# Weighted Composition Operators from Besov Zygmund-Type Spaces into Zygmund-Type Spaces

Authors: Xiangling Zhu; Nanhui Hu
Journal: Journal of Function Spaces
 (2020)

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## Abstract

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


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## Body

## 1. Introduction

LetD denote the open unit disk in the complex plane ℂ and HD the space of all analytic functions in D. For an analytic self-map φ of D and u∈HD, the weighted composition operator uCφ is defined as follows:
(1)uCφfz=uzfφz,f∈HD,z∈D.Whenu=1, uCφ is just the composition operator, denoted by Cφ. In the past several decades, composition operators and weighted composition operators have received much attention and appear in various settings in the literature (see, for example, [2–5, 8, 10, 13, 15, 16, 19]).Letα∈0,∞. The Bloch type space Bα consists of those functions f∈HD for which
(2)∥f∥Bα=∣f0∣+supz∈D1−z2α∣f′z∣<∞.Bα is a Banach space under the above norm. It is known that when α=1, B1=B is the classical Bloch space.For0<β<∞, an f∈HD is said to be in the Zygmund-type space Zβ, if
(3)∥f∥Zβ=∣f0∣+∣f′0∣+supz∈D1−z2β∣f′′z∣<∞.It is easy to check thatZβ is a Banach space under the norm ∥·∥Zβ. When β=1, Z1=Z is the Zygmund space. When β>1, Zβ is just the Bloch type space Bβ−1. In particular, when β=2, Zβ is just the Bloch space B. Hence, the Zygmund space is the space of all f∈HD such that f′∈B with norm
(4)∥f∥Z=∣f0∣+∥f′∥B.LetdA be the normalized area measure on D. For 1<p<∞, the Besov space, denoted by Bp, is the space of all f∈HD such that
(5)bpf≔∫Df′zp1−z2p−2dAz1/p<∞.This space is a Banach space with the following norm∥f∥Bp=∣f0∣+bpf. In particular, B2 is the classical Dirichlet space. Besov spaces are Möbius invariant in the sense that bpf∘ψ=bpf for all f∈Bp and ψ∈AutD, the set of all Möbius maps of D (see [1, 19]).In [4], Colonna and Tjani introduced a new class type space Zp−2p, called the Besov Zygmund-type space, which consists of all f∈HD such that f′∈Bp. Since the Besov space is contained in the Bloch space, it follows that the Besov Zygmund-type space is a subset of the Zygmund space, and hence, it is contained in the disk algebra.Colonna and Li studied the boundedness and compactness of the operatoruCφ:H∞⟶Z and uCφ:Bα0<α<1⟶Z in [2, 3], respectively. Here, H∞ is the space of bounded analytic functions. (See [2, 3, 6–14, 16, 17] for more results of composition operators, weighted composition operators, and related operators on the Zygmund space and Zygmund-type spaces.) Colonna and Tjani characterized the boundedness and compactness of uCφ:Zp−2p⟶Bα in [4].In this work, we give some characterizations for the boundedness, compactness, and the essential norm of the operatoruCφ:Zp−2p⟶Zβ.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. Main Results and Proofs

In this section, we formulate and prove our main results in this paper. For this purpose, we need the following lemmas.Lemma 1.
Suppose1<p<∞. Then, there exists a positive constant C such that
(6)∣f′z∣≤C∥f∥Zp−2plog21−z21−1/p,(7)f′′z≤C∥f∥Zp−2p1−z2,(8)∥f∥∞≤C∥f∥Zp−2pfor every f∈Zp−2p.Proof.
Forf∈Bp, it is well known that
(9)∣fz∣≲∥f∥Bplog21−z21−1/p,z∈D,∥f∥B≲∥f∥Bp.
Then, the inequalities in (6) follow from the definition of the Besov Zygmund-type space. Since the Zygmund space is continuously embedded into H∞, as shown in Lemma 2.1 of [18], we get that ∥f∥∞≤C∥f∥Zp−2p. The proof is complete.Lemma 2 (see [4]).
Let1<p<∞. Every sequence in Zp−2p bounded in norm has a subsequence which converges uniformly in D¯ to a function in Zp−2p.Lemma 3 (see [4]).
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 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 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 the sequence ∥Tfn∥Y⟶0 as n→∞.The following result is a direct consequence of Lemmas2 and 3.Lemma 4.
Let1<p<∞ and 0<β<∞. IfT:Zp−2p⟶Zβ is bounded, then T is compact if and only if ∥Tfk∥Zβ⟶0 as k⟶∞ for any sequence fk in Zp−2p bounded in norm which converge to 0 uniformly in D¯.The following estimates are fundamental in operator theory and function spaces on the unit disk (see ([19], Lemma 3.10)).Lemma 5 (see [19]).
Suppose thatz∈D,c is real, t>−1, and
(10)Ic,tz=∫D1−w2t1−w¯z2+t+cdAw.(i)
Ifc<0, then as a function of z,Ic,t is bounded on D(ii)
Ifc=0, then Ic,tz≈log1/1−z2,as∣z∣⟶1(iii)
Ifc>0, then Ic,tz≈1/1−z2c,as∣z∣⟶1
Now, we are in a position to give the following characterization of bound composition operators fromZp−2p to Zβ.Theorem 6.
Let1<p<∞, 0<β<∞, u∈HD, and φ be an analytic self-map of D. Then, uCφ:Zp−2p⟶Zβ is bounded if and only if u∈Zβ,
(11)supz∈D1−z2β∣2u′zφ′z+uzφ″z∣log21−φz21−1/p<∞,(12)supz∈D1−z2βuzφ′z21−φz2<∞.Proof.
First, suppose thatu∈Zβ, (11) and (12) hold. For arbitrary z∈D and f∈Zp−2p, by Lemma 1, we have
(13)uCφf0≲u0∥f∥Zp−2p,∣uCφf′0∣≲∣u′0∣+∣u0φ′0∣log21−φ021−1/p∥f∥Zp−2p,supz∈D1−z2β∣uCφf″z∣≤supz∈D1−z2βu″zfφz+supz∈D1−z2βuzφ′z2f″φz+supz∈D1−z2β2u′zφ′z+uzφ″zf′φz≲supz∈D1−z2βu″z∥f∥Zp−2p+supz∈D1−z2βuzφ′z21−φz2∥f∥Zp−2p+supz∈D1−z2β∣2u′zφ′z+uzφ″z∣log21−φz21−1/p∥f∥Zp−2p<∞.
Therefore,uCφ:Zp−2p⟶Zβ is bounded.
Conversely, suppose thatuCφ:Zp−2p⟶Zβ is bounded. Applying the operator uCφ to zj with j=0,1,2, and using the boundedness of uCφ, we get that u∈Zβ, uφ∈Zβ, and uφ2∈Zβ. Hence,
(14)supz∈D1−z2β∣2u′zφ′z+uzφ″z∣<∞,(15)supz∈D1−z2β∣uzφ′z2∣<∞.
Fora∈D such that ∣a∣>1−2/e, set
(16)faz=a¯z−1a¯1+log21−a¯z2+1log21−a2−1−1/p−2∫0zlog21−a¯wdwlog21−a2−1/p.
Then,(17)fa′z=log21−a¯z2log21−a2−1−1/p−2log21−a¯zlog21−a2−1/pfa″z=2a¯1−a¯zlog21−a¯zlog21−a2−1−1/p−2a¯1−a¯zlog21−a2−1/p.
Thus, for1−2/e<∣a∣<1, we have
(18)∣fa″z∣≤2∣1−a¯z∣log21−a2−1/p,and by Lemma 5,
(19)∫Dfa″zp1−z2p−2dAz≲1log2/1−a2∫D1−z2p−21−a¯zpdAz≲1.
So,(20)sup1−2/e<∣a∣<1∥fa∥Zp−2p<∞.
Therefore, by the boundedness ofuCφ:Zp−2p⟶Zβ, we get
(21)sup1−2/e<∣a∣<1∥uCφfa∥Zβ<∞.
Sincefa″a=0,∣fa′a∣=log2/1−a21−1/p, for any w∈D such that 1−2/e<∣φw∣<1, we have
(22)1−w2β∣uCφfφw″w∣=1−w2βu″wfφwφw+2u′wφ′w+uwφ″wfφw′φw,which implies that
(23)1−w2β∣2u′wφ′w+uwφ″w∣log2/1−φw21−1/p≲uCφfφw∥Zβ+1−w2β∣u″wfφwφw∣≲∥uCφfφw∥Zβ+∥u∥Zβ∥fφw∥Zp−2p<∞.
By (14), we get
(24)sup∣φw∣≤1−2/e1−w2β∣2u′wφ′w+uwφ″w∣log2/1−φw21−1/p≲supw∈D1−w2β∣2u′wφ′w+uwφ″w∣<∞.
From (23) and (24), we see that (11) holds.
Fora∈D, define
(25)gaz=121−a221−a¯z−1−a231−a¯z2+121−a241−a¯z3.
So,(26)ga′z=a¯21−a221−a¯z2−2a¯1−a231−a¯z3+32a¯1−a241−a¯z4,ga″z=a¯21−a221−a¯z3−6a¯21−a231−a¯z4+6a¯21−a241−a¯z5.
By Lemma5, we see that ga∈Zp−2p and supa∈D∥ga∥Zp−2p<∞. By the boundedness of uCφ:Zp−2p⟶Zβ, we get supa∈D∥uCφga∥Zβ<∞. After a calculation,
(27)gaa=0,ga′a=0,∣ga″a∣=a21−a2.
Hence, for anyw∈D,
(28)1−w2βuwφw2φ′w21−φw2≲∥uCφgφw∥Zβ<∞.
On the one hand, from (28), we obtain
(29)sup∣φw∣>1/21−w2βuwφ′w21−φw2<∞.
On the other hand, by (15), we get
(30)sup∣φw∣≤1/21−w2βuwφ′w21−φw2≲supw∈D1−w2βuwφ′w2<∞.
From (29) and (30), we see that (12) holds. The proof is complete.Next, we estimate the essential norm ofuCφ:Zp−2p⟶Zβ. Recall that the essential norm of uCφ:Zp−2p⟶Zβ is defined as the distance from uCφ to the set of compact operators K:Zp−2p⟶Zβ, that is,
(31)∥uCφ∥e,Zp−2p→Zβ=inf∥uCφ−K∥Zp−2p→Zβ:Kisacompactoperator.Theorem 7.
Let1<p<∞, 0<β<∞, u∈HD, and φ be an analytic self-map of D such that uCφ:Zp−2p⟶Zβ is bounded. Then,
(32)∥uCφ∥e,Zp−2p→Zβ≈maxE,G.
Here,(33)E=limsup∣φz∣→11−z2β∣2u′zφ′z+uzφ″z∣log21−φz21−1/p,G=limsup∣φz∣→11−z2βuzφ′z21−φz2.Proof.
First we prove that(34)∥uCφ∥e,Zp−2p→Zβ≳maxE,G.
Letzjj∈ℕ be a sequence in D such that ∣φzj∣⟶1 as j⟶∞. Define
(35)kjz=φzj¯z−1φzj¯1+log21−φzj¯z2+1log21−φzj2−1−1/p−2∫0zlog21−φzj¯wdwlog21−φzj2−1/p,ljz=121−φzj221−φzj¯z−1−φzj231−φzj¯z2+121−φzj231−φzj¯z3.
From the proof of Theorem6, we see that kj and lj belong to Zp−2p. Moreover, kj and lj converge to 0 uniformly on D¯ as j⟶∞. Hence, for any compact operator K:Zp−2p⟶Zβ, by Lemma 4, we obtain
(36)∥uCφ−K∥Zp−2p→Zβ≳limsupj→∞∥uCφkj∥Zβ−limsupj→∞∥Kkj∥Zβ≳limsupj→∞1−zj2β∣2u′zjφ′zj+uzjφ″zj∣log21−φzj21−1/p+limsupj→∞1−zj2βu″zjkjφzj=limsupj→∞1−zj2β∣2u′zjφ′zj+uzjφ″zj∣log21−φzj21−1/p,∥uCφ−K∥Zp−2p→Zβ≳limsupj→∞∥uCφlj∥Zβ−limsupj→∞∥Klj∥Zβ≳limsupj→∞1−zj2βuzjφ′zj2φzj21−φzj2.
Here, we used the fact thatlimsupj→∞1−zj2βu″zjkjφzj=0 since u∈Zβ and kj converges to 0 uniformly on D¯ as j⟶∞.
Therefore, we get(37)∥uCφ∥e,Zp−2p→Zβ=infK∥uCφ−K∥Zp−2p→Zβ≳limsup∣φz∣→11−z2β∣2u′zφ′z+uzφ″z∣log21−φz21−1/p=E,∥uCφ∥e,Zp−2p→Zβ≳limsup∣φz∣→11−z2βuzφ′z21−φz2=G,as desired.
Next, we prove that(38)∥uCφ∥e,Zp−2p→Zβ≲maxE,G.
Letr∈0,1. Define Kr:HD⟶HD by
(39)Krfz=frz=frz,f∈HD.
It is clear thatKr is compact on Zp−2p and ∥Kr∥Zp−2p→Zp−2p≤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, for each j∈ℕ, uCφKrj:Zp−2p⟶Zβ is compact. Hence,
(40)uCφe,Zp−2p→Zβ≤limsupj→∞uCφ−uCφKrjZp−2p→Zβ.
Thus, we only need to show that(41)limsupj→∞uCφ−uCφKrjZp−2p→Zβ≲maxE,G.
For anyf∈Zp−2p with ∥f∥Zp−2p≤1, from the facts that
(42)limj→∞∣u0fφ0−u0frjφ0∣=0,limj→∞∣u′0f−frj′φ0+u0f−frj′φ0φ′0∣=0,we have
(43)limsupj→∞uCφ−uCφKrjfZβ=limsupj→∞1−z2βu⋅f−frj∘φ″z≤limsupj→∞supz∈D1−z2βu″zf−frjφz+limsupj→∞sup∣φz∣≤rt1−z2β2u′zφ′z+uzφ″zf−frj′φz+limsupj→∞sup∣φz∣>rt1−z2β2u′zφ′z+uzφ″zf−frj′φz+limsupj→∞sup∣φz∣≤rt1−z2βφ′z2uzf−frj″φz+limsupj→∞sup∣φz∣>rt1−z2βφ′z2uzf−frj″φz≔S1+S2+S3+S4+S5,where t∈ℕ is large enough such that rj≥1/2 for all j≥t. Since frj−f⟶0 as j⟶∞, by Lemma 2, we have
(44)S1=limsupj→∞supz∈D1−z2βu″z∣f−frjφz∣≤∥u∥Zβlimsupj→∞supw∈D∣fw−frjw∣=0.
Sincerjf′rj−f′⟶0 and rj2frj′′−f′′⟶0 uniformly on compact subsets of D as j⟶∞, we have
(45)S2=limsupj→∞supφz≤rt1−z2β2u′zφ′z+uzφ″zf−frj′φz≲∥uφ∥Zβ+∥u∥Zβlimsupj→∞supw≤rtf′w−rjf′rjw=0,(46)S4=limsupj→∞sup∣φz∣≤rt1−z2βφ′z2uzf−frj″φz≲uφ2Zβ+uφZβ+uZβlimsupj→∞sup∣w∣≤rtf″w−rj2f′′rjw=0.
Now, we estimateS3. Using Lemma 1 and the fact that ∥f∥Zp−2p≤1, we have
(47)S3=limsupj→∞sup∣φz∣>rt1−z2β2u′zφ′z+uzφ″zf−frj′φz≲limsupj→∞f−frjZp−2psup∣φz∣>rt1−z2β×2u′zφ′z+uzφ′′zlog21−φz21−1/p.
Taking the limit ast⟶∞, we get
(48)S3≲E.
Similarly, again by Lemma1,
(49)S5=limsupj→∞sup∣φz∣>rt1−z2βφ′z2uzf−frj″φz≲limsupj→∞f−frjZp−2psup∣φz∣>rt1−z2βφ′z2uz1−φz2.
Taking the limit ast⟶∞, we get
(50)S5≲G.
Hence, by (43), (44), (45), (46), (48), and (50), we get
(51)limsupj→∞uCφ−uCφKrjZp−2p→Zβ≲maxE,G,which with (40) implies the desired result. The proof is complete.From Theorem7 and the result that ∥uCφ∥e,Zp−2p→Zβ=0 if and only if uCφ:Zp−2p⟶Zβ is compact, we get the following corollary.Corollary 8.
Let1<p<∞, 0<β<∞, u∈HD, and φ be an analytic self-map of D such that uCφ:Zp−2p⟶Zβ is bounded. Then, uCφ:Zp−2p⟶Zβ is compact if and only if
(52)limsup∣φz∣→11−z2βuzφ′z21−φz2=0,limsupφz→11−z2β2u′zφ′z+uzφ″zlog21−φz21−1/p=0.



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