,namely, in θ3, the quotient of two successive terms is not greater than the ratio of the first two terms, and one finds that the series of θ3 is controlled by the geometric series having the same first two terms. Accordingly, equation (65) is applied to produce(68)θ3≤μk+1ξμk+1∑j=0∞μξμk+2−μk+1j=μk+1ξμk+11−μξμk+2−μk+1=μkμxμ1−μxμ2−μ≤μkμ1/2μ1/21−μ1/2μ3/2−μ1/2.
The preceding estimates forθ1,θ2, and θ3 imply(69)h1zz≥μk4=μk+1/24∈μ1/2≥14μ1/21−ξ≥14μ1/21−ξ.
Similarly, one can show that(70)h1ξ¯ξ≥14μ1/21−ξ.
In a completely similar manner, one can prove that if(71)h2ξ=∑j=0∞ξμj,ξ∈D,then 1−ξ2h2ξξ≤λ for all ξ∈D (owing to Lemma 3) and(72)1−ξh2ξξ≤λ,1−μ−k+1/2≤ξ≤1−ξ−k+1,k∈ℕ,we got(73)h2ξξ≥14μ1/21−ξ.
Similarly, one can show that(74)h2ξ¯ξ≥14μ1/21−ξ.
By addition equations (69), (70), (73), and (74), we have(75)h1ξξ+h1ξ¯ξ+h2ξξ+h2ξ¯ξ≽1−ξ2−α,ξ∈D.
Our lemma is therefore proved.Theorem 8.
Letα,p∈0,∞ and ϕ:D⟶D be holomorphic. Then,(i)
The operatorCϕ:ℬHα⟶QHp is bounded if and only if(76)supν∈D∫Dϕ′ξ21−ϕξ22α1−σνξ2pdAξ<∞.(ii)
The operatorCϕ:ℬHα⟶QHp is compact if and only if ϕ∈QHp and(77)limr⟶1supν∈D∫Dϕ′ξ21−ϕξ22α1−σνξ2pχrdAξ=0.(iii)
The operatorCϕ:QHp⟶ℬHα, is bounded if and only if(78)ϕ′ξ21−ϕξ22α1−ξ2α<∞.(iv)
The operatorCϕ:QHp⟶ℬHα is compact if and only if ϕ∈QHp and(79)limr⟶1supν∈Dϕ′ξ21−ϕξ22αχr1−ξ2α=0.proof 1.
(i)
consists of a simple computation and a reformulation of the definition of boundedness from Lemmas1 and 4.(ii)
Letϕ∈QHp and let f and g be analytic on D such that g0=0 and h=f+g.By Lemma1, we are required to show that if a sequence hn⊂D converges to 0 uniformly on compact subsets of D, then Cφhn converges to 0. For each r∈0,1, set B¯r=D/Br. So h°φνξξ+h°φνξ¯ξ tends to 0 uniformly on B¯r. And hence Lemma 4, for a given ε>0, there is an integer N>1 such that as n≥N,(80)supν∈D∫DCϕhnξξ2+Cϕhnξ¯ξ21−σνξ2pχrdAξ≺¯ϵϕQHp2.Meanwhile, one may deduce from equation (70) and the evolution of the derivatives of ℬα functions that for any ε>0, there exists a δ∈0,1, such that r∈δ,1.(81)supw∈D∫DCϕhnξξ2+Cϕhnξ¯ξ21−σνξ2pχrdAξ<ε.These inequalities, when paired with Lemma4, result in CϕhnzQHp⟶0 as n⟶∞.LetCΦ:ℬHα⟶QHp be compact on the other hand. It is obvious that ϕ∈QHp holds, and hence, equation (70) must be demonstrated. Take hnξ=ξn resp. hnξ=ξn/n1−α if α∈1,∞ resp. α∈0,1. Without loss of generality, we only consider α∈0,1. Because hn is norm bounded in ℬHα, it converges to 0 uniformly on compact subsets of D, φnQHp⟶0. We can deduce from Lemma 3 that for every ε>0, there exists an integer N>1 such that as n≥N,(82)n2supν∈D∫Dϕξ2n−2ϕ′ξ21−σνξ2pdAξ<ε.Thus, for eachr∈0,1,(83)N2αr2N−2supν∈D∫Dϕ′ξ21−σνξ2pχrdAξ<ε.By takingr≥N−α/N−1, we get(84)supν∈D∫Dϕ′ξ21−σνξ2pχrdAξ<ε.With equation (76) in mind, we can see that for each h∈Bℬα and for every ε>0, there is a δ=δh,ε (depending on h and ε), such that as r∈δ,1,(85)Th,ϕ,p,r=supν∈D∫Dh°φνξξ+h°φνξ¯ξ21−σνξ2pχrdAξ<ε.In fact, ifhtξ=htξ for h∈Bℬα and t∈0,1, then ht⟶h uniformly on compact subsets of D as t⟶1. Since CΦ:ℬHα⟶QHp is compact, ht°ϕ−h°ϕQHp⟶0 as t⟶1 Furthermore, from Lemma 4, it yields that for every ϵ>0, there is a t∈0,1 to insure(86)supν∈D∫Dht°φνξξ+ht°φνξ¯ξ2−h°φνξξ+h°φνξ¯ξ21−σνξ2pdAξ<ε.As a result, by equation (76), one has arrived to(87)Th,ϕ,p,r≤2ε+2supν∈D∫Dhn°φνξξ+hn°φνξ¯ξ21−σνξ2pχrdAξ<ε,≤2ε+2htξH∞2supν∈D∫Dϕ′ξ21−σνξ2pχrdAξ∈≤2ε1+htξH∞2.SinceCϕ sends Bℬα to a relatively compact subset of QHp, there exists, for every ϵ>0, a finite collection of functions hl,h2,..,hN in Bℬα, such that for each h∈Bℬα, there is a k∈1,2,…,N to guarantee(88)supν∈D∫Dht°φνξξ+ht°φνξ¯ξ2−h°φνξξ+h°φνξ¯ξ21−σνξ2pdAξ<ε.Now, equation (77) is used to induce a δ=max1≤k≤Nδhk,ε so that as r∈δ,1.Thus,(89)suph∈Bℬαsupν∈D∫Dht°φνξξ+ht°φνξ¯ξ2−h°φνξξ+h°φνξ¯ξ21−σνξ2pdAξ≺¯ε.By Lemma4, there are two functions hl,h2∈Bℬα such that(90)h1ξξ+h1ξ¯ξ+h2ξξ+h2ξ¯ξ≽21−ξ2−α,ξ∈D,for allξ∈D. Thus, equation (78) implies(91)supν∈D∫Dϕ′ξ21−ϕξ22α1−σνξ2pχrdAξ≺¯ε,so that equation (70)follows.(iii)
Assume thatCΦ:QHp⟶ℬHα is bounded. Fix ξo∈D. Let ν=ϕξo and pick hνξ=−logl−ν¯ξ. Then, hν∈QHp with hνQHp≺¯1, due to Corollary 2. By the boundedness of Cϕ, we have(92)hνQHp≽|hνϕ0|+CϕhνℬHα≽1−ξ02αϕ′ξ01−ν¯ϕξ0−1,which deduces equation (73). Conversely, if equation (73) holds, then QHp⊆ℬHα gives that for any h∈QHp,(93)CϕhℬHα≺¯hQHpsupν∈D1−ξ2αϕ′ξ1−ϕξ2<∞.In other words,Cϕ is a bounded operator from QHp to ℬHα.(iv)
LetCϕ:QHp⟶ℬHα be compact. Then, ϕ∈ℬHα follows naturally. Now, suppose otherwise that the condition equation (74) fails. Then, there would be a number ε0>0 and a sequence ξn⊂D such that 1−ξ2αϕ′ξ1−ϕξ2−1≥ε0>0 whenever ϕξn>1−1/n. We may assume that wn=ϕξn tend to a point νo∈T. Put hnξ=−log1−ν¯nξ. Accordingly, hn converges to h0 uniformly on compact subsets of D, where h0ξ=−log1−ν¯nξ.
Appealing to these constructions, we obtain(94)Cϕhn−Cϕh0ℬHα≥1−ξn2αhn°φνξξ+hn°φνξ¯ξ2−h0°φνξξ+h0°φνξ¯ξ2=1−ξn2|αϕ′ξnν¯n1−νn2−ν¯01−barν0νn=1−ξn2|αϕ′ξn|1−ϕνξn2ν¯n−ν¯01−ν¯0νϕn≥ε0,for n∈N, so Cϕhn does not converge to Cϕfo in norm. Hence, Cϕ:QHp⟶ℬHα is not compact. This contradicts the above hypothesis.
On the other hand, letϕ∈ℬHα and equation (76) hold. In order to show that Cϕ:QHp⟶ℬHα is compact, it suffices to verify that if hn is a bounded sequence in QHpi.e.,supn∈N<∞ and if it converges to 0 uniformly on any compact subset of D, then Cϕhn approaches 0. By equation (77), we obtain that for any ϵ>0, there exists a δ∈0,1, such that whenever ϕξ>δ,(95)1−ξ2|αCϕhnξξ+Cϕhnξ¯ξ≤εsupn∈ℕhnℬH≺¯εsupn∈ℕhnQHp.
Yet, ifϕξ≤δ, then 1−ξn2αhnξϕξ+hnξ¯ϕξ⟶0 owing to the fact hn tend to 0 uniformly on compact subsets of D, and hence,(96)1−ξ2|αhnξϕξ+hnξ¯ϕξ≤ϕℬH1−δ2−α1−ξn2αhnξϕξ+hnξ¯ϕξ⟶0.
We are done because the prior estimations forceCϕhnℬH⟶0.


## 4. Conclusion

In this article, we studied the boundedness and compactness of composition operators acting on a certain class of harmonic spaces. The author characterizes when the composition operatorCϕ:ℬHα⟶QHp induced by a holomorphic self-map ϕ of D is bounded and compact. Here, QHp is the collection of all harmonic functions of the unit disk such that(97)Qph=supν∈D∫Dhξξ+hξ¯ξ2gpξ,νdAz1/2<∞.



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