# Weighted Composition Operators from the Bloch Space and the Analytic Besov Spaces into the Zygmund Space

Authors: Flavia Colonna; Songxiao Li
Journal: Journal of Operators
 (2013)

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## Abstract

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch spaceℬ and the analytic Besov spaces Bp (with 1<p<∞) into the Zygmund space 𝒵. As a special case, we show that the bounded (resp., compact) composition operators from ℬ, Bp, and H∞ to 𝒵 coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.


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## Body

## 1. Introduction

Let𝔻 be the open unit disk in the complex plane ℂ and H(𝔻) the space of analytic functions on 𝔻. A function f∈H(𝔻) is said to belong to the Bloch spaceℬ if
(1)βf=supz∈𝔻(1-|z|2)|f′(z)|<∞.
Under the norm defined by ∥f∥ℬ=|f(0)|+βf,  ℬ is a conformally invariant Banach space. The functions in the Bloch space satisfy the following growth condition:
(2)|f(z)|≤|f(0)|+12log1+|z|1-|z|βf≤(1+loge1-|z|2)∥f∥ℬ.An important class of Möbius invariant spaces is given by the analytic Besov spacesBp, with 1<p<∞, consisting of the functions f∈H(𝔻) such that
(3)bp(f)p∶=∫𝔻‍(1-|z|2)p-2|f′(z)|pdA(z)<∞,
where dA denotes the normalized area measure on the unit disk. The quantity bp is a seminorm and the Besov norm is defined by
(4)∥f∥Bp=|f(0)|+bp(f).
By Lemma 1.1 of [1], for f∈Bp,
(5)∥f∥ℬ≤C∥f∥Bp.
Thus, the space Bp is continuously embedded in the Bloch space. Moreover, by Theorem 9 in [2], the functions in Bp satisfy the growth condition
(6)|f(z)|≤C(loge1-|z|2)1-1/p,z∈𝔻.Let𝒵 denote the set of all functions f∈H(𝔻)∩C(𝔻¯) such that
(7)∥f∥=sup|f(ei(θ+h))+f(ei(θ-h))-2f(eiθ)|h<∞,
where the supremum is taken over all θ∈ℝ and h>0. By Theorem 5.3 of [3] and the Closed Graph theorem, an analytic function f on 𝔻 belongs to 𝒵 if and only if f′∈ℬ. Furthermore,
(8)∥f∥≍supz∈𝔻(1-|z|2)|f′′(z)|.
The quantities in (8) are just seminorms for the space 𝒵, as they do not distinguish between functions differing by a linear function. The norm
(9)∥f∥𝒵=|f(0)|+|f′(0)|+supz∈𝔻(1-|z|2)|f′′(z)|yields a Banach space structure on𝒵, which is called the Zygmund space. For more information on the Zygmund space on the unit disk, see, for example, [3].LetS(𝔻) denote the set of all analytic self-maps of 𝔻. Each φ∈S(𝔻) induces the composition operator on H(𝔻) defined by
(10)Cφ(f)(z)=f(φ(z))forf∈H(𝔻) and z∈𝔻. We refer the interested reader to [4, 5] for the theory of the composition operators.Letu∈H(𝔻). The multiplication operator Mu is defined as
(11)(Muf)(z)=u(z)f(z)forf∈H(𝔻) and z∈𝔻. The composition product uCφ of  Mu and Cφ yields a linear operator on H(𝔻) called the weighted composition operator with symbolsu  andφ.In recent years, considerable interest has emerged in the study of the weighted composition operators due to the important role they play in the study of the isometries on many Banach spaces of analytic functions, such as the Hardy spaceHp (for 1≤p≤∞, p≠2) [6, 7], the weighted Bergman space [8], and the disk algebra [9].There is a very extensive literature on the composition operators and the weighted composition operators between the Bloch space and other spaces of analytic functions in one and several complex variables. The study on such operators between the Zygmund space and some other spaces of analytic functions is more recent and not yet well developed. The weighted composition operators and related operators from the Zygmund space to Bloch-type spaces have been investigated by the second author and Stević in [10, 11].Research on the composition operators between various Banach spaces of analytic functions has shown that the compact composition operators acting on different Banach spaces are often the same. For example, in [12], it was shown that the composition operator Cφ mapping into the Bloch space is compact as an operator acting on H∞ if and only if it is compact as an operator acting on the analytic Besov space Bp (with 1<p<∞), on the space BMOA, and on the Bloch space itself. Also, an interesting question that arises in operator theory is whether a countable set of functions on the range of the operator exists whose norm approaching 0 is sufficient to characterize the compactness of the operator. In [13], it was shown that the sequence of powers of the symbol {φn} has this property with regards to the composition operator on the Bloch space as well as the space BMOA. In fact, for X=H∞,Bp,BMOA,ℬ, the operator Cφ:X→ℬ is compact if and only if ∥φn∥ℬ→0 as n→∞. For the weighted composition operator uCφ, this is not the case, although a few cases are known in which the sequence of norms ∥uφn∥ℬ is sufficient to characterize boundedness and compactness (see [14, 15], see also [16] for analogous questions for the operators mapping into BMOA). In all examples known to us, the space X is small compared to the codomain of the operator. These results prompted us to investigate analogous questions when the operator maps into the Zygmund space.In this work, besides giving characterizations of the bounded and the compact operatoruCφ from the Besov and the Bloch spaces into 𝒵 in terms of function theoretic conditions on the symbols u and φ, we provide boundedness and the compactness criteria for the operator uCφ, in terms of the norms in 𝒵 of functions in the range of the operator belonging to certain one-parameter families. For both boundedness and compactness, one of these characterizations involves the sequence {∥uφk∥𝒵}.Finally, we show that there are no nontrivial bounded multiplication operators from the Bloch space to the Zygmund space, and that a composition operator fromℬ or Bp into 𝒵 is bounded (resp., compact) if and only if it is bounded (resp., compact) as an operator acting on the space H∞ of bounded analytic functions on 𝔻 under the supremum norm. One of the characterizing conditions for boundedness (resp., compactness, under the boundedness assumption of the operator) of such operators is that ∥φk∥𝒵=O(1) (resp., ∥φk∥𝒵=o(1)) as k→∞. We found this phenomenon quite surprising due to the size of the spaces Bp and ℬ in comparison to that of the Zygmund space.


## 2. Boundedness

We begin the section by characterizing the bounded weighted composition operators fromℬ to the Zygmund space. We note that since the constant function 1 belongs to all spaces under consideration, the boundedness of uCφ requires that u=uCφ1∈𝒵. Thus, we will assume throughout that u∈𝒵.For a fixeda∈𝔻 and for z∈𝔻, set
(12)fa(z)=1-|a|21-a¯z,ga(z)=(1-|a|21-a¯z)2,ha(z)=(loge1-a¯z)2(loge1-|a|2)-1.For an integerk≥0 and z∈𝔻, let pk(z)=zk.Theorem 1.
Letu∈𝒵 and φ∈S(𝔻). Then the following conditions are equivalent.(a)
The operatoruCφ:ℬ→𝒵 is bounded.(b)
M∶=supk≥0∥uCφpk∥𝒵<∞, and H∶=supw∈𝔻∥uCφhφ(w)∥𝒵<∞.(c)
N1∶=supz∈𝔻(1-|z|2)|u(z)φ′(z)2|, N2∶=supz∈𝔻(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|,
(13)A∶=supw∈𝔻∥uCφfφ(w)∥𝒵,B∶=supw∈𝔻∥uCφgφ(w)∥𝒵,
and H are finite.(d)
The quantitiesM1∶=supz∈𝔻(1-|z|2)|u′′(z)|log(e/(1-|φ(z)|2)),
(14)M2∶=supz∈𝔻(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|1-|φ(z)|2,M3∶=supz∈𝔻(1-|z|2)|u(z)φ′(z)2|(1-|φ(z)|2)2
are finite.To prove Theorem1, we will use the following result which follows from the characterization of the bounded weighted composition operators from H∞ to the Zygmund space given in Theorem 1 of [17].Lemma 2.
Letu∈𝒵 and φ∈S(𝔻). Then the following conditions are equivalent.(a)
supk≥0∥uCφpk∥𝒵<∞.(b)
The quantitiesN1, N2, A, and B are finite.(c)
The quantitiesM2 and M3 are finite.Proof.
(a)⇒ (b) Since the sequence {pk} is bounded in ℬ with supremum norms no greater than 1, if uCφ:ℬ→𝒵 is bounded, then for each integer k≥0, we have
(15)∥uCφpk∥𝒵≤∥uCφ∥.
Therefore, the supremum of ∥uCφpk∥𝒵 over all integers k≥0 is finite. Likewise, supw∈𝔻∥hφ(w)∥ℬ=L<∞. Therefore,
(16)∥uCφhφ(w)∥𝒵≤L∥uCφ∥.
(b)⇒ (c) follows at once from Lemma 2.
(c)⇒ (d) Suppose that H, N1, N2, A, and B are finite. By Lemma 2 we see that M2 and M3 are finite. For each nonnegative integer n and each a∈𝔻, a direct calculation shows that
(17)fa(n)(a)=n!a¯n(1-|a|2)n,ga(n)(a)=(n+1)!a¯n(1-|a|2)n,ha(a)=loge1-|a|2,ha(n)(a)=(n+2)!a¯n2(1-|a|2)n,n≥1.
Setting(18)v(z)=2u′(z)φ′(z)+u(z)φ′′(z),z∈𝔻,
for w∈𝔻, from (17) we obtain
(19)(uCφfφ(w))′′(w)=u′′(w)+v(w)φ(w)¯1-|φ(w)|2+2u(w)φ′(w)2φ(w)¯2(1-|φ(w)|2)2,(20)(uCφgφ(w))′′(w)=u′′(w)+2v(w)φ(w)¯1-|φ(w)|2+6u(w)φ′(w)2φ(w)¯2(1-|φ(w)|2)2,(21)(uCφhφ(w))′′(w)=loge1-|w|2u′′(w)+3v(w)φ(w)¯1-|φ(w)|2+12u(w)φ′(w)2φ(w)¯2(1-|φ(w)|2)2.
Subtracting (19) from (20), we get
(22)-(uCφfφ(w))′′(w)+(uCφgφ(w))′′(w)=v(w)φ(w)¯1-|φ(w)|2+4u(w)φ′(w)2φ(w)¯2(1-|φ(w)|2)2.
Hence, from (21) and (22), we deduce
(23)loge1-|φ(w)|2u′′(w)=(uCφhφ(w))′′(w)-3[v(w)φ(w)¯1-|φ(w)|2+4u(w)φ′(w)2φ(w)¯2(1-|φ(w)|2)2]=(uCφhφ(w))′′(w)+3(uCφfφ(w))′′(w)-3(uCφgφ(w))′′(w).
Therefore, taking the modulus and multiplying by (1-|w|2), we obtain
(24)(1-|w|2)loge1-|φ(w)|2|u′′(w)|≤∥uCφhφ(w)∥𝒵+3∥uCφfφ(w)∥𝒵+3∥uCφgφ(w)∥𝒵.
Taking the supremum over allw∈𝔻, we see that M1 is finite.
(d)⇒ (a) Suppose that (d) holds. As a consequence of Theorem 5.1.5 of [18], for each f∈ℬ,
(25)supz∈𝔻(1-|z|2)2|f′′(z)|≤Cβf≤C∥f∥ℬ.
Thus, using (2) and (25), for an arbitrary z in 𝔻, we have
(26)(1-|z|2)|(uCφf)′′(z)|≤(1-|z|2)|u′′(z)||f(φ(z))|+(1-|z|2)|f′(φ(z))||v(z)|+(1-|z|2)|f′′(φ(z))||u(z)φ′(z)2|≤(1-|z|2)|u′′(z)|×(1+loge1-|φ(z)|2)∥f∥ℬ+(1-|z|2)|v(z)|1-|φ(z)|2∥f∥ℬ+C(1-|z|2)|u(z)φ′(z)2|(1-|φ(z)|2)2∥f∥ℬ≤(∥u∥𝒵+M1+M2+CM3)∥f∥ℬ.
In addition, again by (2), we have
(27)|(uCφf)(0)|≤|u(0)|(1+loge1-|φ(0)|2)∥f∥ℬ,|(uCφf)′(0)|≤|u′(0)|(1+loge1-|φ(0)|2+C|u(0)φ′(0)|1-|φ(0)|2)∥f∥ℬ.
The boundedness ofuCφ:ℬ→𝒵 follows from (27) and (26) after taking the supremum over all z∈𝔻.We now analyze the case of the weighted composition operator acting on the analytic Besov spacesBp for 1<p<∞. The case p=1 has been established by the second author in [19]. For 1<p<∞ and a∈𝔻, define
(28)hp,a(z)=(loge1-a¯z)(loge1-|a|2)-1/p,z∈𝔻.
Then, as shown in [12], hp,a∈Bp and ∥hp,a∥Bp is bounded by a constant Lp independent of a.Theorem 3.
Letu∈𝒵, φ∈S(𝔻), and 1<p<∞. Then the following conditions are equivalent.(a)
The operatoruCφ:Bp→𝒵 is bounded.(b)
M∶=supk≥0∥uCφpk∥𝒵<∞, and Hp∶=supw∈𝔻∥uCφhp,φ(w)∥𝒵<∞.(c)
The quantitiesN1, N2, A, and B in Theorem 1 and Hp are finite.(d)
The quantitiesM2 and M3 in Theorem 1 and
(29)M4∶=supz∈𝔻(1-|z|2)|u′′(z)|(loge1-|φ(z)|2)1-1/p
are finite.Proof.
Note that, unlike the case of the Bloch space, the sequence{pk} is unbounded in Bp. Therefore, first condition in (b) is not an immediate consequence of the boundedness of the operator.
(a)⇒ (d) Suppose uCφ:Bp→𝒵 is bounded. Since supw∈𝔻∥hp,φ(w)∥ℬ=Lp<∞, it follows that
(30)∥uCφhp,φ(w)∥𝒵≤Lp∥uCφ∥.
Moreover, for a∈𝔻, fa and ga are in Bp and their Besov norms are bounded by constants independent of a. Indeed, noting that fa=1-a¯La, with La(z)=(a-z)/(1-a¯z), and using the Möbius invariance of the Besov seminorm, we have
(31)∥fa∥Bp=1-|a|2+bp(fa)≤1+|a|bp(La)=1+|a|bp(p1)≤1+(1p-1)1/p.
Likewise, since ga=1-2a¯La+a¯2La2, and observing that bp(p2)≤2bp(p1), again by the Möbius invariance of bp, we obtain
(32)∥ga∥Bp=(1-|a|2)2+bp(ga)≤1+2|a|bp(La)+|a|2bp(La2)=≤1+2bp(p1)+bp(p2)≤1+4(1p-1)1/p.
Therefore, by the boundedness ofuCφ, the quantities A and B are finite.
Next observe that sincep1 and p2 are in Bp and uCφ is bounded, uCφ(p1) and uCφ(p2) are in 𝒵, so arguing as in the proof of Theorem 1 in [17], we see that N1 and N2 are finite as well. By Lemma 2, it follows that M2 and M3 are finite. To prove that M4 is finite, let w∈𝔻 and note that
(33)hp,φ(w)(φ(w))=(loge1-|φ(w)|2)1-1/p,hp,φ(w)′(φ(w))=(loge1-|φ(w)|2)-1/pφ(w)¯1-|φ(w)|2,hp,φ(w)′′(φ(w))=(loge1-|φ(w)|2)-1/pφ(w)¯2(1-|φ(w)|2)2.
Recalling the notation v(w)=2u′(w)φ′(w)+u(w)φ′′(w), a straightforward calculation shows that
(34)(uCφhp,φ(w))′′(w)=u′′(w)(loge1-|φ(w)|2)1-1/p+v(w)φ(w)¯1-|φ(w)|2(loge1-|φ(w)|2)-1/p+u(w)φ′(w)2φ(w)¯2(1-|φ(w)|2)2(loge1-|φ(w)|2)-1/p.
Thus, by the boundedness of uCφ, we have
(35)(1-|w|2)|u′′(w)|(loge1-|φ(w)|2)1-1/p≤∥uCφhp,φ(w)∥𝒵+M2+M3≤Lp∥uCφ∥+M2+M3.
Taking the supremum over allw∈𝔻, it follows that M4 is finite, as desired.
(d)⇒ (a) Assume (d) holds. As a consequence of Theorem 5.1.5 of [18] and (5), for each f∈Bp,
(36)supz∈𝔻(1-|z|2)2|f′′(z)|≤Cβf≤C∥f∥ℬ≤C∥f∥Bp.
Thus, using (6) and (36), for any z∈𝔻, we have
(37)(1-|z|2)|(uCφf)′′(z)|≤C(1-|z|2)|u′′(z)|×(loge1-|φ(z)|2)1-1/p∥f∥Bp+C(1-|z|2)|v(z)|1-|φ(z)|2∥f∥Bp+C(1-|z|2)|u(z)φ′(z)2|(1-|φ(z)|2)2∥f∥Bp≤C(M2+M3+M4)∥f∥Bp.
Moreover, again by (6) and (5), we have
(38)|(uCφf)(0)|≤C|u(0)|(loge1-|φ(0)|2)1-1/p∥f∥Bp,|(uCφf)′(0)|≤C(|u′(0)|(loge1-|φ(0)|2)1-1/pnnnnnnnn+|u(0)φ′(0)|1-|φ(0)|2)∥f∥Bp.
Taking the supremum over allz∈𝔻 and using (38) and (37), we conclude that uCφ:Bp→𝒵 is a bounded operator.
The equivalence of (c) and (d) follows from (34) and Lemma 2. The equivalence of (b) and (c) is an immediate consequence of Lemma 2.


## 3. Compactness

We begin the section with a useful compactness criterion which will be used to characterize the compact weighted composition operators from the Bloch space and the Besov spaces to the Zygmund space. Its proof is based on standard arguments similar to those outlined in Proposition 3.11 of [4].Lemma 4.
Letu∈H(𝔻), φ∈S(𝔻), and 1<p<∞. Then, the operator uCφ  from ℬ (resp., Bp) into 𝒵 is compact if and only if it uCφ:ℬ→𝒵 (resp., uCφ:Bp→𝒵) is bounded and ∥uCφfk∥𝒵→0, as k→∞, whenever {fk} is a bounded sequence in ℬ (resp., Bp) converging to zero uniformly on compact subsets of  𝔻.We will also make use the following result that follows from the characterization of the compact weighted composition operators fromH∞ to 𝒵 given in Theorem 2 of [17].Lemma 5.
Letu∈𝒵 and φ∈S(𝔻) be such that any of the equivalent conditions in Lemma 2 hold. Then the following conditions are equivalent.(a)
limk→∞∥uCφpk∥𝒵=0.(b)
lim|φ(w)|→1∥uCφfφ(w)∥𝒵=lim|φ(w)|→1∥uCφgφ(w)∥𝒵=0.(c)
lim|φ(z)|→1(1-|z|2)|u′′(z)|=lim|φ(z)|→1((1-|z|2)|u(z)φ′(z)2|/(1-|φ(z)|2)2)=0, and
(39)lim|φ(z)|→1(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|1-|φ(z)|2=0.In the next theorem we focus on the weighted composition operators acting on the Bloch space. We will use the one-parameter family{ha:a∈𝔻} introduced in (12).Theorem 6.
Letu∈𝒵, φ∈S(𝔻) and suppose that the operator uCφ:ℬ→𝒵 is bounded. Then the following conditions are equivalent. (a)
The operatoruCφ:ℬ→𝒵 is compact.(b)
limk→∞∥uCφpk∥𝒵=lim|φ(w)|→1∥uCφhφ(w)∥𝒵=0.(c)
lim|φ(w)|→1∥uCφfφ(w)∥𝒵=lim|φ(w)|→1∥uCφgφ(w)∥𝒵=lim|φ(w)|→1∥uCφhφ(w)∥𝒵=0.(d)
lim|φ(z)|→1(1-|z|2)|u′′(z)|log(e/(1-|φ(z)|2))=lim|φ(z)|→1((1-|z|2)|u(z)φ′(z)2|/(1-|φ(z)|2)2) = 0, and
(40)lim|φ(z)|→1(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|1-|φ(z)|2=0.Proof.
(a)⇒ (b) Suppose uCφ:ℬ→𝒵 is a compact operator. Since the sequence {pk} is bounded in ℬ and converges to 0 uniformly on compact subsets of 𝔻, then by Lemma 4,
(41)∥uCφpk∥𝒵⟶0,ask⟶∞.
On the other hand, if {wn} is a sequence in 𝔻 such that |φ(wn)|→1 as n→∞, then {hφ(wn)} is a bounded sequence in ℬ converging to 0 uniformly on compact subsets of 𝔻, so again by Lemma 4, it follows that
(42)∥uCφhφ(wn)∥𝒵⟶0asn⟶∞,
proving (b).
(b)⇒ (c) follows from Lemma 4.
(c)⇒ (d) Suppose that the limits in (c) are 0. Using the inequality (24), we get
(43)(1-|w|2)|u′′(w)|loge1-|φ(w)|2≤3∥uCφfφ(w)∥𝒵+3∥uCφgφ(w)∥𝒵+∥uCφhφ(w)∥𝒵⟶0
as|φ(w)|→1. The desired result now follows from the last formula and Lemma 4.
(d)⇒ (a) Assume (d) holds. Then, in particular,
(44)lim|φ(z)|→1(1-|z|2)|u′′(z)|=0.
Thus, recalling the notation v=2u′φ′+uφ′′, for any ε>0, there is a constant r∈(0,1), such that
(45)(1-|z|2)|u′′(z)|<ε,(1-|z|2)|u′′(z)|loge1-|z|2<ε,(1-|z|2)|u(z)φ′(z)2|(1-|φ(z)|2)2<ε,(1-|z|2)|v(z)|1-|φ(z)|2<ε,
wheneverr<|φ(z)|<1.
Let{fk} be a sequence in ℬ with sup∥fk∥ℬ≤Q and converging to 0 uniformly on compact subsets of 𝔻. In light of Lemma 4, it suffices to show that ∥uCφfk∥𝒵→0 as k→∞. Using (45), for |φ(w)|>r, and recalling (2) and (25), we have
(46)(1-|w|2)|(uCφfk)′′(w)|≤(1-|w|2)|u′′(w)||fk(φ(w))|+(1-|w|2)|fk′(φ(w))||v(w)|+(1-|w|2)|fk′′(φ(w))||u(w)φ′(w)2|≤(1-|w|2)|u′′(w)|×(1+loge1-|φ(w)|2)∥fk∥ℬ+(1-|w|2)|v(w)|1-|φ(w)|2∥fk∥ℬ+C(1-|w|2)|u(w)φ′(w)2|(1-|φ(w)|2)2∥fk∥ℬ≤(3+C)Qε.
By Cauchy's estimate, if {fk} is a sequence which converges to zero on compact subset of 𝔻, then so do the sequences {fk′} and {fk′′}. Hence, for |φ(w)|≤r, we have
(47)(1-|w|2)|(uCφfk)′′(w)|≤(1-|w|2)|u′′(w)||fk(φ(w))|+(1-|w|2)|fk′′(φ(w))||u(w)φ′(w)2|+(1-|w|2)|fk′(φ(w))||v(w)|≤∥u∥𝒵|fk(φ(w))|+N1|fk′′(φ(w))|+N2|fk′(φ(w))|⟶0,ask⟶∞.
Since |(uCφfk)(0)|=|u(0)fk(φ(0))|→0 and
(48)|(uCφfk)′(0)|=|u(0)φ′(0)fk′(φ(0))+u′(0)fk(φ(0))|⟶0
ask→∞, from (46) and (47) we deduce that ∥uCφfk∥𝒵→0, as k→∞. This completes the proof.We now turn our attention to the weighted composition operators mappingBp into the Zygmund space. We will use the one-parameter family {hp,a:a∈𝔻} defined in (28).Theorem 7.
Letu∈𝒵, φ∈S(𝔻), 1<p<∞ and assume the operator uCφ:Bp→𝒵 is bounded. Then the following conditions are equivalent. (a)
The operatoruCφ:Bp→𝒵 is compact;(b)
limk→∞∥uCφpk∥𝒵=lim|φ(w)|→1∥uCφhp,φ(w)∥𝒵=0;(c)
lim|φ(w)|→1∥uCφfφ(w)∥𝒵=lim|φ(w)|→1∥uCφgφ(w)∥𝒵=lim|φ(w)|→1∥uCφhp,φ(w)∥𝒵=0;(d)
lim|φ(z)|→1(1-|z|2)|u′′(z)|(log(e/(1-|φ(z)|2)))1-1/p = lim|φ(z)|→1((1-|z|2)|u(z)φ′(z)2|/(1-|φ(z)|2)2)=0, and
(49)lim|φ(z)|→1(1-|z|2)|2u′(z)φ′(z)+u(z)φ′′(z)|1-|φ(z)|2=0.Proof.
AssumeuCφ:Bp→𝒵 is compact. Since {hp,φ(w)} is bounded in Bp and convergent to 0 uniformly on compact subsets of 𝔻, if {wn} is a sequence in 𝔻 such that |φ(wn)|→1 as n→∞, by Lemma 4 it follows that ∥uCφhp,φ(wn)∥𝒵→0 as n→∞. Similarly, since the sequences {fφ(wn)} and {gφ(wn)} are bounded in Bp and converge to 0 uniformly on compact subsets of 𝔻, it follows that ∥uCφfφ(wn)∥𝒵 and ∥uCφgφ(wn)∥𝒵 converge to 0 as n→∞, proving (c). Therefore, by Lemma 5, it follows that ∥uCφpk∥𝒵→0 as k→∞, so (b) holds as well.
On the other hand, by (34) and Lemma 5, for all w∈𝔻, we have
(50)(1-|w|2)|u′′(w)|(loge1-|φ(w)|2)1-1/p≤∥uCφhp,φ(w)∥𝒵+(1-|w|2)|v(w)|1-|φ(w)|2+(1-|w|2)|u(w)φ′(w)2|(1-|φ(w)|2)2⟶0,as|φ(w)|⟶1,
which proves that (d) holds as well. Thus, to prove the equivalence of (a)–(d), it suffices to show that (d) implies (a). Since the proof is similar to the proof of (d) implies (a) in Theorem6, we omit the details.Observe that conditions (d) in Theorems1 and 6 coincide with the corresponding conditions (d) in Theorems 3 and 7 by taking p=∞. We could indeed have adopted the notation B∞ in place of ℬ, as the Bloch space is widely viewed as a limit of Bp as p→∞, which could have allowed us to unify parts of Theorems 1 and 3 (resp., Theorems 6 and 7) into a single statement taking 1<p≤∞. However, we did not do so because the family of functions {h∞,a} is not suitable to characterize the bounded and the compact weighted composition operators from the Bloch space into the Zygmund space.


## 4. Multiplication and the Composition Operators

In this section we highlight the results concerning the boundedness and the compactness of the multiplication operator and the composition operator, which to the best of our knowledge have not appeared in the literature.For the case of the multiplication operator, the finiteness ofM3 in part (d) of Theorems 1 and 3, combined with the continuous inclusions of H∞ and of Bp into ℬ and Theorems 1 and 2 in [19], yields the following result.Corollary 8.
Foru∈𝒵 and 1≤p<∞, the following statements are equivalent.(a)
Mu:ℬ→𝒵 is bounded.(b)
Mu:ℬ→𝒵 is compact.(c)
Mu:Bp→𝒵 is bounded.(d)
Mu:Bp→𝒵 is compact.(e)
Mu:H∞→𝒵 is bounded.(f)
Mu:H∞→𝒵 is compact.(g)
u is identically 0.For the case of the composition operator, noting thatu=1 implies that u′=u′′=0, from (23), we see that for any w∈𝔻, (Cφhφ(w))′′ is a linear combination of (Cφfφ(w))′′ and (Cφgφ(w))′′. Furthermore, since for a∈𝔻, the function z↦e/(1-a¯z) maps 𝔻 into the right half-plane, the image of the function z↦log(e/(1-a¯z)) is contained in the horizontal strip {w∈ℂ:|Imw|<π/2}. Thus,
(51)|hφ(w)(φ(0))|=(loge1-|φ(w)|2)-1|loge1-φ(w)¯φ(0)|2≤(loge1-|φ(w)|2)-1×[(loge|1-φ(w)¯φ(0)|)2+π24]≤(loge1-|φ(0)|)2+π24,
which does not depend on w. Thus, the boundedness of ∥Cφfφ(w)∥𝒵 and ∥Cφgφ(w)∥𝒵 implies the boundedness of ∥Cφhφ(w)∥𝒵. Moreover, from (34), it follows that for w∈𝔻,
(52)(Cφhp,φ(w))′′(w)=[φ′′(w)φ(w)¯1-|φ(w)|2+φ′(w)2φ(w)¯2(1-|φ(w)|2)2]×(loge1-|φ(w)|2)-1/p.
Therefore, if
(53)supw∈𝔻(1-|w|2)|φ′′(w)|1-|φ(w)|2<∞,supw∈𝔻(1-|w|2)|φ′(w)2|(1-|φ(w)|2)2<∞,
then, arguing as above, we see that |hp,φ(w)(φ(0))| is bounded by a constant independent of w, so
(54)supw∈𝔻∥Cφhp,φ(w)∥𝒵<∞.
From these remarks, Corollary 2 of [17], and Theorem 1 of [19], Theorems 1 and 3 yield the following result.Corollary 9.
Forφ∈S(𝔻) and 1≤p<∞, the following statements are equivalent.(a)
Cφ:ℬ→𝒵 is bounded.(b)
supk≥0∥φk∥𝒵<∞.(c)
φ,φ2∈𝒵, supw∈𝔻∥Cφfφ(w)∥𝒵<∞ and supw∈𝔻∥Cφgφ(w)∥𝒵<∞.(d)
supz∈𝔻((1-z|2)|φ′′(z)|/(1-|φ(z)|2))<∞ and supz∈𝔻((1-|z|2)|φ′(z)|2/(1-|φ(z)|2)2)<∞.(e)
Cφ:Bp→𝒵 is bounded.(f)
Cφ:H∞→𝒵 is bounded.Next note that, by the above remarks, as|φ(w)|→1,
(55)|hφ(w)(φ(0))|≤(loge1-|φ(w)|2)-1×[(loge1-|φ(0)|)2+π24]⟶0,|hp,φ(w)(φ(0))|≤(loge1-|φ(w)|2)-1/p×[(loge1-|φ(0)|)2+π24]1/2⟶0.
Thus, the convergence to 0 of ∥Cφfφ(w)∥𝒵 and ∥Cφgφ(w)∥𝒵 as |φ(w)|→1 implies that ∥Cφhφ(w)∥𝒵 and ∥Cφhp,φ(w)∥𝒵 converge to 0 as well. From Theorems 6 and 7, and from Corollary 3 of [17] and Theorem 2 of [19], we deduce the following result.Corollary 10.
Forφ∈S(𝔻) such that Cφ:ℬ→𝒵 is bounded and for 1≤p<∞, the following statements are equivalent.(a)
Cφ:ℬ→𝒵 is compact.(b)
limk→∞∥φk∥𝒵=0.(c)
lim|φ(w)|→1∥Cφfφ(w)∥𝒵=lim|φ(w)|→1∥Cφgφ(w)∥𝒵=0.(d)
lim|φ(z)|→1((1-z|2)|φ′′(z)|/(1-|φ(z)|2))=limφ|z|→1((1-|z|2)|φ′(z)|2/(1-|φ(z)|2)2)=0.(e)
Cφ:Bp→𝒵 is compact.(f)
Cφ:H∞→𝒵 is compact.


### 4.1. Concluding Remarks

In this section, we established that the boundedness (resp., compactness) of the multiplication and the composition operator fromH∞ to 𝒵 is equivalent to the boundedness (resp., compactness) of the multiplication and composition operator from Bp or ℬ to 𝒵. It is evident that the same can be said for the weighted composition operator mapping between these spaces for any choice of composition symbol when the multiplicative symbol is linear, or for an arbitrary multiplication symbol in the Zygmund space if the range of the composition symbol is relatively compact. This leads to the question of whether this equivalence holds also for completely general weighted composition operators. As a consequence of [17, 19], the answer is affirmative for the case of the spaces H∞ and B1. We suspect this is false for the other Besov spaces or ℬ but have not been able to construct examples to support this claim.


## 4.1. Concluding Remarks

In this section, we established that the boundedness (resp., compactness) of the multiplication and the composition operator fromH∞ to 𝒵 is equivalent to the boundedness (resp., compactness) of the multiplication and composition operator from Bp or ℬ to 𝒵. It is evident that the same can be said for the weighted composition operator mapping between these spaces for any choice of composition symbol when the multiplicative symbol is linear, or for an arbitrary multiplication symbol in the Zygmund space if the range of the composition symbol is relatively compact. This leads to the question of whether this equivalence holds also for completely general weighted composition operators. As a consequence of [17, 19], the answer is affirmative for the case of the spaces H∞ and B1. We suspect this is false for the other Besov spaces or ℬ but have not been able to construct examples to support this claim.



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*Source: 154029-2013-05-19.xml*