# Boundedness and Compactness of Specific Weighted Composition Operators on the Zygmund-Orlicz Space

Authors: Hang Zhou
Journal: Journal of Function Spaces
 (2022)

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## Abstract

We sayφ∈U if a constant Cφ>1 could be found, such that φst≤Cφsφt,s>1,t>0. Under the assumption φ∈U, boundedness and compactness of weighted composition operators on the Zygmund-Orlicz space Zφ are investigated in this paper.


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## Body

## 1. Introduction

The composition operator, defined by an analytic self-map of the unit diskϕ, could be defined as
(1)Cϕf=f∘ϕ,f∈HD,where HD is, as usual, the collection of all analytic functions on D. For ϕ∈SD, ϕn denotes its iterates of n times, n=1,2,⋯. Specially, ϕ0 turns to the identity mapping. Moreover, ϕ−n appears when the analytic self-map ϕ is invertible.For a given analytic functionψ∈HD, the multiplication operator Mψf=ψ·f can be induced. Then, the weighted composition operator ψCϕ is induced by
(2)ψCϕfz=ψzfϕz,f∈HD,z∈D.Theory of (weighted) composition operators has been establishing since the last century. Properties including boundedness, compactness, essential norm, and spectral properties are always the highlights of research of composition operators. To study the composition operators on classical spaces of analytic functions, book [1] is a good reference. Moreover, this theory is also established on the basis of theory of analytic functions (on the unit disk), which is basically a convenient tool.Recall that Zygmund space is defined as(3)Z=f∈HD:fZ=supz∈D1−z2f′′z<∞.Zygmund space is a Banach space of holomorphic functions endowed with the normf=f0+f′0+fZ.Generally, for a positive continuous functionμz on D, μ−Zygmund space Zμ is defined by
(4)Zμ=f∈HD:fZμ=supz∈Dμzf′′z<∞.Zμ becomes a Banach spaces with fμ=f0+fZμ.It is well-known that Hardy space, Bergman space, and Bloch space are important in the history of spaces of analytic functions. Motivated by the classical Orlicz space, the Hardy-Orlicz space, Bergman-Orlicz space, and Bloch-Orlicz space are studied in the recent years (see, e.g., [2–11]).Motivated by the same approach, for a Young’s functionφ, Zygmund-Orlicz space Zφ is defined as
(5)Zφ=f∈HD:supz∈D1−z2φλf′′z<∞,where λ>0 is a positive number depending of f.By the similar discussion ofφ with the function Ψt=∫0tφx/xdx,t≥0, we conduct that Zφ becomes a Banach space with the norm
(6)fZφ≔f0+f′0+fφ,where
(7)fφ=infk>0:Sφf′′k≤1,Sφf≔supz∈D1−z2φfz.Specially, we sayφ∈U if there exists a constant Cφ>1 such that
(8)φst≤Cφsφt,s>1,t>0.Note that this also implies that(9)sφ−1t≤φ−1st≤Cφsφ−1t,s<1,t>0,sφ−1t≥φ−1st≥1Cφsφ−1t,s>1,t>0.Under the hypothesisφ∈U, the boundedness and compactness of weighted composition operators on the Zygmund-Orlicz space are investigated, respectively.


## 2. Auxiliary

Basic properties ofZφ are given in this section. Since most of them are similar arguments with the ones of Bloch-Orlicz space(see [3]), we omit the proofs.Proposition 1.
For eachf∈Zφ,
(10)Sφf′′fZφ≤Sφf′′fφ≤1.Proposition 2.
The Zygmund-Orlicz space is isometrically equal to theμ-Zygmund space, where
(11)μz=1φ−11/1−z2.In other words,(12)fZφ=f0+f′0+supz∈Dμzf′′zholds for each f∈Zφ.The proposition above is of vital significance, which states thatZφ is isometrically equal to μ-Zygmund space.For eachα>0,f∈Zφ,z∈D, by Proposition 2, we have that
(13)fz≤2+φ−112fZφlog41−z2≲log41−z2,(14)f′z≤1+φ−112fZφlog41−z2≲log41−z2,(15)f′′z≤φ−11fZφ11−z2≲11−z2.It follows the following.Proposition 3.
The evaluation functionalez is continuous on Zφ, where z∈D is fixed.Moreover, the equivalent condition below is easily conducted.Corollary 4.
The equivalent condition(16)Sφf′′≤1⇔fφ≤1holds for each f∈Zφ.Specially, for two real numbersA and B, we say A°B if there exists a constant C≠0 such that A≤CB.


## 3. Boundedness of Weighted Composition Operator onZφ

Under the assumptionϕ∈U, the boundedness of ψCϕ on the Zygmund-Orlicz is investigated in this section.Basic properties of the testing functions entailed to the proof of the boundedness of the composition operatorCϕ are described in Lemma 5.Lemma 5.
For a fixeda∈D such that ϕa≠0,1, suppose that
(17)faz=1a¯∫0z1−a221−a¯t2−1−a21−a¯tdt,f~az=∫0z3log4/1−a¯t2log4/1−a2−2log4/1−a¯t3log4/1−a22dt,where z∈D. Then,
(i)
fa∈Zφ and f~a,f~′a∈Zφ, where φ∈U(ii)
f′′aa=1/1−a2, f~′aa=log1/1−a2, and fa′a=f~′′aa=0.Proof.
Observing that(18)supz∈Dμzf′′az=supz∈D1φ−11/1−z221−a221−a¯z3+1−a21−a¯z2≲supz∈D1−a21−z21−a¯z2+21−a221−z21−a¯z3≲1,supz∈Dμzf~′′az=supz∈D1φ−11/1−z2·3a1−a¯z2log4/1−a¯zlog4/1−a2−log4/1−a¯z2log4/1−a22≲1.
Moreover,(19)supz∈Dμzf~′′′az=supz∈D1φ−11/1−z2a¯21−a¯z2log4/1−a238−34log4/1−a¯zlog4/1−a2≲1.Theorem 6.
Forφ∈U, ψCϕ is bounded on Zφ if and only if
(20)M1≔supz∈Dψ′′zlog4/1−ϕz2φ−11/1−z2<∞,M2≔supz∈D2ψ′zϕ′z+ψzϕ′′zlog4/1−ϕz2φ−11/1−z2<∞,M3≔supz∈Dψzϕ′z21/1−ϕz2φ−11/1−z2<∞.Proof.
Suppose thatM1,M2,M3<∞. For any f∈Zφ, we have that
(21)SφψCϕf′′MfZφ=supz∈D1−z2φψ′′zfϕz+2ψ′zϕ′z+ψzϕ′′zf′ϕzMfZφ+ψzϕ′z2f′′ϕzMfZφ≤supz∈D1−z2φψ′′zfϕzMfZφ+2ψ′zϕ′z+ψzϕ′′zf′ϕzMfZφ+ψzϕ′z2f′′ϕzMfZφ≤supz∈D1−z2φψ′′zlog4/1−ϕz22+φ−11/2fZφM∥f∥Zφ+2ψ′zϕ′z+ψzϕ′′zlog4/1−ϕz21+φ−11/2fZφMfZφ+ψzϕ′z21/1−ϕz2φ−11fZφMfZφ≲supz∈D1−z2φ2+φ−11/2M1+1+φ−11/2M2+φ−11M3M=1,·where M is a number which satisfies 2+φ−11/2M1+1+φ−11/2M2+φ−11M3≤M and (15), (14), and (13) are employed. Note that, by (13), (14), and (15), fϕ0≲fZφ,f′ϕ0≲fZφ, and f′′ϕ0≲fZφ.Then, the boundedness of theψCϕ on Zφ is guaranteed by ψCϕfφ≤MfZφ and the above estimations.Conversely, ifψCϕ is bounded on Zφ, then there exists a constant C≥0 such that ψCϕfZφ≤CfZφ for each 0≠f∈Zφ. It follows by Proposition 1 that
(22)SφψCϕf′′CfZφ≤SφψCϕf′′CϕfZφ≤1holds for each f∈Zφ, which is equivalent with
(23)supz∈D1−z2,·φψ′′zfϕz+2ψ′zϕ′z+ψzϕ′′zf′ϕz+ψzϕ′z2f′′ϕzCfZφ≤1.Takingf0z=1∈Zφ into the above equality, we have that
(24)L1≔supz∈Dψ′′zφ−11/1−z2<∞.Takingf^0=z∈Zφ into it, we have that
(25)ψ′′zϕz+2ψ′zϕ′z+ψzϕ′′zφ−11/1−z2≤CfBαφ.This implies that(26)L2≔supz∈D2ψ′zϕ′z+ψzϕ′′zφ−11/1−z2≤supz∈Dψ′′zϕzφ−11/1−z2+supz∈Dψ′′zϕz+2ψ′zϕ′z+ψzϕ′′zφ−11/1−z2≤supz∈Dψ′′zφ−11/1−z2+supz∈Dψ′′zϕz+2ψ′zϕ′z+ψzϕ′′zφ−11/1−z2<∞.Takingf~0=1/2z2∈Zφ into the same equality again, we have thatThis leads to(27)L3≔supz∈Dψzϕ′z2φ−11/1−z2≤supz∈Dψ′′z1/2ϕz2+2ψ′zϕ′z+ψzϕ′′zϕz+ψzϕ′z2φ−11/1−z2+supz∈Dψ′′z1/2ϕz2φ−11/1−z2+supz∈D2ψ′zϕ′z+ψzϕ′′zϕzφ−11/1−z2≤supz∈Dψ′′z1/2ϕz2+2ψ′zϕ′z+ψzϕ′′zϕz+ψzϕ′z2φ−11/1−z2+supz∈Dψ′′zφ−11/1−z2+supz∈D2ψ′zϕ′z+ψzϕ′′zφ−11/1−z2<∞.Sincefϕa∈Zφ, we obtain that
(28)SφψCϕfϕa′′CfϕaZφ≤1.This leads to(29)supz∈D1−z2,·φψ′′z/ϕa∫0ϕz1−ϕa22/1−ϕa¯t2−1−ϕa2/1−ϕa¯tdt+1/1−ϕa2ψzϕ′2zCfϕaZφ≤1.Note that(30)supz∈Dψ′′zφ−11/1−z21ϕa∫0ϕz1−ϕa221−ϕa¯t2−1−ϕa21−ϕa¯tdt≤6supz∈Dψ′′zφ−11/1−z2<∞,where (24) is used in the last inequality. Combining the two equality above, we prove that
(31)M3=supz∈Dψzϕ′z21/1−z2φ−11/1−z2<∞.Sincef~ϕa∈Zφ, we obtain that
(32)SφψCϕf~ϕa′′Cf~ϕaZφ≤1.Note that(33)supz∈D1φ−11/1−z2,·ψ′′z∫0ϕz3log4/1−ϕa¯t2log4/1−ϕa2−2log4/1−ϕa¯t3log4/1−ϕa22dt<∞.Combing the above equality and (24), we obtain that
(34)M2=supz∈D2ψ′zϕ′z+ψzϕ′′zlog4/1−ϕz2φ−11/1−z2<∞.Moreover, by the factf~′ϕa∈Zφ, 3.3 and some similar estimations, we have that
(35)M1=supz∈Dψ′′zlog4/1−ϕz2φ−11/1−z2<∞.This completes the proof.


## 4. Compactness of Weighted Composition Operator onZφ

Theorem 7.
Forφ∈U, the weighted composition operator ψCϕ is compact on Zφ if and only if ψCϕ is bounded on Zφ,
(36)limϕz⟶1−ψ′′zlog4/1−ϕz2φ−11/1−z2=0,(37)limϕz⟶1−2ψ′zϕ′z+ψzϕ′′zlog4/1−ϕz2φ−11/1−z2=0,(38)limϕz⟶1−ψzϕ′z21/1−ϕz2φ−11/1−z2=0.Proof.
Suppose thatψCϕ is bounded on Zφ with (36), (37), and (38) hold. Observe that L1<∞, L2<∞, and L3<∞ (defined in the proof of Theorem 6) by the boundedness of ψCϕ.For everyε>0, there exists a 0<r<1 such that for ϕz>r,
(39)ψ′′zlog4/1−ϕz2φ−11/1−z2<ε2,2ψ′zϕ′z+ψzϕ′′zlog4/1−ϕz2φ−11/1−z2<ε2,∣ψzϕ′z2∣1/1−ϕz2φ−11/1−z2<ε2.Choose a sequencefnn⊂Zφ which satisfies supn∈ℕfnZφ≤K and fn converges to zero uniformly on any compact subsets of the unit disk as n⟶∞, where K is a fixed constant. Then by the compactness of ψCϕ, we only should conduct that limn⟶∞ψCϕfnZφ=0.Note thatlimn⟶∞fn0=0 implies that limn⟶∞fnkz=0 uniformly on any compact subsets of D, k∈ℕ. It follows that
(40)ψCϕfnZφ=ψfnϕ0+supz∈DψCϕfn′′zφ−11/1−z2≤ψfnϕ0+supz∈D:ϕz≤rψ′′zfnϕz+2ψ′zϕ′z+ψzϕ′′zfn′ϕz+ψzϕ′z2f′′nϕzφ−11/1−z2+supz∈D:ϕz>r∣ψ′′zfnϕz+2ψ′zϕ′z+ψzϕ′′zfn′ϕz+ψzϕ′z2f′′nϕzφ−11/1−z2≲ψfnϕ0+L1supw∈Dfnw+L2supw∈Dfn′w+L3supw∈Df′′nw+supz∈D:ϕz>rlog4/1−ϕz2fZφψ′′zφ−11/1−z2+supz∈D:ϕz>rlog4/1−ϕz2fZφ2ψ′zϕ′z+ψzϕ′′zφ−11/1−z2+supz∈D:ϕz>r1/1−ϕz2fZφψzϕ′z2φ−11/1−z2≲ε,where the second inequality is calculated by (13), (14), and (15).Then, we conclude thatψCϕ is compact on Zφ.Conversely, suppose thatψCϕ is compact on Zφ. Then, ψCϕ is bounded on Zφ. Let znn be a sequence in D which satisfies limn⟶∞ϕzn=1. (If such sequence does not exist, then the proof is completed.)Note thatfϕznn is bounded uniformly in Zφ and uniformly converges to 0 on any compact subset of D. Thus, we have
(41)limn⟶∞ψCϕfϕznZφ=0.By the similar calculations shown in the proof of boundedness ofψCϕ, we conclude that
(42)limϕz⟶1−ψzϕ′z21/1−ϕz2φ−11/1−z2=0.Note thatf~ϕznn is bounded uniformly in Zφ and uniformly converges to zero on any compact subset of the unit disk as n⟶∞. Thus, we haveNote thatf~′ϕznn is bounded uniformly in Zφ and uniformly converges to zero on any compact subset of the unit disk as n⟶∞. Thus, we have
(43)limϕz⟶1−ψ′′zlog4/1−ϕz2φ−11/1−z2=0.This completes the proof.


## 5. Conclusions

In this paper, the boundedness and compactness of weighted composition operatorsψCϕ on the Zygmund-Orlicz space are investigated whenever φ∈U, which are concluded in the following.
(i)
Forφ∈U, ψCϕ is bounded on Zφ if and only if
(44)M1≔supz∈Dψ′′zlog4/1−ϕz2φ−11/1−z2<∞,M2≔supz∈D2ψ′zϕ′z+ψzϕ′′zlog4/1−ϕz2φ−11/1−z2<∞,(45)M3≔supz∈Dψzϕ′z21/1−ϕz2φ−11/1−z2<∞.Forφ∈U, the weighted composition operator ψCϕ is compact on Zφ if and only if ψCϕ is bounded on Zφ,
(46)limϕz⟶1−ψ′′zlog4/1−ϕz2φ−11/1−z2=0,limϕz⟶1−2ψ′zϕ′z+ψzϕ′′zlog4/1−ϕz2φ−11/1−z2=0,limϕz⟶1−∣ψzϕ′z2∣1/1−ϕz2φ−11/1−z2=0.



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*Source: 5042795-2022-06-02.xml*