# Weighted Composition Operators from Derivative Hardy Spaces inton-th Weighted-Type Spaces

Authors: Nanhui Hu
Journal: Journal of Mathematics
 (2021)

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## Abstract

The boundedness, compactness, and the essential norm of weighted composition operators from derivative Hardy spaces inton-th weighted-type spaces are investigated in this paper.


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## Body

## 1. Introduction

LetHD denote the space of analytic functions on the open unit disk D. Let SD denote the set of all analytic self-maps of D. Let φ∈SD. The composition operator Cφ with the symbol φ is defined by(1)Cφfz=fφz,f∈HD.Letφ∈SD and ψ∈HD. The weighted composition operator ψCφ is defined on HD by(2)ψCφfz=ψzfφz,f∈HD.It is important to give function theoretic descriptions whenψ and φ induce a bounded or compact weighted composition operator on various function spaces. See references [1, 2] for more information of this research field.For0<p<∞, the Hardy space, denoted by ℋp, consists of all functions f∈HD such that (see [3])(3)fℋpp=sup0<r<112π∫02πfreiθpdθ<∞.As usual,ℋ∞ denotes the space of bounded analytic functions in D. If f′∈ℋp, we say that f belongs to the derivative Hardy space, denoted by Sp. When p>1, the space Sp is a Banach space under the norm defined by(4)fSpp=f0p+f′ℋpp.Whenp=2, S2 is a Hilbert space. In [4], Roan started the study of composition operators on the space Sp. In [5], MacCluer investigated composition operators on the space Sp in terms of Carleson measure. The boundedness and compactness of weighted composition operators on Sp were studied in [6]. See references [4–7] and references therein for more study of composition operators and weighted composition operators on the space Sp.Ifμ is a radial, positive, and continuous function on D, then μ is called a radial weight. Let μ be a radial weight and n∈ℕ, the set of all positive integers. Let Wμn denote the n-th weighted space, which consists of all f∈HD such that(5)fWμn=∑k=0n−1fk0+supz∈Dμzfnz<∞.It is a Banach space with the norm⋅Wμn. When n=1 and n=2, Wμn becomes the Bloch-type space ℬμ and the Zygmund-type space Zμ, respectively. Furthermore, when μz=1−z2, ℬμ=ℬ is the Bloch space and Zμ=Z is the Zygmund space. For some results on the space Wμn, see references [2, 8–13].Letn,k∈ℕ0 with k≤n. Recall that the partial Bell polynomials are defined as follows:(6)Bn,kx1,x2,…,xn−k+1=∑n!j1!j2!,…,jn−k+1!x11!j1x22!j2,…,xn−k+1n−k+1!jn−k+1,where the sum taken over all sequences j1,j2,…,jn−k+1 of nonnegative integers such that the following two conditions hold:(7)j1+j2+⋯+jn−k+1=kj1+2j2+⋯+n−k+1jn−k+1=n.See reference [14] for more information about Bell polynomials.In [15], Stević studied the boundedness and compactness of the composition operator from Aαp to Wμn on the unit disk. In [12], Stević studied the boundedness and compactness of weighted composition operators from ℋ∞ and the Bloch space to Wμn. See references [8–10] for more characterizations for weighted composition operators from ℋ∞ and the Bloch space to Wμn. In [16], Zhu and Du studied the boundedness, compactness, and essential norm of weighted composition operators from weighted Bergman spaces with doubling weight Aωp to Wμn. Recall that the essential norm of a bounded linear operator T:X⟶Y is its distance to the set of compact operators K mapping X into Y, that is,(8)Te,X⟶Y=infT−KX⟶Y:Kisacompact operator.Here,XandY are Banach spaces and ⋅X⟶Y denotes the operator norm. In [17], Colonna and Tjani studied the boundedness and compactness of weighted composition operators from derivative Hardy spaces Sp to Bloch-type space ℬμ.In this paper, we use Bell polynomials to study the boundedness and compactness of weighted composition operators from derivative Hardy spacesSp to n-th weighted-type spaces Wμn. Moreover, we give an estimate for the essential norm of weighted composition operators from Sp to Wμn.Throughout the paper, we denote byC a positive constant which may differ from one occurrence to the next. In addition, we say that A≲B if there exists a constant C such that A≤CB. The symbol A≈B means that A≲B≲A.


## 2. Boundedness

In this section, the boundedness of weighted composition operators fromSp to Wμn is characterized.Lemma 1.
Suppose1<p<∞ and k∈ℕ. Then, there exists a positive constant C such that(9)f∞≤CfSp,fkz≤CfSp1−z2k−1+1/p,for every f∈Sp.Proof.
The first inequality follows from the fact thatSp are contained in the disk algebra for p>1. In addition, it is well known that for every f∈ℋp, there exists a positive constant C such that(10)fkz≤Cfℋp1−z2k+1/p,which implies the second inequality.
For anya∈D, 1<p<∞, and j∈1,2,…,n+1, set(11)fj,az=1−a2j1−a¯zj+1/p−1,z∈D.
After a calculation, for eachj∈1,2,…,n+1,(12)fj,a∈Spsupa∈Dfj,aSp<∞.Lemma 2.
Let1<p<∞ and 0≠a∈D. For any i∈1,…,n, there exist constants c2i,…,cn+1i such that(13)vi,a=f1,a+∑j=2n+1cjifj,a∈Sp,vi,aa=0,vi,aka=a¯i1−a2i+1/p−1,k=i,0,k≠i.
Moreover,vi,a converges to 0 uniformly in D¯.Proof.
The proof is similar to the proof of Theorem 1 in [16]. Hence, we omit the details of the proof.
For the simplicity of this paper, we define(14)Iinz=∑l=innlψn−lzBl,iφ′z,φ″z,…,φl−i+1z.
From the definition, we see that, for example,(15)B0,0x1=1,B1,0x1,x2=0,B1,1x1=x1,B2,0x1,x2,x3=0,B2,1x1,x2=x2,B2,2x1=x12.
This fact will be used in the proof of the following theorem.
Now, we are in a position to state and prove the first result in this paper.Theorem 1.
Letn∈ℕ, 1<p<∞, φ∈SD, ψ∈HD, and μ be a weight. Then, the operator ψCφ:Sp⟶Wμn is bounded if and only if ψ∈Wμn and(16)∑i=1nsupz∈Dμz∑l=innlψn−lzBl,iφ′z,φ″z,…,φl−i+1z1−φz2i+1/p−1<∞.Proof (sufficiency).
Letf∈Sp. After a calculation, we have (see, e.g., [12])(17)ψCφfnz=∑i=0nfiφz∑l=innlψn−lzBl,iφ′z,…,φl−i+1z.
SinceB0,0φ′z=1 and Bl,0φ′z,…,φl+1z=0l∈ℕ, we see that I0nz=ψnz. Hence, by (14) and Lemma 1, we get(18)μzψCφfnz≤μz+μz∑i=1nfiφzIinz,(19)≲fSpψWμn+fSp∑i=1nsupz∈DμzIinz1−φz2i+1/p−1.
From (18), for each j∈0,1,…,n−1,(20)ψCφfj0≤fφ0ψj0+∑i=1jfiφ0Iij0≲fSpψj0+fSp∑i=1jIij01−φ02i+1/p−1.
Therefore, from (19) and (20) and the fact that ψ∈Wμn, we get that ψCφ:Sp⟶Wμn is bounded.
Necessity. Assume that ψCφ:Sp⟶Wμn is bounded. It is clear that ψ∈Wμn, and for each j∈1,…,n+1 and a∈D,(21)supa∈DψCφfj,aWμn<∞,by the boundedness of ψCφ:Sp⟶Wμn.
Next, we show that fori=1,2,…,n,(22)supz∈DμzIinz<∞.
Applying the operatorψCφ for h1z=z, by (17) and (14), we obtain(23)supz∈DμzI0nzφz+I1nz=supz∈DμzψCφh1nz≤ψCφh1Wμn<∞.
Therefore, by the triangle inequality and the boundedness ofφ, we obtain(24)supz∈DμzI1nz<∞.
Now, we assume that for1≤i≤j−1j≤n, supz∈DμzIinz<∞. To get the desired result, we only need to show that(25)supz∈DμzIjnz<∞.
Applying the operatorψCφ for hjz=zj, we obtain(26)supz∈DμzφjzI0nz+∑k=1jjj−1,…,j−k+1φzj−k,Iknz≤ψCφhjWμn<∞.
Hence, from the boundedness ofφ and triangle inequality again, we get the desired result.
For anyi∈1,…,n and φa≠0, by Lemma 2,(27)μaφaiIina1−φa2i+1/p−1≤supa∈DψCφvi,φaWμn≤supa∈DψCφf1,aWμn+∑j=2n+1cjisupa∈DψCφfj,aWμn<∞,where cji are independent of the choice of a. From the last inequality and (22), for any i∈1,…,n, we get(28)supφa>1/2μaIina1−φa2i+1/p−1≲supa∈DψCφf1,aWμn+∑j=2n+1cjisupa∈DψCφfj,aWμn<∞,supφa≤1/2μaIina1−φa2i+1/p−1≲supφa≤1/2μaIina<∞.
Therefore,(29)∑i=1nsupz∈DμzIinz1−φz2i+1/p−1<∞.
The proof is complete.
Letn=1. We get the following result, which was first obtained in [17].Corollary 1.
Let1<p<∞, φ∈SD, ψ∈HD, and μ be a weight. Assume that ψCφ:Sp⟶ℬμ is bounded if and only if ψ∈ℬμ and(30)supz∈Dμzψzφ′z1−φz21/p<∞.
Letn=2. We get the following result.Corollary 2.
Let1<p<∞, φ∈SD, ψ∈HD, and μ be a weight. Then, the operator ψCφ:Sp⟶Zμ is bounded if and only if ψ∈Zμ,(31)supz∈Dμzψzφ″z+2ψ′zφ′z1−φz21/p<∞supz∈Dμzψzφ′2z1−φz21+1/p<∞.


## 3. Essential Norm

In this section, we obtain some estimates for the essential norm of the weighted composition operatorψCφ:Sp⟶Wμn, and we need the following lemma.Lemma 3.
(see [17]). Let X 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(1)
The point evaluation functionals onY are continuous(2)
For every sequencefn in the unit ball of X, there exists f∈X and a subsequence fnj such that fnj⟶f uniformly on D¯(3)
The operatorT:X⟶Y is continuous if X has the supremum norm and Y is given the topology of uniform convergence on compact sets
Then,T is a compact operator if and only if, given a bounded sequence fn in X such that fn⟶0 uniformly on D¯, then\ TfnY⟶0 as n⟶∞.Lemma 4.
(see [17]). Let 1<p<∞. Every sequence in Sp bounded in norm has a subsequence which converges uniformly in D¯ to a function in Sp.
The following result is a direct consequence of Lemma3 and Lemma 4.Lemma 5.
Letn∈ℕ, 1<p<∞, and μ be a weight. If T is a bounded linear operator from Sp into Wμn, then T is compact if and only if TfnWμn⟶0 as n⟶∞ for any sequence fn in Sp bounded in norm which convergences to 0 uniformly in D¯.Theorem 2.
Letn∈ℕ, 1<p<∞, φ∈SD, ψ∈HD, and μ be a weight such that ψCφ:Sp⟶Wμn is bounded. Then,(32)ψCφe,Sp⟶Wμn≈∑i=1nlimsupφz⟶1μzIinz1−φz2i+1/p−1.Proof.
First, we prove that(33)∑i=1nlimsupφz⟶1μzIinz1−φz2i+1/p−1≲ψCφe,Sp⟶Wμn.
Ifsupz∈Dφz<1, there is a number δ∈0,1 such that supz∈Dφz<δ. In this case, (32) is vacuously satisfied and the desired result follows.
Assume thatsupz∈Dφz=1. Let zjj∈ℕ be a sequence in D such that φzj⟶1 as j⟶∞. Since ψCφ:Sp⟶Wμn is bounded, for any compact operator K:Sp⟶Wμn and i∈1,…,n, by using Lemma 2 and Lemma 5, we obtain(34)ψCφ−KSp⟶Wμn>˜limsupj⟶∞ψCφvi,φzjWμn−limsupj⟶∞Kvi,φzjWμn>˜limsupj⟶∞μzjφzjiIinzj1−φzj2i+1/p−1.
Hence,(35)ψCφe,Sp⟶Wμn>˜limsupj⟶∞μzjφzjiIinzj1−φzj2i+1/p−1,which implies that(36)ψCφe,Sp⟶Wμn>˜limsupφz⟶1μzIinz1−φz2i+1/p−1,as desired.
Now, we show that(37)ψCφe,Sp⟶Wμn≲∑i=1nlimsupφz⟶1μzIinz1−φz2i+1/p−1.
Letr∈0,1 and define Krfz=frz=frz. Then, Kr:Sp⟶Sp is compact and KrSp⟶Sp≤1. It is clear that fr⟶f uniformly on compact subsets of D as r⟶1. Let rj⊂0,1 be a sequence such that rj⟶1 as j⟶∞. Then, for any positive integer j, the operator ψCφKrj:Sp⟶Wμn is compact. By the definition of the essential norm, we get(38)ψCφe,Sp⟶Wμn≤limsupj⟶∞ψCφ−ψCφKrjSp⟶Wμn.
Hence, it is sufficient to show that(39)limsupj⟶∞ψCφ−ψCφKrjSp⟶Wμn≲∑i=1nlimsupφz⟶1μzIinz1−φz2i+1/p−1.
For anyf∈Sp such that fSp≤1,(40)ψCφ−ψCφKrjfWμn≤∑t=0n−1∑i=0tf−frjiφ0Iit0+supz∈Dμz∑i=0nf−frjiφzIinz≤∑t=0n−1∑i=0tf−frjiφ0Iit0︸Ω0+supz∈Dμzf−frjzI0nz︸Ω1+supφz≤rNμz∑i=1nf−frjiφzIinz︸Ω2+supφz>rNμz∑i=1nf−frjiφzIinz︸Ω3,where N∈ℕ such that rj≥2/3 for all j≥N. Since for any nonnegative integer s, f−frjs⟶0 uniformly on compact subsets of D as j⟶∞. It is clear that(41)limsupj⟶∞Ω0=0limsupj⟶∞Ω2=0.
From Lemma4,(42)limj⟶∞Ω1≤ψWμnlimj⟶∞supz∈Df−frjz=0,while(43)Ω3≤∑i=1nsupφz>rNμzfiφzIinz︸Pi+∑i=1nsupφz>rNμzrjifirjφzIinz︸Qi.
For anyi∈1,…,n, by Lemma 1 and Lemma 2,(44)Pi=supφz>rNμz1−φz2i+1/p−1fiφzφziφziIinz1−φz2i+1/p−1≲fSpsupφz>rNψCφvi,φzWμn≲sup|a|>rNψCφf1,aWμn+∑k=2n+1ckisup|a|>rNψCφfk,aWμn≲∑k=1n+1sup|a|>rNψCφfk,aWμn.
Taking the limit asN⟶∞, by Lemma 1, we get(45)limsupj⟶∞Pi≲limsupφz⟶1μzIinz1−φz2i+1/p−1.
By the same reason, we have(46)limsupj⟶∞Qi≲limsupφz⟶1μzIinz1−φz2i+1/p−1.
Hence, by (40)–(43), (45), and (46), we obtain(47)limsupj⟶∞ψCφ−ψCφKrjSp⟶Wμn=limsupj⟶∞supfSp≤1ψCφ−ψCφKrjfWμn≲∑i=1nlimsupφz⟶1μzIinz1−φz2i+1/p−1.
Hence, from the last two inequalities, we obtain (39). The proof is complete.
From Theorem2 and the well-known result that Te,X⟶Y=0 if and only if T:X⟶Y is compact, we obtain the following corollary.Corollary 3.
Letn∈ℕ, 1<p<∞, φ∈SD, ψ∈HD, and μ be a weight. Assume that ψCφ:Sp⟶Wμn is bounded, then the operator ψCφ:Sp⟶Wμn is compact if and only if(48)∑i=1nlimsupφz⟶1μzIinz1−φz2i+1/p−1=0.
Letn=1. We get the following result, which was first obtained in [17].Corollary 4.
Let1<p<∞, φ∈SD, ψ∈HD, and μ be a weight such that ψCφ:Sp⟶ℬμ is bounded. Then, the operator ψCφ:Sp⟶ℬμ is compact if and only if(49)limsupφz⟶1μzψzφ′z1−φz21/p=0.
Letn=2. We get the following result.Corollary 5.
Let1<p<∞, φ∈SD, ψ∈HD, and μ be a weight such that ψCφ:Sp⟶Zμ is bounded. Then, the operator ψCφ:Sp⟶Zμ is compact if and only if (50)limsupφz⟶1μzψzφ″z+2ψ′zφ′z1−φz21/p=0limsupφz⟶1μzψzφ′2z1−φz21+1/p=0.



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