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

1. Introduction

Let D 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.

When u=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]).

Let α∈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.

For 0<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 that Zβ 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.

Let dA 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 operator uCφ: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 operator uCφ:Zp−2p⟶Zβ.

Throughout the paper, we denote by C 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.

Suppose 1<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.

For f∈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 [<xref ref-type="bibr" rid="B4">4</xref>]).

Let 1<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 [<xref ref-type="bibr" rid="B4">4</xref>]).

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

the point evaluation functionals on Y are continuous

for every sequence fn in the unit ball of X that exists f∈X and a subsequence fnj such that fnj⟶f uniformly on D¯

the operator T: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 Lemmas 2 and 3.

Lemma 4.

Let 1<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 [<xref ref-type="bibr" rid="B19">19</xref>]).

Suppose that z∈D,c is real, t>−1, and
(10)Ic,tz=∫D1−w2t1−w¯z2+t+cdAw.

If c<0, then as a function of z,Ic,t is bounded on D

If c=0, then Ic,tz≈log1/1−z2,as∣z∣⟶1

If c>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 from Zp−2p to Zβ.

Theorem 6.

Let 1<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 that u∈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 that uCφ: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∣<∞.

For a∈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.

Thus, for 1−2/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 of uCφ:Zp−2p⟶Zβ, we get
(21)sup1−2/e<∣a∣<1∥uCφfa∥Zβ<∞.

Since fa″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.

For a∈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 Lemma 5, 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 any w∈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 of uCφ: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.

Let 1<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.

Let zjj∈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 Theorem 6, 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 that limsupj→∞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.

Let r∈0,1. Define Kr:HD⟶HD by
(39)Krfz=frz=frz,f∈HD.

It is clear that Kr 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 any f∈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 estimate S3. 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 as t⟶∞, we get
(48)S3≲E.

Similarly, again by Lemma 1,
(49)S5=limsupj→∞sup∣φz∣>rt1−z2βφ′z2uzf−frj″φz≲limsupj→∞f−frjZp−2psup∣φz∣>rt1−z2βφ′z2uz1−φz2.

Taking the limit as t⟶∞, 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 Theorem 7 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.

Let 1<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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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