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

Abstract

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space ℬ and the analytic Besov spaces 𝐵_𝑝 (with 1<𝑝<∞) into the Zygmund space 𝒵. As a special case, we show that the bounded (resp., compact) composition operators from ℬ, 𝐵_𝑝, 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.

1. Introduction

Let 𝔻 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 space ℬ if 𝛽_𝑓=sup_𝑧∈𝔻1−|𝑧|²𝑓^′(𝑧)<∞. (1) Under the norm defined by ‖𝑓‖_ℬ=|𝑓(0)|+𝛽_𝑓,  ℬ is a conformally invariant Banach space. The functions in the Bloch space satisfy the following growth condition: 𝑓(𝑧≤𝑓(0+1))2|𝑧|log1+|𝑧|𝛽1−^𝑓≤𝑒1+log1−|𝑧|²𝑓_ℬ. (2)

An important class of Möbius invariant spaces is given by the analytic Besov spaces 𝐵_𝑝, with 1<𝑝<∞, consisting of the functions 𝑓∈𝐻(𝔻) such that 𝑏_𝑝𝑓^𝑝∶=_𝔻1−|𝑧|^2𝑝−2𝑓^′(𝑧)^𝑝𝑑𝐴(𝑧)<∞, (3) where 𝑑𝐴 denotes the normalized area measure on the unit disk. The quantity 𝑏_𝑝 is a seminorm and the Besov norm is defined by 𝑓_𝐵𝑝=𝑓(0)+𝑏_𝑝𝑓. (4) By Lemma 1.1 of [1], for 𝑓∈𝐵_𝑝, 𝑓_ℬ≤𝐶𝑓_𝐵𝑝. (5) Thus, the space 𝐵_𝑝 is continuously embedded in the Bloch space. Moreover, by Theorem 9 in [2], the functions in 𝐵_𝑝 satisfy the growth condition 𝑓(𝑧𝑒)≤𝐶log1−|𝑧|^21−1/𝑝,𝑧∈𝔻. (6)

Let 𝒵 denote the set of all functions 𝑓∈𝐻(𝔻)∩𝐶(𝔻) such that 𝑓𝑓𝑒=sup^{𝑖(𝜃+ℎ)}+𝑓𝑒^{𝑖(𝜃−ℎ)}−2𝑓𝑒^𝑖𝜃ℎ<∞, (7) 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, 𝑓≍sup_𝑧∈𝔻1−|𝑧|²𝑓^′′(𝑧). (8) 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 𝑓_𝒵=𝑓(0)+𝑓^′(0)+sup_𝑧∈𝔻1−|𝑧|²𝑓^′′(𝑧) (9)

yields a Banach space structure on 𝒵, which is called the Zygmund space. For more information on the Zygmund space on the unit disk, see, for example, [3].

Let 𝑆(𝔻) denote the set of all analytic self-maps of 𝔻. Each 𝜑∈𝑆(𝔻) induces the composition operator on 𝐻(𝔻) defined by 𝐶_𝜑𝑓(𝑧)=𝑓𝜑(𝑧) (10)

for 𝑓∈𝐻(𝔻) and 𝑧∈𝔻. We refer the interested reader to [4, 5] for the theory of the composition operators.

Let 𝑢∈𝐻(𝔻). The multiplication operator 𝑀_𝑢 is defined as 𝑀_𝑢𝑓(𝑧)=𝑢(𝑧)𝑓(𝑧) (11)

for 𝑓∈𝐻(𝔻) and 𝑧∈𝔻. The composition product 𝑢𝐶_𝜑 of  𝑀_𝑢 and 𝐶_𝜑 yields a linear operator on 𝐻(𝔻) called the weighted composition operator with symbols 𝑢  and 𝜑.

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 space 𝐻^𝑝 (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 𝐶_𝜑 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 𝐵_𝑝 (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 {𝜑^𝑛} has this property with regards to the composition operator on the Bloch space as well as the space 𝐵𝑀𝑂𝐴. In fact, for 𝑋=𝐻^∞,𝐵_𝑝,𝐵𝑀𝑂𝐴,ℬ, the operator 𝐶_𝜑∶𝑋→ℬ is compact if and only if ‖𝜑^𝑛‖_ℬ→0 as 𝑛→∞. For the weighted composition operator 𝑢𝐶_𝜑, this is not the case, although a few cases are known in which the sequence of norms ‖𝑢𝜑^𝑛‖_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 operator 𝑢𝐶_𝜑 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 𝑢𝐶_𝜑, 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 {‖𝑢𝜑^𝑘‖_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 from ℬ or 𝐵_𝑝 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 ‖𝜑^𝑘‖_𝒵=𝑂(1) (resp., ‖𝜑^𝑘‖_𝒵=𝑜(1)) as 𝑘→∞. We found this phenomenon quite surprising due to the size of the spaces 𝐵_𝑝 and ℬ in comparison to that of the Zygmund space.

2. Boundedness

We begin the section by characterizing the bounded weighted composition operators from ℬ to the Zygmund space. We note that since the constant function 1 belongs to all spaces under consideration, the boundedness of 𝑢𝐶_𝜑 requires that 𝑢=𝑢𝐶_𝜑1∈𝒵. Thus, we will assume throughout that 𝑢∈𝒵.

For a fixed 𝑎∈𝔻 and for 𝑧∈𝔻, set 𝑓_𝑎(𝑧)=1−|𝑎|²1−𝑎𝑧,𝑔_𝑎(𝑧)=1−|𝑎|²1−𝑎𝑧²,ℎ_𝑎(𝑧)=𝑒log1−𝑎𝑧²𝑒log1−|𝑎|²⁻¹. (12)

For an integer 𝑘≥0 and 𝑧∈𝔻, let 𝑝_𝑘(𝑧)=𝑧^𝑘.

Theorem 1. Let 𝑢∈𝒵 and 𝜑∈𝑆(𝔻). Then the following conditions are equivalent.(a)The operator 𝑢𝐶_𝜑∶ℬ→𝒵 is bounded.(b)𝑀∶=sup_𝑘≥0‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵<∞, and 𝐻∶=sup_𝑤∈𝔻‖𝑢𝐶_𝜑ℎ_𝜑(𝑤)‖_𝒵<∞.(c)𝑁₁∶=sup_𝑧∈𝔻(1−|𝑧|²)|𝑢(𝑧)𝜑^′(𝑧)²|, 𝑁₂∶=sup_𝑧∈𝔻(1−|𝑧|²)|2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)|, 𝐴∶=sup_𝑤∈𝔻𝑢𝐶_𝜑𝑓_{𝜑(𝑤)𝒵},𝐵∶=sup_𝑤∈𝔻𝑢𝐶_𝜑𝑔_{𝜑(𝑤)𝒵}, (13) and 𝐻 are finite. (d)The quantities 𝑀₁∶=sup_𝑧∈𝔻(1−|𝑧|²)|𝑢^′′(𝑧)|log(𝑒/(1−|𝜑(𝑧)|²)), 𝑀₂∶=sup_𝑧∈𝔻1−|𝑧|²2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)1−𝜑(𝑧)²,𝑀₃∶=sup_𝑧∈𝔻1−|𝑧|²𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)²² (14) are finite.

To prove Theorem 1, 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. Let 𝑢∈𝒵 and 𝜑∈𝑆(𝔻). Then the following conditions are equivalent.(a)sup_𝑘≥0‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵<∞. (b)The quantities 𝑁₁, 𝑁₂, 𝐴, and 𝐵 are finite.(c)The quantities 𝑀₂ and 𝑀₃ are finite.

Proof. (a) ⇒ (b) Since the sequence {𝑝_𝑘} is bounded in ℬ with supremum norms no greater than 1, if 𝑢𝐶_𝜑∶ℬ→𝒵 is bounded, then for each integer 𝑘≥0, we have 𝑢𝐶_𝜑𝑝_𝑘𝒵≤𝑢𝐶_𝜑. (15) Therefore, the supremum of ‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵 over all integers 𝑘≥0 is finite. Likewise, sup_𝑤∈𝔻‖ℎ_𝜑(𝑤)‖_ℬ=𝐿<∞. Therefore, 𝑢𝐶_𝜑ℎ_{𝜑(𝑤)𝒵}≤𝐿𝑢𝐶_𝜑. (16)
(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 𝑓_𝑎^(𝑛)(𝑎)=𝑛!𝑎^𝑛1−|𝑎|^2𝑛,𝑔_𝑎^(𝑛)(𝑎)=(𝑛+1)!𝑎^𝑛1−|𝑎|^2𝑛,ℎ_𝑎(𝑎)𝑒=log1−|𝑎|²,ℎ_𝑎^(𝑛)(𝑎)=(𝑛+2)!𝑎^𝑛21−|𝑎|^2𝑛,𝑛≥1. (17)
Setting 𝑣(𝑧)=2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧),𝑧∈𝔻, (18) for 𝑤∈𝔻, from (17) we obtain 𝑢𝐶_𝜑𝑓_𝜑(𝑤)^′′(𝑤)=𝑢^′′(𝑤)+𝑣(𝑤)𝜑(𝑤)1−𝜑(𝑤)²+2𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)²1−𝜑(𝑤)²²,𝑢𝐶_𝜑𝑔_𝜑(𝑤)^′′(𝑤)=𝑢^′′(𝑤)+2𝑣(𝑤)𝜑(𝑤)1−𝜑(𝑤)²+6𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)²1−𝜑(𝑤)²²,𝑢𝐶_𝜑ℎ_𝜑(𝑤)^′′(𝑤)𝑒=log1−|𝑤|²𝑢^′′(𝑤)+3𝑣(𝑤)𝜑(𝑤)1−𝜑(𝑤)²+12𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)²1−𝜑(𝑤)²². (19)(20)(21) Subtracting (19) from (20), we get −𝑢𝐶_𝜑𝑓_𝜑(𝑤)^′′(𝑤)+𝑢𝐶_𝜑𝑔_𝜑(𝑤)^′′(𝑤)=𝑣(𝑤)𝜑(𝑤)1−𝜑(𝑤)²+4𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)²1−𝜑(𝑤)²². (22) Hence, from (21) and (22), we deduce 𝑒log1−𝜑(𝑤)²𝑢^′′(𝑤)=𝑢𝐶_𝜑ℎ_𝜑(𝑤)^′′(𝑤)𝑣(𝑤)−3𝜑(𝑤)1−𝜑(𝑤)²+4𝑢(𝑤)𝜑′(𝑤)²𝜑(𝑤)²1−𝜑(𝑤)²²=𝑢𝐶_𝜑ℎ_𝜑(𝑤)^′′(𝑤)+3𝑢𝐶_𝜑𝑓_𝜑(𝑤)^′′(𝑤)−3𝑢𝐶_𝜑𝑔_𝜑(𝑤)^′′(𝑤). (23) Therefore, taking the modulus and multiplying by (1−|𝑤|²), we obtain 1−|𝑤|²𝑒log1−𝜑(𝑤)²𝑢^′′(𝑤≤)𝑢𝐶_𝜑ℎ_{𝜑(𝑤)𝒵}+3𝑢𝐶_𝜑𝑓_{𝜑(𝑤)𝒵}+3𝑢𝐶_𝜑𝑔_{𝜑(𝑤)𝒵}. (24)
Taking the supremum over all 𝑤∈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, sup_𝑧∈𝔻1−|𝑧|²²𝑓^′′(𝑧)≤𝐶𝛽_𝑓≤𝐶𝑓_ℬ. (25) Thus, using (2) and (25), for an arbitrary 𝑧 in 𝔻, we have 1−|𝑧|²𝑢𝐶_𝜑𝑓^′′(𝑧≤|𝑧|)1−²𝑢^′′(𝑧𝑓𝜑(𝑧)+|𝑧|)1−²𝑓^′𝜑(𝑧)𝑣(𝑧+|𝑧||)|1−²𝑓^′′𝜑(𝑧)𝑢(𝑧)𝜑^′(𝑧)²≤1−|𝑧|²𝑢^′′(𝑧×𝑒)1+log1−𝜑(𝑧)²𝑓_ℬ+1−|𝑧|²|𝑣(𝑧)|1−𝜑(𝑧)²𝑓_ℬ|𝑧|+𝐶1−²𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)²²𝑓_ℬ≤‖𝑢‖_𝒵+𝑀₁+𝑀₂+𝐶𝑀₃𝑓_ℬ. (26) In addition, again by (2), we have 𝑢𝐶_𝜑𝑓(0≤|𝑢(0𝑒))|1+log1−𝜑(0)²𝑓_ℬ,𝑢𝐶_𝜑𝑓^′(0≤𝑢_′)(0𝑒)1+log1−𝜑(0)²𝑢(0)𝜑+𝐶^′(0)1−𝜑(0)²𝑓_ℬ. (27)
The boundedness of 𝑢𝐶_𝜑∶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 spaces 𝐵_𝑝 for 1<𝑝<∞. The case 𝑝=1 has been established by the second author in [19]. For 1<𝑝<∞ and 𝑎∈𝔻, define ℎ_𝑝,𝑎(𝑧)=𝑒log1−𝑒𝑎𝑧log1−|𝑎|^2−1/𝑝,𝑧∈𝔻. (28) Then, as shown in [12], ℎ_𝑝,𝑎∈𝐵_𝑝 and ‖ℎ_𝑝,𝑎‖_𝐵𝑝 is bounded by a constant 𝐿_𝑝 independent of 𝑎.

Theorem 3. Let 𝑢∈𝒵, 𝜑∈𝑆(𝔻), and 1<𝑝<∞. Then the following conditions are equivalent.(a)The operator 𝑢𝐶_𝜑∶𝐵_𝑝→𝒵 is bounded.(b)𝑀∶=sup_𝑘≥0‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵<∞, and 𝐻_𝑝∶=sup_𝑤∈𝔻‖𝑢𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)}‖_𝒵<∞.(c)The quantities 𝑁₁, 𝑁₂, 𝐴, and 𝐵 in Theorem 1 and 𝐻_𝑝 are finite.(d)The quantities 𝑀₂ and 𝑀₃ in Theorem 1 and 𝑀₄∶=sup_𝑧∈𝔻1−|𝑧|²𝑢^′′(𝑧𝑒)log1−𝜑(𝑧)^21−1/𝑝 (29) are finite.

Proof. Note that, unlike the case of the Bloch space, the sequence {𝑝_𝑘} is unbounded in 𝐵_𝑝. Therefore, first condition in (b) is not an immediate consequence of the boundedness of the operator.
(a) ⇒ (d) Suppose 𝑢𝐶_𝜑∶𝐵_𝑝→𝒵 is bounded. Since sup_𝑤∈𝔻‖ℎ_{𝑝,𝜑(𝑤)}‖_ℬ=𝐿_𝑝<∞, it follows that 𝑢𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)𝒵}≤𝐿_𝑝𝑢𝐶_𝜑. (30) Moreover, for 𝑎∈𝔻, 𝑓_𝑎 and 𝑔_𝑎 are in 𝐵_𝑝 and their Besov norms are bounded by constants independent of 𝑎. Indeed, noting that 𝑓_𝑎=1−𝑎𝐿_𝑎, with 𝐿_𝑎(𝑧)=(𝑎−𝑧)/(1−𝑎𝑧), and using the Möbius invariance of the Besov seminorm, we have 𝑓_𝑎𝐵𝑝=1−|𝑎|²+𝑏_𝑝𝑓_𝑎≤1+|𝑎|𝑏_𝑝𝐿_𝑎=1+|𝑎|𝑏_𝑝𝑝₁1≤1+𝑝−1^1/𝑝. (31) Likewise, since 𝑔_𝑎=1−2𝑎𝐿_𝑎+𝑎²𝐿²_𝑎, and observing that 𝑏_𝑝(𝑝₂)≤2𝑏_𝑝(𝑝₁), again by the Möbius invariance of 𝑏_𝑝, we obtain 𝑔_𝑎𝐵𝑝=1−|𝑎|²²+𝑏_𝑝𝑔_𝑎≤1+2|𝑎|𝑏_𝑝𝐿_𝑎+|𝑎|²𝑏_𝑝𝐿²_𝑎=≤1+2𝑏_𝑝𝑝₁+𝑏_𝑝𝑝₂1≤1+4𝑝−1^1/𝑝. (32)
Therefore, by the boundedness of 𝑢𝐶_𝜑, the quantities 𝐴 and 𝐵 are finite.
Next observe that since 𝑝₁ and 𝑝₂ are in 𝐵_𝑝 and 𝑢𝐶_𝜑 is bounded, 𝑢𝐶_𝜑(𝑝₁) and 𝑢𝐶_𝜑(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 ℎ_{𝑝,𝜑(𝑤)}𝜑(𝑤)=𝑒log1−𝜑(𝑤)^21−1/𝑝,ℎ^′_{(𝑤)𝑝,𝜑}𝜑(𝑤)=𝑒log1−𝜑(𝑤)^2−1/𝑝𝜑(𝑤)1−𝜑(𝑤)²,ℎ_(𝑤)^′′_𝑝,𝜑𝜑(𝑤)=𝑒log1−𝜑(𝑤)^2−1/𝑝𝜑(𝑤)²1−𝜑(𝑤)²². (33) Recalling the notation 𝑣(𝑤)=2𝑢^′(𝑤)𝜑^′(𝑤)+𝑢(𝑤)𝜑^′′(𝑤), a straightforward calculation shows that 𝑢𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)}^′′(𝑤)=𝑢^′′(𝑤)𝑒log1−𝜑(𝑤)^21−1/𝑝+𝑣(𝑤)𝜑(𝑤)1−𝜑(𝑤)²𝑒log1−𝜑(𝑤)^2−1/𝑝+𝑢(𝑤)𝜑^′(𝑤)²𝜑(𝑤)²1−𝜑(𝑤)²²𝑒log1−𝜑(𝑤)^2−1/𝑝. (34) Thus, by the boundedness of 𝑢𝐶_𝜑, we have 1−|𝑤|²𝑢^′′(𝑤𝑒)log1−𝜑(𝑤)^21−1/𝑝≤𝑢𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)𝒵}+𝑀₂+𝑀₃≤𝐿_𝑝𝑢𝐶_𝜑+𝑀₂+𝑀₃. (35)
Taking the supremum over all 𝑤∈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∈𝐵_𝑝, sup_𝑧∈𝔻1−|𝑧|²²𝑓^′′(𝑧)≤𝐶𝛽_𝑓≤𝐶𝑓_ℬ≤𝐶𝑓_𝐵𝑝. (36) Thus, using (6) and (36), for any 𝑧∈𝔻, we have 1−|𝑧|²𝑢𝐶_𝜑𝑓^′′(𝑧|𝑧|)≤𝐶1−²𝑢^′′(𝑧×𝑒)log1−𝜑(𝑧)^21−1/𝑝𝑓_𝐵𝑝|𝑧|+𝐶1−²|𝑣(𝑧)|1−𝜑(𝑧)²𝑓_𝐵𝑝|𝑧|+𝐶1−²𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)²²𝑓_𝐵𝑝≤𝐶𝑀₂+𝑀₃+𝑀₄𝑓_𝐵𝑝. (37) Moreover, again by (6) and (5), we have 𝑢𝐶_𝜑𝑓0|𝑢(0𝑒()≤𝐶)|log1−𝜑(0)^21−1/𝑝𝑓_𝐵𝑝,𝑢𝐶_𝜑𝑓^′(0𝑢)≤𝐶^′(0𝑒)log1−𝜑(0)^21−1/𝑝+𝑢(0)𝜑^′(0)1−𝜑(0)²𝑓_𝐵𝑝. (38)
Taking the supremum over all 𝑧∈𝔻 and using (38) and (37), we conclude that 𝑢𝐶_𝜑∶𝐵_𝑝→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. Let 𝑢∈𝐻(𝔻), 𝜑∈𝑆(𝔻), and 1<𝑝<∞. Then, the operator 𝑢𝐶_𝜑  from ℬ (resp., 𝐵_𝑝) into 𝒵 is compact if and only if it 𝑢𝐶_𝜑∶ℬ→𝒵 (resp., 𝑢𝐶_𝜑∶𝐵_𝑝→𝒵) is bounded and ‖𝑢𝐶_𝜑𝑓_𝑘‖_𝒵→0, as 𝑘→∞, whenever {𝑓_𝑘} is a bounded sequence in ℬ (resp., 𝐵_𝑝) 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 from 𝐻^∞ to 𝒵 given in Theorem 2 of [17].

Lemma 5. Let 𝑢∈𝒵 and 𝜑∈𝑆(𝔻) be such that any of the equivalent conditions in Lemma 2 hold. Then the following conditions are equivalent.(a)lim_𝑘→∞‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵=0. (b)lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑𝑓_𝜑(𝑤)‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑𝑔_𝜑(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 lim_{𝜑(𝑧)→1}1−|𝑧|²2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)1−𝜑(𝑧)²=0. (39)

In the next theorem we focus on the weighted composition operators acting on the Bloch space. We will use the one-parameter family {ℎ_𝑎∶𝑎∈𝔻} introduced in (12).

Theorem 6. Let 𝑢∈𝒵, 𝜑∈𝑆(𝔻) and suppose that the operator 𝑢𝐶_𝜑∶ℬ→𝒵 is bounded. Then the following conditions are equivalent. (a)The operator 𝑢𝐶_𝜑∶ℬ→𝒵 is compact. (b)lim_𝑘→∞‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑ℎ_𝜑(𝑤)‖_𝒵=0. (c)lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑𝑓_𝜑(𝑤)‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑𝑔_𝜑(𝑤)‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑ℎ_𝜑(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 lim_{𝜑(𝑧)→1}1−|𝑧|²2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)1−𝜑(𝑧)²=0. (40)

Proof. (a) ⇒ (b) Suppose 𝑢𝐶_𝜑∶ℬ→𝒵 is a compact operator. Since the sequence {𝑝_𝑘} is bounded in ℬ and converges to 0 uniformly on compact subsets of 𝔻, then by Lemma 4, 𝑢𝐶_𝜑𝑝_𝑘𝒵⟶0,as𝑘⟶∞. (41) On the other hand, if {𝑤_𝑛} is a sequence in 𝔻 such that |𝜑(𝑤_𝑛)|→1 as 𝑛→∞, then {ℎ_{𝜑(𝑤𝑛)}} is a bounded sequence in ℬ converging to 0 uniformly on compact subsets of 𝔻, so again by Lemma 4, it follows that 𝑢𝐶_𝜑ℎ_{𝜑(𝑤𝑛)𝒵}⟶0as𝑛⟶∞, (42)
proving (b).
(b) ⇒ (c) follows from Lemma 4.
(c) ⇒ (d) Suppose that the limits in (c) are 0. Using the inequality (24), we get 1−|𝑤|²𝑢^′′(𝑤𝑒)log1−𝜑(𝑤)²≤3𝑢𝐶_𝜑𝑓_{𝜑(𝑤)𝒵}+3𝑢𝐶_𝜑𝑔_{𝜑(𝑤)𝒵}+𝑢𝐶_𝜑ℎ_{𝜑(𝑤)𝒵}⟶0 (43)
as |𝜑(𝑤)|→1. The desired result now follows from the last formula and Lemma 4.
(d) ⇒ (a) Assume (d) holds. Then, in particular, lim_{𝜑(𝑧)→1}1−|𝑧|²𝑢^′′(𝑧)=0. (44) Thus, recalling the notation 𝑣=2𝑢^′𝜑^′+𝑢𝜑^′′, for any 𝜀>0, there is a constant 𝑟∈(0,1), such that 1−|𝑧|²𝑢^′′(𝑧|𝑧|)<𝜀,1−²𝑢^′′(𝑧𝑒)log1−|𝑧|²|𝑧|<𝜀,1−²𝑢(𝑧)𝜑^′(𝑧)²1−𝜑(𝑧)²²|𝑧|<𝜀,1−²|𝑣(𝑧)|1−𝜑(𝑧)²<𝜀, (45)
whenever 𝑟<|𝜑(𝑧)|<1.
Let {𝑓_𝑘} be a sequence in ℬ with sup‖𝑓_𝑘‖_ℬ≤𝑄 and converging to 0 uniformly on compact subsets of 𝔻. In light of Lemma 4, it suffices to show that ‖𝑢𝐶_𝜑𝑓_𝑘‖_𝒵→0 as 𝑘→∞. Using (45), for |𝜑(𝑤)|>𝑟, and recalling (2) and (25), we have 1−|𝑤|²𝑢𝐶_𝜑𝑓_𝑘^′′(𝑤≤|𝑤|)1−²𝑢^′′(𝑤)𝑓_𝑘𝜑(𝑤)+1−|𝑤|²𝑓^′_𝑘𝜑(𝑤)𝑣(𝑤+|𝑤||)|1−²𝑓_𝑘^′′𝜑(𝑤)𝑢(𝑤)𝜑^′(𝑤)²≤1−|𝑤|²𝑢^′′(𝑤×𝑒)1+log1−𝜑(𝑤)²𝑓_𝑘ℬ+1−|𝑤|²|𝑣(𝑤)|1−𝜑(𝑤)²𝑓_𝑘ℬ|𝑤|+𝐶1−²𝑢(𝑤)𝜑^′(𝑤)²1−𝜑(𝑤)²²𝑓_𝑘ℬ≤(3+𝐶)𝑄𝜀. (46) By Cauchy's estimate, if {𝑓_𝑘} is a sequence which converges to zero on compact subset of 𝔻, then so do the sequences {𝑓^′_𝑘} and {𝑓_𝑘^′′}. Hence, for |𝜑(𝑤)|≤𝑟, we have 1−|𝑤|²𝑢𝐶_𝜑𝑓_𝑘^′′(𝑤≤|𝑤|)1−²𝑢^′′(𝑤)𝑓_𝑘𝜑(𝑤)+1−|𝑤|²𝑓_𝑘^′′𝜑(𝑤)𝑢(𝑤)𝜑^′(𝑤)²+1−|𝑤|²𝑓^′_𝑘𝜑(𝑤)𝑣(𝑤≤‖𝑢‖|)|_𝒵𝑓_𝑘𝜑(𝑤)+𝑁₁𝑓_𝑘^′′𝜑(𝑤)+𝑁₂𝑓^′_𝑘𝜑(𝑤)⟶0,as𝑘⟶∞. (47) Since |(𝑢𝐶_𝜑𝑓_𝑘)(0)|=|𝑢(0)𝑓_𝑘(𝜑(0))|→0 and 𝑢𝐶_𝜑𝑓_𝑘^′(0=𝑢(0)𝜑_′)(0)𝑓^′_𝑘𝜑(0)+𝑢^′(0)𝑓_𝑘𝜑(0)⟶0 (48)
as 𝑘→∞, from (46) and (47) we deduce that ‖𝑢𝐶_𝜑𝑓_𝑘‖_𝒵→0, as 𝑘→∞. This completes the proof.

We now turn our attention to the weighted composition operators mapping 𝐵_𝑝 into the Zygmund space. We will use the one-parameter family {ℎ_𝑝,𝑎∶𝑎∈𝔻} defined in (28).

Theorem 7. Let 𝑢∈𝒵, 𝜑∈𝑆(𝔻), 1<𝑝<∞ and assume the operator 𝑢𝐶_𝜑∶𝐵_𝑝→𝒵 is bounded. Then the following conditions are equivalent. (a)The operator 𝑢𝐶_𝜑∶𝐵_𝑝→𝒵 is compact; (b)lim_𝑘→∞‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)}‖_𝒵=0; (c)lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑𝑓_𝜑(𝑤)‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑𝑔_𝜑(𝑤)‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝑢𝐶_𝜑ℎ_{𝑝,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 lim_{𝜑(𝑧)→1}1−|𝑧|²2𝑢^′(𝑧)𝜑^′(𝑧)+𝑢(𝑧)𝜑^′′(𝑧)1−𝜑(𝑧)²=0. (49)

Proof. Assume 𝑢𝐶_𝜑∶𝐵_𝑝→𝒵 is compact. Since {ℎ_{𝑝,𝜑(𝑤)}} is bounded in 𝐵_𝑝 and convergent to 0 uniformly on compact subsets of 𝔻, if {𝑤_𝑛} is a sequence in 𝔻 such that |𝜑(𝑤_𝑛)|→1 as 𝑛→∞, by Lemma 4 it follows that ‖𝑢𝐶_𝜑ℎ_{𝑝,𝜑(𝑤𝑛)}‖_𝒵→0 as 𝑛→∞. Similarly, since the sequences {𝑓_{𝜑(𝑤𝑛)}} and {𝑔_{𝜑(𝑤𝑛)}} are bounded in 𝐵_𝑝 and converge to 0 uniformly on compact subsets of 𝔻, it follows that ‖𝑢𝐶_𝜑𝑓_{𝜑(𝑤𝑛)}‖_𝒵 and ‖𝑢𝐶_𝜑𝑔_{𝜑(𝑤𝑛)}‖_𝒵 converge to 0 as 𝑛→∞, proving (c). Therefore, by Lemma 5, it follows that ‖𝑢𝐶_𝜑𝑝_𝑘‖_𝒵→0 as 𝑘→∞, so (b) holds as well.
On the other hand, by (34) and Lemma 5, for all 𝑤∈𝔻, we have 1−|𝑤|²𝑢^′′(𝑤𝑒)log1−𝜑(𝑤)^21−1/𝑝≤𝑢𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)𝒵}+1−|𝑤|²|𝑣(𝑤)|1−𝜑(𝑤)²+1−|𝑤|²𝑢(𝑤)𝜑^′(𝑤)²1−𝜑(𝑤)²²⟶0,as𝜑(𝑤)⟶1, (50)
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 Theorem 6, we omit the details.

Observe that conditions (d) in Theorems 1 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 𝐵_𝑝 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 of 𝑀₃ in part (d) of Theorems 1 and 3, combined with the continuous inclusions of 𝐻^∞ and of 𝐵_𝑝 into ℬ and Theorems 1 and 2 in [19], yields the following result.

Corollary 8. For 𝑢∈𝒵 and 1≤𝑝<∞, the following statements are equivalent.(a)𝑀_𝑢∶ℬ→𝒵 is bounded.(b)𝑀_𝑢∶ℬ→𝒵 is compact.(c)𝑀_𝑢∶𝐵_𝑝→𝒵 is bounded.(d)𝑀_𝑢∶𝐵_𝑝→𝒵 is compact.(e)𝑀_𝑢∶𝐻^∞→𝒵 is bounded.(f)𝑀_𝑢∶𝐻^∞→𝒵 is compact.(g)𝑢 is identically 0.

For the case of the composition operator, noting that 𝑢=1 implies that 𝑢^′=𝑢^′′=0, from (23), we see that for any 𝑤∈𝔻, (𝐶_𝜑ℎ_𝜑(𝑤))^′′ is a linear combination of (𝐶_𝜑𝑓_𝜑(𝑤))^′′ and (𝐶_𝜑𝑔_𝜑(𝑤))^′′. Furthermore, since for 𝑎∈𝔻, the function 𝑧↦𝑒/(1−𝑎𝑧) maps 𝔻 into the right half-plane, the image of the function 𝑧↦log(𝑒/(1−𝑎𝑧)) is contained in the horizontal strip {𝑤∈ℂ∶|Im𝑤|<𝜋/2}. Thus, ℎ_𝜑(𝑤)𝜑(0)=𝑒log1−𝜑(𝑤)²⁻¹𝑒log1−𝜑(𝑤)𝜑(0)²≤𝑒log1−𝜑(𝑤)²⁻¹×𝑒log1−𝜑(𝑤)𝜑(0)²+𝜋²4≤𝑒log1−𝜑(0)²+𝜋²4, (51) which does not depend on 𝑤. Thus, the boundedness of ‖𝐶_𝜑𝑓_𝜑(𝑤)‖_𝒵 and ‖𝐶_𝜑𝑔_𝜑(𝑤)‖_𝒵 implies the boundedness of ‖𝐶_𝜑ℎ_𝜑(𝑤)‖_𝒵. Moreover, from (34), it follows that for 𝑤∈𝔻, 𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)}^′′(𝑤)=𝜑^′′(𝑤)𝜑(𝑤)1−𝜑(𝑤)²+𝜑^′(𝑤)²𝜑(𝑤)²1−𝜑(𝑤)²²×𝑒log1−𝜑(𝑤)^2−1/𝑝. (52) Therefore, if sup_𝑤∈𝔻1−|𝑤|²𝜑^′′(𝑤)1−𝜑(𝑤)²<∞,sup_𝑤∈𝔻1−|𝑤|²𝜑^′(𝑤)²1−𝜑(𝑤)²²<∞, (53) then, arguing as above, we see that |ℎ_{𝑝,𝜑(𝑤)}(𝜑(0))| is bounded by a constant independent of 𝑤, so sup_𝑤∈𝔻𝐶_𝜑ℎ_{𝑝,𝜑(𝑤)𝒵}<∞. (54) From these remarks, Corollary 2 of [17], and Theorem 1 of [19], Theorems 1 and 3 yield the following result.

Corollary 9. For 𝜑∈𝑆(𝔻) and 1≤𝑝<∞, the following statements are equivalent.(a)𝐶_𝜑∶ℬ→𝒵 is bounded.(b)sup_𝑘≥0‖𝜑^𝑘‖_𝒵<∞.(c)𝜑,𝜑²∈𝒵, sup_𝑤∈𝔻‖𝐶_𝜑𝑓_𝜑(𝑤)‖_𝒵<∞ and sup_𝑤∈𝔻‖𝐶_𝜑𝑔_𝜑(𝑤)‖_𝒵<∞.(d)sup_𝑧∈𝔻((1−𝑧|²)|𝜑^′′(𝑧)|/(1−|𝜑(𝑧)|²))<∞ and sup_𝑧∈𝔻((1−|𝑧|²)|𝜑^′(𝑧)|²/(1−|𝜑(𝑧)|²)²)<∞.(e)𝐶_𝜑∶𝐵_𝑝→𝒵 is bounded.(f)𝐶_𝜑∶𝐻^∞→𝒵 is bounded.

Next note that, by the above remarks, as |𝜑(𝑤)|→1, ℎ_𝜑(𝑤)𝜑(0)≤𝑒log1−𝜑(𝑤)²⁻¹×𝑒log1−𝜑(0)²+𝜋²4ℎ⟶0,_{𝑝,𝜑(𝑤)}𝜑(0)≤𝑒log1−𝜑(𝑤)^2−1/𝑝×𝑒log1−𝜑(0)²+𝜋²4^1/2⟶0. (55) Thus, the convergence to 0 of ‖𝐶_𝜑𝑓_𝜑(𝑤)‖_𝒵 and ‖𝐶_𝜑𝑔_𝜑(𝑤)‖_𝒵 as |𝜑(𝑤)|→1 implies that ‖𝐶_𝜑ℎ_𝜑(𝑤)‖_𝒵 and ‖𝐶_𝜑ℎ_{𝑝,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. For 𝜑∈𝑆(𝔻) such that 𝐶_𝜑∶ℬ→𝒵 is bounded and for 1≤𝑝<∞, the following statements are equivalent.(a)𝐶_𝜑∶ℬ→𝒵 is compact.(b)lim_𝑘→∞‖𝜑^𝑘‖_𝒵=0.(c)lim_{|𝜑(𝑤)|→1}‖𝐶_𝜑𝑓_𝜑(𝑤)‖_𝒵=lim_{|𝜑(𝑤)|→1}‖𝐶_𝜑𝑔_𝜑(𝑤)‖_𝒵=0.(d)lim_{|𝜑(𝑧)|→1}((1−𝑧|²)|𝜑^′′(𝑧)|/(1−|𝜑(𝑧)|²))=lim_{𝜑|𝑧|→1}((1−|𝑧|²)|𝜑^′(𝑧)|²/(1−|𝜑(𝑧)|²)²)=0.(e)𝐶_𝜑∶𝐵_𝑝→𝒵 is compact.(f)𝐶_𝜑∶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 from 𝐻^∞ to 𝒵 is equivalent to the boundedness (resp., compactness) of the multiplication and composition operator from 𝐵_𝑝 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.

Acknowledgments

The second author is supported by the Foundation for Scientific and Technological Innovation in Higher Education of Guangdong (no. 2012KJCX0096), National Natural Science Foundation of China (no. 11001107), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. LYM11117).