# Weighted Composition Operators from Dirichlet-Zygmund Spaces into Zygmund-Type Spaces and Bloch-Type Spaces

Authors: Xiangling Zhu
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The boundedness, compactness, and essential norm of weighted composition operators from Dirichlet-Zygmund spaces into Zygmund-type spaces and Bloch-type spaces are investigated in this paper.


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## Body

## 1. Introduction

LetHD denote the space of all analytic functions in the open unit disk D. For 1≤p<∞, the Dirichlet type space Dp−1p is the set of all f∈HD such that
(1)fDp−1pp=f0p+∫Df′zp1−z2p−1dAz<∞,where dAz=1/πdxdy is the normalized Lebesgue area measure. Dp−1p is a Banach space under the norm ·Dp−1p. If f′∈Dp−1p, we say that f belongs to the Dirichlet-Zygmund space, denoted by Zp−1p. To the best of our knowledge, this is the first work to study the Dirichlet-Zygmund space.Recall that the spaceB1, called the minimal Möbius invariant space, is the space of all f∈HD that admit the representation fz=∑j=1∞bjσtjz for some sequence bj in l1 and tj∈D. The norm on f∈B1 is defined by
(2)fB1=inf∑j=1∞bj:fz=∑j=1∞bjσtjz.Here,σaz=a−z/1−a¯z. For any f∈B1, the authors in [1] showed that there exists a constant C>0 such that
(3)C−1∫Df′′zdAz≤f−f0−f′0zB1≤C∫Df′′zdAz.Therefore,Z01 is in fact the space B1.We callv:D⟶ℝ+ a weight, if v is a continuous, strictly positive and bounded function. v is called radial, if vz=vz for all z∈D. Let v be a radial weight. Recall that the Zygmund-type space Zv is the space that consists of all f∈HD such that
(4)fZv=f0+f′0+supz∈Dvzf′′z<∞.Zv is a Banach space under the norm ·Zv. We say that f belongs to the Bloch-type space Bv, if
(5)fBv=f0+supz∈Dvzf′z<∞.When vz=1−z2, Zv=Z is called the Zygmund space, and Bv=B is called the Bloch space, respectively. In particular, Zv is just the Bloch space when vz=1−z22.The weighted space, denoted byHv∞, is the set of all f∈HD such that
(6)fHv∞=supz∈Dvzfz<∞.When vz=vαz=1−z2α0≤α<∞, we denote Hv∞ by Hvα∞. In particular, when α=0, Hv0∞=H∞ is just the bounded analytic function space.We denote bySD the set of all analytic self-maps of D for simplicity. Let φ∈SD and ψ∈HD. The weighted composition operator ψCφ is defined as follows.
(7)ψCφfz=ψzfφz,f∈HD,z∈D.When ψ=1, ψCφ is called the composition operator, denoted by Cφ. See [2, 3] for more results about the theory of composition operators and weighted composition operators.For anyφ∈SD, by the Schwarz-Pick lemma, we see that Cφ:B⟶B is bounded. It was shown in [4] that Cφ:B⟶B is compact if and only if limj⟶∞φjB=0. Motivated by [4], Colonna and Li in [5, 6] studied the operators ψCφ:H∞⟶Z and ψCφ:Lipα⟶Z by ψφjZ and j−αψφjZ, respectively. Here, Lipα is the Lipschtiz space. The composition operator on the space B1 was extensively studied in [1]. In [7], Colonna and Li studied the boundedness and compactness of weighted composition operators from the minimal Möbius invariant space B1Z01 to the Bloch space B. In [8], Li studied the boundedness and compactness of the weighted composition operator ψCφ:B1Z01⟶Z. See [5, 6, 8–17] for more results for composition operators, weighted composition operators, and related operators on the Zygmund space and Zygmund-type spaces.In this paper, we follow the methods of [17] and give some characterizations for the boundedness, compactness, and essential norm of the operator ψCφ:Zp−1p⟶Zμ and ψCφ:Zp−1p⟶Bμ.We denoted byC a positive constant which may differ from one occurrence to the next. In addition, we will use the following notations throughout this paper: A≈B means that there exists a constant C such that A≤CB, while A≈B means that A≲B≲A.


## 2. Main Results and Proofs

In this section, we formulate and prove our main results in this paper.Lemma 1.
Suppose1<p<∞. Then, there exists a positive constant C such that
(8)f′z≤CfZp−1p1−z21/p,f′′z≤CfZp−1p1−z21+1/p,and f∞≤CfZp−1p for every f∈Zp−1p.Proof.
Supposer>0 and g∈HD. Then, there exists a constant C>0 such that
(9)gzp≤C1−z2α+2∫Dz,rgwp1−w2αdAw,which implies that
(10)gz≤CgDp−1p1−z21/pandg′z≤CgDp−1p1−z21+1/p.
The inequalities in (8) hold. Here, Dz,r is the hyperbolic disk (see [3]). From (8), we see that Zp−1p are contained in the disk algebra for p>1. Hence, we get that f∞≤CfZp−1p.Lemma 2.
Let1<p<∞. If f∈Zp−1p, then for all t∈0,1 and z∈D\0, there exists a positive constant C such that
(11)fz−ftzz≤CfZp−1p1−z1−1/p.Proof.
Fixf∈Zp−1p. Let t∈0,1 and z∈D\0. By Lemma 1,
(12)fz−ftzz≤∫1t/zzf′szds≤∫11/zzf′szds≤CfZp−1p∫11/zz1−s2z21/pds≤CfZp−1p1−z1−1/p,as desired.Using Lemma2 and similarly to the proof of Lemma 7 in [18], we get the following lemma.Lemma 3.
Let1<p<∞. Every sequence in Zp−1p bounded in norm has a subsequence which converges uniformly in D¯ to a function in Zp−1p.Lemma 4 (see [5]).
LetX be a Banach space that is continuously contained in the disk algebra, and let Y be any Banach space of analytic functions on D. Suppose that
(i)
The point evaluation functionals onY are continuous(ii)
For every sequencefn in the unit ball of X that exists an f∈X and a subsequence fnj such that fnj⟶f uniformly on D¯(iii)
The operatorT:X⟶Y is continuous if X has the supremum norm and Y is given by the topology of uniform convergence on compact setsThen,T is a compact operator if and only if, given a bounded sequence fn in X such that fn⟶0 uniformly on D¯, then the sequence TfnY⟶0 as n⟶∞.The following result is a direct consequence of Lemmas3 and 4.Lemma 5.
Let1<p<∞ and μ be a weight. If T:Zp−1p⟶Zv is bounded, then T is compact if and only if TfkZv⟶0 as k⟶∞ for any sequence fk in Zp−1p bounded in norm which converge to 0 uniformly in D¯.Theorem 6.
Letv be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1<p<∞, ψ∈HD, and φ∈SD. Then, the following statements are equivalent.
(i)
The operatorψCφ:Zp−1p⟶Zv is bounded(ii)
ψ∈Zv,(13)P≔supz∈Dvz2ψ′zφ′z+ψzφ′′z1−φz21/p<∞,and
(14)Q≔supz∈Dvzψzφ′z21−φz21/p+1<∞,(iii)
ψ∈Zv,
supj≥1j1/p2ψ′φ′+ψφ′′φj−1Hv∞<∞ and supj≥1j1/p+1ψφ′2φj−1Hv∞<∞.Proof.
ii⇒i. For any z∈D and f∈Zp−1p, by Lemma 1, we have
(15)ψCφf0≲ψ0fZp−1p,ψCφf′0≲ψ′0+ψ0φ′01−φ021/pfZp−1p,vzψCφf′′z≤vzψ′′zfφz+vzf′′φzψzφ′z2+vzf′φz2ψ′zφ′z+ψzφ′′z≲vzψ′′zfZp−1p+vz2ψ′zφ′z+ψzφ′′z1−φz21/pφzZp−1p+vzψzφ′z21−φz21/p+1fZp−1p.
Hence,(16)ψCφfZv=ψCφf0+ψCφf′0+supz∈DvzψCφf′′z≲ψ0+ψ′0+ψ0φ′01−φ021/p+P+QfZp−1p<∞.
Therefore,ψCφ:Zp−1p⟶Zv is bounded.
i⇒ii. Applying the operator ψCφ to zj with j=0,1,2 and using the boundedness of ψCφ, we get that ψ∈Zv, ψφ∈Zv, and ψφ2∈Zv. Hence, we obtain
(17)supz∈Dvz2ψ′zφ′z+ψzφ′′z<∞,supz∈Dvzψzφ′z2<∞.
For anya∈D, set
(18)faz=1−a21−a¯z1/p,gaz=1−a221−a¯z1+p/p,z∈D.
It is easy to check that(19)supa∈DfaZp−1p<∞andsupa∈DgaZp−1p<∞.
Therefore, by the boundedness ofψCφ:Zp−1p⟶Zv and arbitrary of a∈D, we get
(20)supa∈DψCφfaZv<∞andsupa∈DψCφgaZv<∞.
Forw∈D, we get
(21)ψCφfφw′′w=ψ′′w1−φw21−1/p+2ψ′wφ′w+ψwφ′′wφw¯p1−φw21/p+1+pψwφ′w2φw¯2p21−φw21/p+1,(22)ψCφgφw′′w=ψ′′w1−φw21−1/p+1+pp2ψ′wφ′w+ψwφ′′wφw¯1−φw21/p+1+p1+2pp2ψwφ′w2φw¯21−φw21/p+1.
From (21) and (22), we obtain
(23)−1+pψCφfφw′′w+ψCφgφw′′w+pψ′′w1−φw21−1/p=1+pψwφ′w2φw¯2p1−φw21/p+1,(24)2ψ′wφ′w+ψwφ′′wφw¯1−φw21/p=−ψCφfφw′′w+ψCφgφw′′w−21+pψwφ′w2φw¯2p1−φw21/p+1=1+2pψCφfφw′′w−ψCφgφw′′w−2pψ′′w1−φw21−1/p.
From (24), we get
(25)supw∈Dvw2ψ′wφ′w+ψwφ′′wφw1−φw21/p≤1+2pψCφfφwZv+ψCφgφwZv+2pψZv<∞.
On one hand, from (25), we obtain
(26)supφw>1/2vw2ψ′wφ′w+ψwφ′′w1−φw21/p<∞.
On the other hand, from the fact thatψ,ψφ∈Zv, we get
(27)supφw≤1/2vw2ψ′wφ′w+ψwφ′′w1−φw21/p≤431/psupz∈Dvw2ψ′wφ′w+ψwφ′′w≲supz∈DψφZv+2ψZv<∞.
From (26) and (27), we see that P is finite. Using similar arguments, we see that Q is also finite.
ii⇔iii. From [19], we see that the inequality in is equivalent to the operator 2ψ′φ′+ψφ′′Cφ:Hv1/p∞⟶Hv∞ is bounded. By [20], the boundedness of 2ψ′φ′+ψφ′′Cφ is equivalent to
(28)supj≥12ψ′φ′+ψφ′′φj−1Hv∞zj−1Hv1/p∞<∞.
From [21], we get limj⟶∞j1/pzj−1Hv1/p∞=2/pep, which together with (28) imply that
(29)supj≥1j1/p2ψ′φ′+ψφ′′φj−1Hv∞≈supj≥1j1/p2ψ′φ′+ψφ′′φj−1Hv∞j1/pzj−1Hv1/p∞<∞.
Similarly, the inequality in is equivalent to(30)supj≥1j1/p+1ψφ′2φj−1Hv∞≈supj≥1j1/p+1ψφ′2φj−1Hv∞j1/p+1zj−1Hv1/p+1∞=supj≥1ψφ′2φj−1Hv∞zj−1Hv1/p+1∞<∞.
The proof is complete.Next, we consider the essential norm ofψCφ:Zp−1p⟶Zv. Recall that the essential norm of T:X⟶Y is its distance to the set of compact operators K:X⟶Y, that is,
(31)Te,X⟶Y=infT−KX⟶Y:Kisacompactoperator.Here,X,Y are Banach spaces, and T is a bounded linear operator.Theorem 7.
Letv be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1<p<∞, ψ∈HD, and φ∈SD. Suppose that ψCφ:Zp−1p⟶Zv is bounded. Then,
(32)ψCφe,Zp−1p⟶Zv≈maxE,G≈maxM,T.Here,(33)E≔limsupφz⟶1vz2ψ′zφ′z+ψzφ′′z1−φz21/p,G≔limsupφz⟶1vzψzφ′z21−φz21/p+1,M≔limsupj⟶∞j1/p2ψ′φ′+ψφ′′φj−1Hv∞,T≔limsupj⟶∞j1/p+1ψφ′2φj−1Hv∞.Proof.
First we show thatψCφe,Zp−1p⟶Zv≳maxE,G. Let zjj∈ℕ be a sequence in the unit disk such that φzj⟶1 as j⟶∞. Define
(34)kjz=1−φzj21−φzj¯z1/p−1+2p1+3p−1+p2p1+2p1−φzj221−φzj¯z1/p+1+1+p1+2p1−φzj231−φzj¯z1/p+2,mjz=1−φzj21−φzj¯z1/p−21−φzj221−φzj¯z1/p+1+1−φzj231−φzj¯z1/p+2.
After a calculation, we get allkj and mj belong to Zp−1p and
(35)kjφzj=0,k′′jφzj=0,k′jφzj=p1+2pφzj1−φzj21/p,mjφzj=0,m′jφzj=0,m′′jφzj=2φzj21−φzj21/p+1.
Moreover,kj and mj converge to 0 uniformly on D¯ as j⟶∞. Hence, for any compact operator K:Zp−1p⟶Zv, by Lemma 5, we get
(36)ψCφ−KZp−1p⟶Zv≳limsupj⟶∞ψCφkjZv−limsupj⟶∞KkjZv≳limsupj⟶∞vzj2ψ′zjφ′zj+ψzjφ′′zjφzj1−φzj21/p,ψCφ−KZp−1p⟶Zv≳limsupj⟶∞ψCφmjZv−limsupj⟶∞KmjZv≳limsupj⟶∞vzjψzjφ′zj2φzj21−φzj21/p+1.
Hence,(37)ψCφe,Zp−1p⟶Zv=infKψCφ−KZp−1p⟶Zv≳limsupj⟶∞vzj2ψ′zjφ′zj+ψzjφ′′zjφzj1−φzj21/p=limsupφz⟶1vz2ψ′zφ′z+ψzφ′′z1−φz21/p=E,ψCφe,Zp−1p⟶Zv≳limsupφz⟶1vzψzφ′z21−φz21/p+1=G,as desired.
Next, we show that(38)ψCφe,Zp−1p⟶Zv≲maxE,G.
Letr∈0,1. Define Kr:HD⟶HD by
(39)Krfz=frz=frz,f∈HD.
It is clear thatKr is compact on Zp−1p and KrZp−1p⟶Zp−1p≤1. Moreover, fr−f⟶0 uniformly on compact subsets of D as r⟶1. Let rj⊂0,1 such that rj⟶1 as j⟶∞. Then, ψCφKrj:Zp−1p⟶Zv is compact for each j∈ℕ. Hence,
(40)ψCφe,Zp−1p⟶Zv≤limsupj⟶∞ψCφ−ψCφKrjZp−1p⟶Zv.
Thus, we only need to prove that(41)limsupj⟶∞ψCφ−ψCφKrjZp−1p⟶Zv≲maxE,G.
For anyf∈Zp−1p with fZp−1p≤1, by the facts that
(42)limj⟶∞ψ0fφ0−ψ0frjφ0=0,limj⟶∞ψ′0f−frj′φ0+ψ0f−frj′φ0φ′0=0,we have
(43)limsupj⟶∞ψCφ−ψCφKrjfZv=limsupj⟶∞vzψ·f−frj∘φ′′z≤limsupj⟶∞supφz≤rtvzf−frj′φz2ψ′zφ′z+ψzφ′′z+limsupj⟶∞supφz>rtvzf−frj′φz2ψ′zφ′z+ψzφ′′z+limsupj⟶∞supz∈Dvzf−frjφzψ′′z+limsupj⟶∞supφz≤rtvzf−frj′′φzφ′z2ψz+limsupj⟶∞supφz>rtvzf−frj′′φzφ′z2ψz≔P1+P2+P3+P4+P5,where t∈ℕ is large enough such that rj≥1/2 for all j≥t. Since frj−f⟶0, rjf′rj−f′⟶0, and rj2frj′′−f′′⟶0 uniformly on compact subsets of D as j⟶∞, by Lemma 3, we obtain
(44)P3=limsupj⟶∞supz∈Dvzf−frjφzψ′′z≤ψZvlimsupj⟶∞supw∈Dfw−frjw=0,(45)P1=limsupj⟶∞supφz≤rtvzf−frj′φz2ψ′zφ′z+ψzφ′′z≲ψφZv+ψZvlimsupj⟶∞supw≤rtf′w−rjf′rjw=0,(46)P4=limsupj⟶∞supφz≤rtvzf−frj′′φzφ′z2ψz≲ψφ2Zv+ψφZv+ψZvlimsupj⟶∞supw≤rtf′′w−rj2f′′rjw=0.
Using Lemma1 and fZp−1p≤1, we obtain
(47)P2=limsupj⟶∞supφz>rtvzf−frj′φz2ψ′zφ′z+ψzφ′′z≲limsupj⟶∞f−frjZp−1psupφz>rtvz2ψ′zφ′z+ψzφ′′z1−φz21/p.
Taking the limit ast⟶∞, we get
(48)P2≲E.
Similarly,(49)P5=limsupj⟶∞supφz>rtvzf−frj′′φzφ′z2ψz≲limsupj⟶∞f−frjZp−1psupφz>rtvzφ′z2ψz1−φz21/p+1.
Taking the limit ast⟶∞, we get
(50)P5≲G.
Hence, by (43), (44), (45), (46), (48), and (50), we get
(51)limsupj⟶∞ψCφ−ψCφKrjZp−1p⟶Zv≲maxE,G,which with (40) implies the desired result.
Finally, we prove that(52)ψCφe,Zp−1p⟶Zv≈maxM,T.
On one hand, by the proof of Theorem6, we see that the boundedness of ψCφ:Zp−1p⟶Zv is equivalent to the boundedness of 2ψ′φ′+ψφ′′Cφ:Hv1/p∞⟶Hv∞ and ψφ′2Cφ:Hv1/p+1∞⟶Hv∞. From [19, 20], we have
(53)2ψ′φ′+ψφ′′Cφe,Hv1/p∞⟶Hv∞≈M,ψφ′2Cφe,Hv1/p+1∞⟶Hv∞≈T.
Hence,(54)ψCφe,Zp−1p⟶Zv≲2ψ′φ′+ψφ′′Cφe,Hv1/p∞⟶Hv∞+ψφ′2Cφe,Hv1/p+1∞⟶Hv∞≲M+T≲maxM,T.
On the other hand, from [19, 21], we have
(55)ψCφe,Zp−1p⟶Zv≳E=2ψ′φ′+ψφ′′Cφe,Hv1/p∞⟶Hv∞=limsupj⟶∞2ψ′φ′+ψφ′′φj−1Hv∞zj−1Hv1/p∞≈M,ψCφe,Zp−1p⟶Zv≳G=ψφ′2Cφe,Hv1/p+1∞⟶Hv∞=limsupj⟶∞ψφ′2φj−1Hv∞zj−1Hv1/p+1∞≈T.
Therefore,(56)ψCφe,Zp−1p⟶Zv≳maxM,T.
The proof is complete.From Theorem7 and the well-known result that Te,X⟶Y=0 if and only if T:X⟶Y is compact, we get the following corollary.Corollary 8.
Letv be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1<p<∞, ψ∈HD, and φ∈SD. Suppose that ψCφ:Zp−1p⟶Zv is bounded. Then, the following statements are equivalent:
(i)
The operatorψCφ:Zp−1p⟶Zv is compact(ii)
limsupφz⟶1vzψzφ′z2/1−φz21/p+1=0 and limsupφz⟶1vz2ψ′zφ′z+ψzφ′′z/1−φz21/p=0(iii)
limsupj⟶∞j1/p+1ψφ′2φj−1Hv∞=limsupj⟶∞j1/p2ψ′φ′+ψφ′′φj−1Hv∞=0
Similarly to the above proof, we can get the characterizations of the boundedness, compactness, and essential norm of the weighted composition operatorψCφ:Zp−1p⟶Bv as follows. The details are left to the interested readers.Theorem 9.
Letv be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1<p<∞, ψ∈HD, and φ∈SD. Then, the following statements are equivalent.
(i)
ψCφ:Zp−1p⟶Bv is bounded(ii)
ψ∈Bv and(57)supz∈Dvzψzφ′z1−φz21/p<∞.(iii)
ψ∈Bv and supj≥1j1/pψφ′φj−1Hv∞<∞Theorem 10.
Letv be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1<p<∞, ψ∈HD, and φ∈SD. Suppose that ψCφ:Zp−1p⟶Bv is bounded. Then,
(58)ψCφe,Zp−1p⟶Bv≈limsupφz⟶1vzψzφ′z1−φz21/p≈limsupj⟶∞j1/pψφ′φj−1Hv∞.Corollary 11.
Letv be a radial, nonincreasing weight tending to zero at the boundary of D. Let 1<p<∞, ψ∈HD, and φ∈SD. Suppose that ψCφ:Zp−1p⟶Bv is bounded. Then, the following statements are equivalent:
(i)
The operatorψCφ:Zp−1p⟶Bv is compact(ii)
limsupφz⟶1vzψzφ′z/1−φz21/p=0(iii)
limsupj⟶∞j1/pψφ′φj−1Hv∞=0



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