Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

Abstract

The logarithmic Bloch space ℬ_log is the Banach space of analytic functions on the open unit disk 𝔻 whose elements 𝑓 satisfy the condition ‖𝑓‖=sup_𝑧∈𝔻(1−|𝑧|²)log(2/(1−|𝑧|²))|𝑓′(𝑧)|<∞. In this work we characterize the bounded and the compact weighted composition operators from the Hardy space 𝐻^𝑝 (with 1≤𝑝≤∞) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping 𝐻^𝑝 into the little logarithmic Bloch space defined as the subspace of ℬ_log consisting of the functions 𝑓 such that lim_|𝑧|→1(1−|𝑧|²)log(2/(1−|𝑧|²))|𝑓′(𝑧)|=0.

1. Introduction

Let 𝑋 and 𝑌 be Banach spaces of analytic functions on a domain Ω in ℂ, 𝜓 an analytic function on Ω, and let 𝜑 be an analytic function mapping Ω into itself. The weighted composition operator with symbols 𝜓 and 𝜑 from 𝑋 to 𝑌 is the operator 𝑊_𝜓,𝜑 with range in 𝑌 defined by𝑊_𝜓,𝜑𝑓=𝑀_𝜓𝐶_𝜑𝑓=𝜓(𝑓∘𝜑),for𝑓∈𝑋,(1.1) where 𝑀_𝜓 denotes the multiplication operator with symbol 𝜓, and 𝐶_𝜑 denotes the composition operator with symbol 𝜑.

Let 𝐻(𝔻) be the set of analytic functions on 𝔻={𝑧∈ℂ∶|𝑧|<1}. For 0<𝑝<∞ the Hardy space 𝐻^𝑝 is the space consisting of all 𝑓∈𝐻(𝔻) such that‖𝑓‖^𝑝𝐻_𝑝=sup_0<𝑟<1^02𝜋||𝑓𝑟𝑒^𝑖𝜃||^𝑝𝑑𝜃2𝜋<∞.(1.2) Let 𝐻^∞ denote the space of all 𝑓∈𝐻(𝔻) for which ‖𝑓‖_∞=sup_𝑧∈𝔻|𝑓(𝑧)|<∞.

The Bloch space ℬ on the open unit disk 𝔻 is the Banach space consisting of the analytic functions 𝑓 on 𝔻 such that‖𝑓‖_𝛽=sup_𝑧∈𝔻1−|𝑧|²||𝑓^(𝑧)||<∞.(1.3) The Bloch norm is given by ‖𝑓‖_ℬ=|𝑓(0)|+‖𝑓‖_𝛽. Using the Schwarz-Pick lemma, it is easy to see that the Hardy space 𝐻^∞ is contained in ℬ and ‖𝑓‖_𝛽≤‖𝑓‖_∞. The inclusion is proper, as the function 𝑓(𝑧)=log(1+𝑧)/(1−𝑧) shows.

The little Bloch space, denoted by ℬ₀, is defined as the set of the analytic functions 𝑓 on 𝔻 such that lim_|𝑧|→1(1−|𝑧|²)|𝑓′(𝑧)|=0. It is well known that ℬ₀ is a closed separable subspace of ℬ. The interested reader is referred to [1] for more information on the Bloch space.

The logarithmic Bloch space ℬ_log is defined as the set of functions 𝑓 on 𝔻 such that‖𝑓‖=sup_𝑧∈𝔻1−|𝑧|²log21−|𝑧|²||𝑓^(𝑧)||<∞.(1.4) It is a Banach space under the norm defined by ‖𝑓‖_ℬlog=|𝑓(0)|+‖𝑓‖. Clearly, if 𝑓∈ℬ_log, then lim_|𝑧|→1(1−|𝑧|²)|𝑓′(𝑧)|=0, so ℬ_log is a subset of the little Bloch space.

The little logarithmic Bloch space, denoted by ℬ_log,0, is defined as the subspace of ℬ_log whose elements 𝑓 satisfy the conditionlim_|𝑧|→11−|𝑧|²log21−|𝑧|||𝑓^(𝑧)||=0.(1.5)

The space ℬ_log arises in connection to the study of certain operators with symbol. Arazy [2] proved that the multiplication operator 𝑀_𝜓 is bounded on the Bloch space if and only if 𝜓∈ℬ_log∩𝐻^∞. In [3], Brown and Shields extended this result to the little Bloch space.

The space ℬ_log also arises in the study of Hankel operators on the Bergman one space. The Bergman space 𝐴¹ on 𝔻 is defined to be the set of analytic functions 𝑓 on 𝔻 whose modulus is Lebesgue integrable over 𝔻.

The Hankel operator 𝐻_𝑓 on 𝐴¹ is defined as 𝐻_𝑓𝑔=(𝐼−𝑃)(𝑓𝑔), where 𝐼 is the identity operator, and 𝑃 is the standard Bergman projection from 𝐿¹ into 𝐴¹. In [4], Attele showed that 𝐻_𝑓 is bounded on 𝐴¹ if and only if 𝑓∈ℬ_log.

The study of operators with symbol on the logarithmic Bloch space began with the characterizations of the bounded and the compact composition operators given in [5] by Yoneda. In [6], Galanopoulos extended these results to the weighted composition operators on ℬ_log. He also introduced a class of Banach spaces 𝑄^𝑝log (𝑝>0) closely related to ℬ_log and studied the Taylor coefficients of the functions in ℬ_log. In [7], Ye characterized the bounded and the compact weighted composition operators on the little logarithmic Bloch space ℬ_log,0. See [8, 9] for the study of the weighted composition operators on Bloch spaces and weighted Bloch spaces.

In this paper, we characterize the bounded and the compact weighted composition operators from the Hardy space 𝐻^𝑝 (with 1≤𝑝≤∞) to the logarithmic Bloch space ℬ_log as well as to its subspace ℬ_log,0. The paper consists of five sections. Specifically, in Section 2, we consider the bounded weighted composition operators mapping 𝐻^∞ into ℬ_log and ℬ_log,0. In particular, we show that‖‖𝑊_𝜓,𝜑‖‖∼sup_{𝑛∈ℕ∪{0}}‖𝜓𝜑^𝑛‖_ℬlog,(1.6) where the notation 𝐴∼𝐵 stands for 𝑐₁𝐴≤𝐵≤𝑐₂𝐴, for some positive constants 𝑐₁ and 𝑐₂. In Section 3, we look at the issue of compactness of such operators.

In Section 4, we characterize the bounded and the compact weighted composition operators mapping 𝐻^𝑝 into ℬ_log in the case when 1≤𝑝<∞. Finally, in Section 5, we study the operators mapping 𝐻^𝑝 into ℬ_log,0.

2. Boundedness of 𝑊_𝜓,𝜑 from 𝐻^∞ into ℬ_log and ℬ_log,0

In the following theorem, we give two characterizations of boundedness when the operator maps 𝐻^∞ into ℬ_log.

Theorem 2.1. Let 𝜓 be an analytic function on 𝔻, and let 𝜑 be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator 𝑊_𝜓,𝜑∶𝐻^∞→ℬ_log is bounded.(b)sup_{𝑛∈ℕ∪{0}}‖𝜓𝜑^𝑛‖_ℬlog<∞.(c)𝜓∈ℬ_log and 𝜎_𝜓,𝜑=sup_𝑧∈𝔻((1−|𝑧|²)|𝜓(𝑧)𝜑′(𝑧)|/1−|𝜑(𝑧)|²)log(2/(1−|𝑧|²))<∞.

Proof. (a) ⇒ (b). For 𝑛∈ℕ, the function 𝑝_𝑛(𝑧)=𝑧^𝑛 is bounded and ‖𝑝_𝑛‖_∞=1. Therefore, if 𝑊_𝜓,𝜑 is bounded, then ‖𝜓𝜑^𝑛‖_ℬlog≤‖𝑊_𝜓,𝜑‖.
(b) ⇒ (c) Let 𝐶 be an upper bound for ‖𝜓𝜑^𝑛‖_ℬlog, 𝑛≥0. Taking 𝑛=0, we deduce that ‖𝜓‖_ℬlog≤𝐶, so 𝜓∈ℬ_log.
For 𝑁∈ℕ and 𝑛≥2, define the sets 𝐸_𝑁=𝑧∈𝔻∶||𝜑(𝑧)||≤1−1Δ_𝑛𝑁,=𝑧∈𝔻∶1−1𝑛−1≤||𝜑(𝑧)||≤1−1𝑛.(2.1)
Fix an integer 𝑁>2, and 𝑧∈𝔻. For 𝑧∈𝐸_𝑁, by the product rule, we have 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²≤1−|𝑧|²||(𝜓𝜑)^(𝑧)||+||𝜓^(𝑧)𝜑(𝑧)||1−(1−1/𝑁)²log21−|𝑧|²≤11−(1−1/𝑁)²≤2𝐶(‖𝜓𝜑‖+‖𝜓‖)1−(1−1/𝑁)².(2.2)
In the proof of Theorem 2 of [10], it was shown that inf_{𝑧∈Δ𝑛}𝑛||𝜑(𝑧)||^𝑛−11−||𝜑(𝑧)||≥1𝑒.(2.3)
For |𝜑(𝑧)|>1−1/𝑁, there exists 𝑛>𝑁 such that 𝑧∈Δ_𝑛. So 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²≤1−|𝑧|²||𝜓(𝑧)𝑛𝜑(𝑧)^𝑛−1𝜑^(𝑧)||1−||𝜑(𝑧)||𝑛||𝜑(𝑧)||^𝑛−1log21−|𝑧|²≤𝑒1−|𝑧|²log21−|𝑧|²+1−|𝑧|₂||(𝜓𝜑^𝑛)^(𝑧)||log21−|𝑧|²≤2𝑒𝐶.||𝜓^(𝑧)𝜑(𝑧)^𝑛||(2.4) From (2.2) and (2.4), we deduce that 𝜎_𝜓,𝜑 is finite.
(c) ⇒ (a) Let 𝑓∈𝐻^∞ with ‖𝑓‖_∞≤1 and pick 𝑧∈𝔻. Then 1−|𝑧|²log21−|𝑧|²||𝜓^(𝑧)𝑓(𝜑(𝑧))||≤‖𝜓‖,(2.5) and, since ‖𝑓‖_𝛽≤‖𝑓‖_∞≤1, 1−|𝑧|²log21−|𝑧|²=1−|𝑧|₂||𝜓(𝑧)𝑓^(𝜑(𝑧))𝜑^(𝑧)||||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²1−||𝜑(𝑧)||²||𝑓^≤1−|𝑧|₂(𝜑(𝑧))||||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|².(2.6) Thus, by (2.5) and (2.6), we deduce that ‖𝑊_𝜓,𝜑𝑓‖_ℬlog≤‖𝜓‖_ℬlog+𝜎_𝜓,𝜑, completing the proof.

We next turn our attention to the weighted composition operators mapping into the little logarithmic Bloch space.

Theorem 2.2. Let 𝜓 be an analytic function on 𝔻, and let 𝜑 be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator 𝑊_𝜓,𝜑∶𝐻^∞→ℬ_log,0 is bounded.(b)For each integer 𝑛≥0,𝜓𝜑^𝑛∈ℬ_log,0 and sup_𝑛∈ℕ‖𝜓𝜑^𝑛‖_ℬlog<∞.(c)𝜓∈ℬ_log,0 and lim_|𝑧|→1((1−|𝑧|²)|𝜓(𝑧)𝜑^(𝑧)|/(1−|𝜑(𝑧)|²))log(2/(1−|𝑧|²))=0.

Proof. (a) ⇒ (b) is proved as in the case of the operator mapping into ℬ_log.
(b) ⇒ (c) Suppose that (b) holds. If 𝐸_𝑁=𝔻 for some integer 𝑁>1, then for all 𝑧∈𝔻, we have 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²≤11−(1−1/𝑁)²1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||log21−|𝑧|²≤11−(1−1/𝑁)²1−|𝑧|²||(𝜓𝜑)^(𝑧)||+||𝜓^(𝑧)𝜑(𝑧)||log21−|𝑧|²≤11−(1−1/𝑁)²1−|𝑧|²||(𝜓𝜑)^(𝑧)||log21−|𝑧|²+11−(1−1/𝑁)²1−|𝑧|²||𝜓^(𝑧)||log21−|𝑧|²⟶0,as|𝑧|⟶1.(2.7) If 𝐸_𝑁 is properly contained in 𝔻, then for 𝑧∈𝔻⧵𝐸_𝑁, arguing as in the proof of (b) ⇒ (c) in Theorem 2.1, there exists 𝑘≥𝑁 such that 𝑧∈Δ_𝑘, so that 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²≤1−|𝑧|²||𝜓(𝑧)𝑘𝜑(𝑧)^𝑘−1𝜑^(𝑧)||1−||𝜑(𝑧)||𝑘||𝜑(𝑧)||^𝑘−1log21−|𝑧|²≤𝑒1−|𝑧|²log21−|𝑧|²|||𝜓𝜑_𝑘+1−|𝑧|₂^(𝑧)|||log21−|𝑧|²=𝐼+𝐼𝐼,||𝜓^(𝑧)||(2.8) where 𝐼=𝑒(1−|𝑧|²)log(2/(1−|𝑧|²))|(𝜓𝜑^𝑘)′(𝑧)| and 𝐼𝐼=𝑒(1−|𝑧|²)log(2/(1−|𝑧|²))|𝜓′(𝑧)|.
By the assumption of 𝜓𝜑^𝑛∈ℬ_log,0, we have 𝐼≤𝑒sup_𝑛≥𝑘1−|𝑧|²log21−|𝑧|²||(𝜓𝜑^𝑛)^(𝑧)||⟶0(2.9) as |𝑧|→1. On the other hand, since 𝜓∈ℬ_log,0, 𝐼𝐼→0 as |𝑧|→1. Therefore, lim_|𝑧|→11−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²=0.(2.10)
(c) ⇒ (a) Assume that (c) holds. To prove that 𝑊_𝜓,𝜑 is bounded, it suffices to show that 𝑊_𝜓,𝜑𝑓∈ℬ_log,0 for each 𝑓∈𝐻^∞, since the boundedness of the operator can be shown as in the proof of Theorem 2.1. Since 𝜓∈ℬ_log,0, for 𝑓∈𝐻^∞ and 𝑧∈𝔻, we have 1−|𝑧|²||𝜓^(𝑧)𝑓(𝜑(𝑧))||log21−|𝑧|²≤1−|𝑧|²||𝜓^(𝑧)||log21−|𝑧|²‖𝑓‖_∞⟶0,(2.11) as |𝑧|→1. On the other hand, by (2.6), 1−|𝑧|²||𝜓(𝑧)𝑓^(𝜑(𝑧))𝜑^(𝑧)||log21−|𝑧|²≤1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²‖𝑓‖_∞⟶0,(2.12) as |𝑧|→1. Hence, 1−|𝑧|²|||𝑊_𝜓,𝜑𝑓^(𝑧)|||log21−|𝑧|²⟶0(2.13) as |𝑧|→1, completing the proof.

In Section 3, we shall prove that all bounded weighted composition operators from 𝐻^∞ into ℬ_log,0 are compact.

3. Compactness of 𝑊_𝜓,𝜑 from 𝐻^∞ into ℬ_log and ℬ_log,0

The following criterion for compactness follows by a standard argument similar, for example, to that outlined in Proposition 3.11 of [11].

Lemma 3.1. Let 𝜓 be analytic on 𝔻, 𝜑 an analytic self-map of 𝔻, 1≤𝑝≤∞. The operator 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is compact if and only if for any bounded sequence {𝑓_𝑛}_𝑛∈ℕ in 𝐻^𝑝 which converges to zero uniformly on compact subsets of 𝔻, we have ‖𝑊_𝜓,𝜑𝑓_𝑛‖_ℬlog→0 as 𝑛→∞.

The proof of the following result is similar to the proof of Lemma 1 of [12]. Hence we omit it.

Lemma 3.2. A closed set K in ℬ_log,0 is compact if and only if it is bounded and satisfies the following: lim_|𝑧|→1sup_𝑓∈𝐾1−|𝑧|²||𝑓^(𝑧)||log21−|𝑧|²=0.(3.1)

We now introduce two one-parameter families of functions which will be used to characterize the compactness of the operators under consideration.

Fix 𝑎∈𝔻 and, for 𝑧∈𝔻, define𝑓_𝑎(𝑧)=1−|𝑎|²1−𝑎𝑧^1/2,𝑔_𝑎(𝑧)=1−|𝑎|²1−𝑎𝑧.(3.2)

Theorem 3.3. Let 𝜓 be analytic on 𝔻, 𝜑 an analytic self-map of 𝔻, and assume that 𝑊_𝜓,𝜑∶𝐻^∞→ℬ_log is bounded. Then the following conditions are equivalent: (a)𝑊_𝜓,𝜑∶𝐻^∞→ℬ_log is compact.(b)lim_{|𝜑(𝑤)|→1}‖𝑊_𝜓,𝜑𝑓_𝜑(𝑤)‖_ℬlog=0 and lim_{|𝜑(𝑤)|→1}‖𝑊_𝜓,𝜑𝑔_𝜑(𝑤)‖_ℬlog=0.(c)lim_{|𝜑(𝑤)|→1}((1−|𝑤|²)|𝜓(𝑤)𝜑′(𝑤)|/1−|𝜑(𝑤)|²)log(2/(1−|𝑤|²))=0 and lim_{|𝜑(𝑤)|→1}(1−|𝑤|²)|𝜓′(𝑤)|log(2/(1−|𝑤|²))=0.(d)lim_𝑛→∞‖𝜓𝜑^𝑛‖_ℬlog=0 and lim_{|𝜑(𝑤)|→1}(1−|𝑤|²)|𝜓^(𝑤)|log(2/(1−|𝑤|²))=0.

Proof. We begin by showing that (a), (b), and (c) are equivalent.
(a)⇒(b) Suppose that 𝑊_𝜓,𝜑 is compact and that {𝑤_𝑛} is a sequence in 𝔻 such that |𝜑(𝑤_𝑛)|→1 as 𝑛→∞. Since the sequences {𝑓_{𝜑(𝑤𝑛)}} and {𝑔_{𝜑(𝑤𝑛)}} are bounded in 𝐻^∞ and converge to 0 uniformly on compact subsets of 𝔻, by Lemma 3.1, it follows that ‖𝑊_𝜓,𝜑𝑓_{𝜑(𝑤𝑛)}‖_ℬlog→0 and ‖𝑊_𝜓,𝜑𝑔_{𝜑(𝑤𝑛)}‖_ℬlog→0 as 𝑛→∞.
(b)⇒(c) Assume that (b) holds. Fix 𝑤∈𝔻. A straightforward calculation shows that 𝜓𝑓_𝜑(𝑤)∘𝜑^(𝑤)=𝜓^(𝑤)+𝜓(𝑤)𝜑(𝑤)𝜑^(𝑤)𝜓𝑔_𝜑(𝑤)21−||𝜑(𝑤)||²,∘𝜑^(𝑤)=𝜓^(𝑤)+𝜓(𝑤)𝜑(𝑤)𝜑^(𝑤)1−||𝜑(𝑤)||².(3.3) Eliminating 𝜓′(𝑤), we obtain that 𝜓(𝑤)𝜑(𝑤)𝜑^(𝑤)21−||𝜑(𝑤)||²=𝜓𝑔^𝜑(𝑤)∘𝜑^(𝑤)−𝜓𝑓_𝜑(𝑤)∘𝜑^(𝑤).(3.4) Thus, for |𝜑(𝑤)|>𝑟∈(0,1), 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²log21−|𝑤|²≤2𝑟‖‖𝑊^𝜓,𝜑𝑓_𝜑(𝑤)‖‖_ℬlog+‖‖𝑊_𝜓,𝜑𝑔_𝜑(𝑤)‖‖_ℬlog.(3.5) Taking the limit as |𝜑(𝑤)|→1, we deduce that 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²log21−|𝑤|²⟶0.(3.6) On the other hand, using (3.3), we obtain that 1−|𝑤|²||𝜓^(𝑤)||log21−|𝑤|²≤‖‖𝑊_𝜓,𝜑𝑔_𝜑(𝑤)‖‖_ℬlog+1−|𝑤|²||𝜓(𝑤)𝜑(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²log21−|𝑤|².(3.7) Taking the limit as |𝜑(𝑤)|→1, we obtain that 1−|𝑤|²||𝜓^(𝑤)||log21−|𝑤|²⟶0,(3.8) proving (c).
(c)⇒(a) Suppose that (c) holds. Let {𝑓_𝑛} be a bounded sequence in 𝐻^∞ converging to 0 uniformly on compact subsets of 𝔻. Set 𝐶=sup_𝑛∈ℕ‖𝑓_𝑛‖_∞. Then, given 𝜀>0, there exists 𝑟∈(0,1) such that for |𝜑(𝑤)|>𝑟, 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²log21−|𝑤|²<𝜀1−|𝑤|₂2𝐶,||𝜓^(𝑤)||log21−|𝑤|²<𝜀2𝐶.(3.9) Then, for 𝑤∈𝔻, noting that (1−|𝜑(𝑤)|²)|𝑓^𝑛(𝜑(𝑤))|≤‖𝑓_𝑛‖_∞≤𝐶, we obtain that 1−|𝑤|²|||𝜓𝑓_𝑛∘𝜑^(𝑤)|||log21−|𝑤|²≤1−|𝑤|²||𝜓^(𝑤)𝑓_𝑛(𝜑(𝑤))||+||𝜓(𝑤)𝑓^𝑛(𝜑(𝑤))𝜑^(𝑤)||log21−|𝑤|²≤1−|𝑤|²||𝜓^(𝑤)||‖‖𝑓_𝑛‖‖_∞log21−|𝑤|²+1−|𝑤|²1−||𝜑(𝑤)||²||𝜓(𝑤)𝑓^𝑛(𝜑(𝑤))𝜑^(𝑤)||1−||𝜑(𝑤)||²log21−|𝑤|²≤𝐶1−|𝑤|²||𝜓^(𝑤)||+1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²log1−2|𝑤|².(3.10) Thus, for |𝜑(𝑤)|>𝑟, we have 1−|𝑤|²|||𝜓𝑓_𝑛∘𝜑^(𝑤)|||log21−|𝑤|²<𝜀.(3.11) On the other hand, for |𝜑(𝑤)|≤𝑟, 1−|𝑤|²|||𝜓𝑓_𝑛∘𝜑^(𝑤)|||log21−|𝑤|²≤‖𝜓‖||𝑓_𝑛(𝜑(𝑤))||+𝜎_𝜓,𝜑||𝑓^𝑛(𝜑(𝑤))||.(3.12) Thus, by the uniform convergence to 0 of 𝑓_𝑛 and 𝑓^𝑛 on compact sets, we see that (3.11) holds also in this case for 𝑛 sufficiently large. Hence, ‖𝜓(𝑓_𝑛∘𝜑)‖≤𝜀 for 𝑛 sufficiently large. Since |𝜓(0)𝑓_𝑛(𝜑(0))|→0, we deduce that ‖𝜓(𝑓_𝑛∘𝜑)‖_ℬlog→0 as 𝑛→∞. Therefore, 𝑊_𝜓,𝜑 is compact.
(a) ⇒ (d) Suppose that 𝑊_𝜓,𝜑 is compact. Since the sequence {𝑝_𝑛} defined by 𝑝_𝑛(𝑧)=𝑧^𝑛, 𝑧∈𝔻 is bounded in 𝐻^∞ and converges to 0 uniformly on compact subsets, by Lemma 3.1, it follows that ‖𝜓𝜑^𝑛‖_ℬlog=‖𝑊_𝜓,𝜑𝑝_𝑛‖_ℬlog→0 as 𝑛→∞. The second condition in (d) follows from the equivalence of (a) and (c).
(d) ⇒ (c) Fix 𝜀>0 and choose 𝑁>2 such that ‖𝜓𝜑^𝑛‖_ℬlog<𝜀/2 for all 𝑛≥𝑁 and (1−|𝑤|²)|𝜓′(𝑤)|log(2/(1−𝑤|²))<𝜀/2 for |𝜑(𝑤)|>1−1/𝑁.
For |𝜑(𝑤)|>1−1/𝑁, there exists 𝑛>𝑁 such that 𝑧∈Δ_𝑛. Using the product rule, we may write the following 𝜓(𝑤)𝑛𝜑(𝑤)^𝑛−1𝜑^(𝑤)=(𝜓𝜑^𝑛)^(𝑤)−𝜓^(𝑤)𝜑(𝑤)^𝑛,(3.13) so that 1−|𝑤|²||𝜓(𝑤)𝑛𝜑(𝑤)^𝑛−1𝜑^(𝑤)||log21−|𝑤|²≤‖𝜓𝜑^𝑛‖_ℬlog+1−|𝑤|²||𝜓^(𝑤)𝜑(𝑤)^𝑛||log21−|𝑤|².(3.14) The left-hand side of (3.14) can be written as 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²1−||𝜑(𝑤)||²𝑛||𝜑(𝑤)||^𝑛−1log1−2|𝑤|²,(3.15) which, by (2.3), is bounded below by 1𝑒1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²log21−|𝑤|².(3.16) Thus, from (3.14), we deduce that 1𝑒1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²log21−|𝑤|²≤‖𝜓𝜑^𝑛‖_ℬlog+1−|𝑤|²||𝜓^(𝑤)||log21−|𝑤|²<𝜀,(3.17) proving (c). The equivalence of statements (a)–(d) is now established.

Next, we characterize the compact weighted composition operators from 𝐻^∞ into ℬ_log,0.

Theorem 3.4. Let 𝜓 be an analytic function on 𝔻, and let 𝜑 be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator 𝑊_𝜓,𝜑∶𝐻^∞→ℬ_log,0 is compact.(b)For each integer 𝑛≥0, 𝜓𝜑^𝑛∈ℬ_log,0 and lim_𝑛→∞‖𝜓𝜑^𝑛‖_ℬlog=0.(c)𝜓∈ℬ_log,0 andlim_|𝑧|→11−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²=0.(3.18)

Proof. (a) ⇒ (b) is immediate.
(b) ⇒ (c) It suffices to show that if (b) holds, then 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²⟶0(3.19) as |𝑧|→1. Fix 𝜀>0 and let 𝑁>2 be an integer such that ‖𝜓𝜑^𝑛‖_ℬlog<𝜀/𝑒 for all 𝑛≥𝑁. Observe that, since for 𝑧∈𝔻, ||𝜓(𝑧)𝜑^(𝑧)||≤||(𝜓𝜑)^(𝑧)||+||𝜓^(𝑧)𝜑(𝑧)||≤||(𝜓𝜑)^(𝑧)||+||𝜓^(𝑧)||,(3.20) and, by assumption, the functions 𝜓𝜑 and 𝜓 are in ℬ_log,0, we have lim_|𝑧|→11−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||log21−|𝑧|²=0.(3.21) Therefore, there is 𝑟∈(0,1), such that for |𝑧|>𝑟, 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||log21−|𝑧|²<(2𝑁−1)𝜀𝑁².(3.22) Thus, if 𝑧∈𝐸_𝑁, and |𝑧|>𝑟, then 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²≤𝑁²2𝑁−11−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||log21−|𝑧|²<𝜀.(3.23) On the other hand, if 𝑧∉𝐸_𝑁, then there exists 𝑛>𝑁 such that 𝑧∈Δ_𝑛, so, as shown in the proof of (d) implies (c) of Theorem 3.3, we have 1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²≤𝑒‖𝜓𝜑^𝑛‖_ℬlog+1−|𝑧|²||𝜓^(𝑧)||log2<𝜀+𝑒1−|𝑧|₂1−|𝑧|²||𝜓^(𝑧)||log21−|𝑧|²⟶𝜀,(3.24) as |𝑧|→1. Since 𝜀 is arbitrary, the result follows.
(c) ⇒ (a) Let {𝑓_𝑛} be a bounded sequence in 𝐻^∞ converging to 0 uniformly on compact subsets, and let 𝐶=sup_𝑛∈ℕ‖𝑓_𝑛‖_∞. We wish to show that 𝑊_𝜓,𝜑𝑓_𝑛∈ℬ_log,0 and ‖𝑊_𝜓,𝜑𝑓_𝑛‖_ℬlog→0 as 𝑛→∞. As shown in the proof of (c) implies (a) of Theorem 3.3, for 𝑧∈𝔻 and 𝑛∈ℕ, 1−|𝑧|²|||𝑊_𝜓,𝜑𝑓_𝑛^(𝑧)|||log21−|𝑧|²≤𝐶1−|𝑧|²||𝜓^(𝑧)||log21−|𝑧|²+𝐶1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²log21−|𝑧|²⟶0,(3.25) as |𝑧|→1. Thus, 𝑊_𝜓,𝜑𝑓_𝑛∈ℬ_log,0. The convergence to 0 of ‖𝑊_𝜓,𝜑𝑓_𝑛‖_ℬlog is proved as in the case of the operator mapping into ℬ_log.

From Theorems 2.2 and 3.4, we obtain the following result.

Corollary 3.5. Let 𝜓 be an analytic function on 𝔻, and let 𝜑 be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator 𝑊_𝜓,𝜑∶𝐻^∞→ℬ_log,0 is bounded.(b)The operator 𝑊_𝜓,𝜑∶𝐻^∞→ℬ_log,0 is compact.(c)For each integer 𝑛≥0, 𝜓𝜑^𝑛∈ℬ_log,0 and lim_𝑛→∞‖𝜓𝜑^𝑛‖_ℬlog=0.(d)𝜓∈ℬ_log,0 and lim_|𝑧|→1((1−|𝑧|²)|𝜓(𝑧)𝜑′(𝑧)|/(1−|𝜑(𝑧)|²))log(2/(1−|𝑧|²))=0.

In the special cases when 𝜑 is the identity, respectively, 𝜓 is identically 1, we obtain the following results.

Corollary 3.6. Let 𝜓 be analytic on 𝔻. The following statements are equivalent: (a)𝑀_𝜓∶𝐻^∞→ℬ_log is bounded,(b)𝑀_𝜓∶𝐻^∞→ℬ_log,0 is bounded,(c)𝜓 is identically 0.

Corollary 3.7. Let 𝜑 be an analytic self map of 𝔻. Then the following statements are equivalent: (a)𝐶_𝜑∶𝐻^∞→ℬ_log is bounded,(b)sup_𝑛∈ℕ‖𝜑^𝑛‖_ℬlog<∞,(c)sup_𝑧∈𝔻((1−|𝑧|²)|𝜑′(𝑧)|/(1−|𝜑(𝑧)|²))log(2/(1−|𝑧|²))<∞.

Corollary 3.8. Let 𝜑 be an analytic self map of 𝔻. Then the following statements are equivalent: (a)𝐶_𝜑∶𝐻^∞→ℬ_log is compact,(b)lim_𝑛→∞‖𝜑^𝑛‖_ℬlog=0,(c)lim_{|𝜑(𝑧)|→1}((1−|𝑧|²)|𝜑′(𝑧)|/(1−|𝜑(𝑧)|²))log(2/(1−|𝑧|²))=0.

Corollary 3.9. Let 𝜑 be an analytic self map of 𝔻. Then the following statements are equivalent: (a)𝐶_𝜑∶𝐻^∞→ℬ_log,0 is bounded,(b)𝐶_𝜑∶𝐻^∞→ℬ_log,0 is compact,(c)𝜑∈ℬ_log,0 and lim_𝑛→∞‖𝜑^𝑛‖_ℬlog=0,(d)lim_|𝑧|→1((1−|𝑧|²)|𝜑′(𝑧)|/(1−|𝜑(𝑧)|²))log(2/(1−|𝑧|²))=0.

4. 𝑊_𝜓,𝜑 from 𝐻^𝑝(1≤𝑝<∞) into ℬ_log

We begin this section with two useful point evaluation estimates that will be needed to prove our results.

Lemma 4.1 (See [11]). Let 0<𝑝<∞. Then for any 𝑓∈𝐻^𝑝, 𝑧∈𝔻, ||𝑓(𝑧)||≤‖𝑓‖^𝐻𝑝1−|𝑧|²^1/𝑝.(4.1)

Lemma 4.2 (See [13]). Let 0<𝑝<∞. Then for any 𝑓∈𝐻^𝑝, 𝑧∈𝔻, ||𝑓^(𝑧)||≤𝐶‖𝑓‖^𝐻𝑝1−|𝑧|²^1+1/𝑝.(4.2)

Fix 1≤𝑝<∞ and 𝑎∈𝔻. For 𝑧∈𝔻, define the functions𝑓_𝑎(𝑧)=1−|𝑎|²^2−1/𝑝1−𝑎𝑧²,𝑔_𝑎(𝑧)=1−|𝑎|²²1−𝑎𝑧^2+1/𝑝.(4.3) Then 𝑓_𝑎,𝑔_𝑎∈𝐻^𝑝 and the norms ‖𝑓_𝑎‖_𝐻^𝑝 and ‖𝑔_𝑎‖_𝐻^𝑝 are bounded by constants only dependent of 𝑝. In addition, a straightforward calculation shows that𝑓_𝑎(𝑎)=𝑔_𝑎(𝑎)=11−|𝑎|²^1/𝑝𝑓_𝑎,(𝑎)=2𝑎1−|𝑎|²^1+1/𝑝𝑔_𝑎,(𝑎)=(2+1/𝑝)𝑎1−|𝑎|²^1+1/𝑝.(4.4)

We use these two families of functions to characterize the bounded and the compact weighted composition operators from 𝐻^𝑝 to ℬ_log.

Theorem 4.3. Let 1≤𝑝<∞, 𝜓 analytic on 𝔻 and let 𝜑 be an analytic self-map of 𝔻. Then the following conditions are equivalent: (a)𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is bounded,(b)𝜓𝜑∈ℬ_log, 𝐴=sup_𝑤∈𝔻‖𝑊_𝜓,𝜑𝑓_𝜑(𝑤)‖_ℬlog<∞ and 𝐵=sup_𝑤∈𝔻‖𝑊_𝜓,𝜑𝑔_𝜑(𝑤)‖_ℬlog<∞,(c)𝑥_𝜓,𝜑=sup_𝑧∈𝔻1−|𝑧|²||𝜓^(𝑧)||1−||𝜑(𝑧)||²^1/𝑝log21−|𝑧|²𝑦<∞,_𝜓,𝜑=sup_𝑧∈𝔻1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²<∞.(4.5)

Proof. (a) ⇒ (b) Assume that 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is bounded. Then 𝜓𝜑∈ℬ_log and for each 𝑤∈𝔻, ‖‖‖‖𝑊𝑊_{𝜓,𝜑𝜓,𝜑}𝑓_𝜑(𝑤)‖‖_ℬlog≤‖‖𝑊_𝜓,𝜑‖‖‖‖𝑓_𝜑(𝑤)‖‖_𝐻𝑝≤𝐶‖‖𝑊_𝜓,𝜑‖‖,𝑔_𝜑(𝑤)‖‖_ℬlog≤‖‖𝑊_𝜓,𝜑‖‖‖‖𝑔_𝜑(𝑤)‖‖_𝐻𝑝≤𝐶‖‖𝑊_𝜓,𝜑‖‖,(4.6) for some constant 𝐶, so 𝐴 and 𝐵 are finite.
(b) ⇒ (c) Suppose that 𝜓𝜑∈ℬ_log, and the quantities 𝐴 and 𝐵 are finite. From (4.4), for 𝑤∈𝔻, we have 𝜓𝑓_𝜑(𝑤)∘𝜑^(𝑤)=𝜓^(𝑤)1−||𝜑(𝑤)||²^1/𝑝+2𝜓(𝑤)𝜑^(𝑤)𝜑(𝑤)1−||𝜑(𝑤)||²^1+1/𝑝,(4.7) whence 1−|𝑤|²||𝜓^(𝑤)||1−||𝜑(𝑤)||²^1/𝑝log21−|𝑤|²≤1−|𝑤|²log21−|𝑤|²|||𝜓𝑓_𝜑(𝑤)∘𝜑^(𝑤)|||+21−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)𝜑(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²≤‖‖𝑊_𝜓,𝜑𝑓_𝜑(𝑤)‖‖_ℬlog+21−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)𝜑(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²≤𝐴+21−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)𝜑(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|².(4.8) Moreover, 𝜓𝑔_𝜑(𝑤)∘𝜑^(𝑤)=𝜓^(𝑤)1−||𝜑(𝑤)||²^1/𝑝+(2+1/𝑝)𝜓(𝑤)𝜑^(𝑤)𝜑(𝑤)1−||𝜑(𝑤)||²^1+1/𝑝.(4.9) Therefore, subtracting (4.7) from (4.9) and taking the modulus, we obtain 1𝑝||𝜓(𝑤)𝜑^(𝑤)𝜑(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝≤|||𝜓𝑓_𝜑(𝑤)∘𝜑^(𝑤)|||+|||𝜓𝑔_𝜑(𝑤)∘𝜑^(𝑤)|||,(4.10) which yields that 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)𝜑(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²≤𝑝(𝐴+𝐵).(4.11) Consequently, from (4.8), we deduce that 1−|𝑤|²||𝜓^(𝑤)||1−||𝜑(𝑤)||²^1/𝑝log21−|𝑤|²≤(1+2𝑝)𝐴+2𝑝𝐵.(4.12) Taking the supremum over all 𝑤∈𝔻, we see that 𝑥_𝜓,𝜑 is finite.
Fix 𝑟∈(0,1). If |𝜑(𝑤)|>𝑟, then from (4.11) we have 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²<𝑝(𝐴+𝐵)𝑟.(4.13) On the other hand, since 𝜓𝜑∈ℬ_log, if |𝜑(𝑤)|≤𝑟, then, 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²≤1−|𝑤|²||(𝜓𝜑)^(𝑤)||1−𝑟²^1+1/𝑝log21−|𝑤|²+1−|𝑤|²||𝜓^(𝑤)𝜑(𝑤)||1−𝑟²1−||𝜑(𝑤)||²^1/𝑝log21−|𝑤|²≤‖𝜓𝜑‖1−𝑟²^1+1/𝑝+𝑥^𝜓,𝜑1−𝑟²<∞.(4.14) Taking the supremum over all 𝑤∈𝔻, it follows that 𝑦_𝜓,𝜑 is finite as well.
(c) ⇒ (a) Suppose that 𝑥_𝜓,𝜑 and 𝑦_𝜓,𝜑 are finite. For arbitrary 𝑧 in 𝔻 and 𝑓∈𝐻^𝑝, by Lemmas 4.1 and 4.2, we have 1−|𝑧|²log21−|𝑧|²|||𝑊_𝜓,𝜑≤1−|𝑧|₂𝑓^(𝑧)|||log21−|𝑧|²≤⎛⎜⎜⎜⎝1−|𝑧|₂||𝜓^(𝑧)||||𝑓(𝜑(𝑧))||+||𝑓^(𝜑(𝑧))||||𝜓(𝑧)𝜑^(𝑧)||||𝜓^(𝑧)||1−||𝜑(𝑧)||²^1/𝑝log21−|𝑧|²+𝐶1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²⎞⎟⎟⎟⎠‖𝑓‖_𝐻𝑝≤𝑥_𝜓,𝜑+𝐶𝑦_𝜓,𝜑‖𝑓‖_𝐻^𝑝.(4.15)
Taking the supremum over all 𝑧∈𝔻 and applying Lemma 4.1, we obtain that ‖‖𝑊_𝜓,𝜑𝑓‖‖_ℬlog≤=||⎛⎜⎜⎜⎝𝜓(0)𝑓||𝜓(𝜑(0()0||))||+‖‖𝑊_𝜓,𝜑𝑓‖‖1−||𝜑(0)||²^1/𝑝+𝑥_𝜓,𝜑+𝐶𝑦_𝜓,𝜑⎞⎟⎟⎟⎠‖𝑓‖_𝐻𝑝.(4.16) The boundedness of the operator 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log follows by taking the supremum over all 𝑓∈𝐻^𝑝.

Theorem 4.4. Let 1≤𝑝<∞, 𝜓 analytic on 𝔻, 𝜑 an analytic self-map of 𝔻, and assume that 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is bounded. Then the following conditions are equivalent: (a)𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is compact,(b)lim_{|𝜑(𝑤)|→1}‖𝑊_𝜓,𝜑𝑓_𝜑(𝑤)‖_ℬlog=0 and lim_{|𝜑(𝑤)|→1}‖𝑊_𝜓,𝜑𝑔_𝜑(𝑤)‖_ℬlog=0,(c)lim_{||||𝜑(𝑤)→1}1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²lim_{||||𝜑(𝑤)→1}=0,1−|𝑤|²||𝜓^(𝑤)||1−||𝜑(𝑤)||²^1/𝑝log21−|𝑤|²=0.(4.17)

Proof. (a)⇒(b) Suppose that 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is compact. Let {𝑤_𝑛} be a sequence in 𝔻 such that lim_𝑛→∞|𝜑(𝑤_𝑛)|=1. Observe that the sequences {𝑓_{𝜑(𝑤𝑛)}} and {𝑔_{𝜑(𝑤𝑛)}} are bounded in 𝐻^𝑝 and converge to 0 uniformly on compact subsets of 𝔻. By Lemma 3.1, it follows that ‖𝑊_𝜓,𝜑𝑓_{𝜑(𝑤𝑛)}‖_ℬlog→0 and ‖𝑊_𝜓,𝜑𝑔_{𝜑(𝑤𝑛)}‖_ℬlog→0 as 𝑛→∞, proving (b).
(b)⇒(c) Assume that the limits in (b) are 0. Using the inequality (4.10), we obtain that 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²≤𝑝‖‖𝑊^𝜓,𝜑𝑓_𝜑(𝑤)‖‖_ℬlog+‖‖𝑊_𝜓,𝜑𝑔_𝜑(𝑤)‖‖_ℬlog||𝜑(𝑤)||⟶0(4.18) as |𝜑(𝑤)|→1. Moreover, using (4.8), we deduce that 1−|𝑤|²||𝜓^(𝑤)||1−||𝜑(𝑤)||²^1/𝑝log21−|𝑤|²⟶0(4.19) as |𝜑(𝑤)|→1.
(c)⇒(a) Suppose that (c) holds. Let {𝑓_𝑛} be a bounded sequence in 𝐻^𝑝 converging to 0 uniformly on compact subsets of 𝔻. Set 𝐶=sup_𝑛∈ℕ‖𝑓_𝑛‖_𝐻^𝑝. Then, given 𝜀>0, there exists 𝑟∈(0,1) such that 1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²<𝜀1−|𝑤|₂2𝐶,||𝜓^(𝑤)||1−||𝜑(𝑤)||²^1/𝑝log21−|𝑤|²<𝜀2𝐶,(4.20) for |𝜑(𝑤)|>𝑟. Therefore, again by Lemmas 4.1 and 4.2, and (4.20), for |𝜑(𝑤)|>𝑟, we have 1−|𝑤|²log21−|𝑤|²|||𝜓𝑓_𝑛≤1−|𝑤|₂∘𝜑^(𝑤)|||log21−|𝑤|²≤‖‖𝑓||𝜓^(𝑤)𝑓_𝑛(𝜑(𝑤))||+||𝜓(𝑤)𝑓^𝑛(𝜑(𝑤))𝜑^(𝑤)_𝑛||‖‖_𝐻𝑝⎛⎜⎜⎜⎝1−|𝑤|₂||𝜓^(𝑤)||1−||𝜑(𝑤)||²^1/𝑝log21−|𝑤|²+1−|𝑤|²||𝜓(𝑤)𝜑^(𝑤)||1−||𝜑(𝑤)||²^1+1/𝑝log21−|𝑤|²<𝜀.⎞⎟⎟⎟⎠(4.21) On the other hand, for |𝜑(𝑤)|≤𝑟, by the uniform convergence to 0 of 𝑓_𝑛 and 𝑓^𝑛 on compact sets, we have 1−|𝑤|²log21−|𝑤|²|||𝜓𝑓_𝑛∘𝜑^(𝑤)|||≤𝑥_𝜓,𝜑||𝑓_𝑛(𝜑(𝑤))||+𝑦_𝜓,𝜑||𝑓^(𝜑(𝑤))||⟶0,(4.22) as 𝑛→∞. Since |𝜓(0)𝑓_𝑛(𝜑(0))|→0, we conclude that ‖𝑊_𝜓,𝜑𝑓_𝑛‖_ℬlog→0 as 𝑛→∞. Consequently, by Lemma 3.1, the operator 𝑊_𝜓,𝜑 is compact.

As a consequence of Theorems 4.3 and 4.4, noting that for 𝑤∈𝔻,𝐶_𝜑𝑔_𝜑(𝑤)^(𝑤)=212+1𝑝𝐶^𝜑𝑓_𝜑(𝑤)^(𝑤)=(2+1/𝑝)𝜑^(𝑤)𝜑(𝑤)1−||𝜑(𝑤)||²^1+1/𝑝,(4.23) we obtain the following characterizations of the bounded and the compact composition operators from 𝐻^𝑝 into ℬ_log.

Corollary 4.5. Let 𝜑 be an analytic self-map of 𝔻, and 1≤𝑝<∞. The following statements are equivalent: (a)𝐶_𝜑∶𝐻^𝑝→ℬ_log is bounded,(b)sup_𝑤∈𝔻‖𝐶_𝜑𝑓_𝜑(𝑤)‖_ℬlog<∞,(c)sup_𝑤∈𝔻‖𝐶_𝜑𝑔_𝜑(𝑤)‖_ℬlog<∞,(d)sup_𝑤∈𝔻((1−|𝑤|²)|𝜑′(𝑤)|/(1−|𝜑(𝑤)|²)^1+1/𝑝)log(2/(1−|𝑤|²))<∞.

Corollary 4.6. Let 𝜑 be an analytic self-map of 𝔻, and 1≤𝑝<∞. If 𝐶_𝜑∶𝐻^𝑝→ℬ_log is bounded, then the following statements are equivalent: (a)𝐶_𝜑∶𝐻^𝑝→ℬ_log is compact,(b)lim_{|𝜑(𝑤)|→1}‖𝐶_𝜑𝑓_𝜑(𝑤)‖_ℬlog=0,(c)lim_{|𝜑(𝑤)|→1}‖𝐶_𝜑𝑔_𝜑(𝑤)‖_ℬlog=0,(d)lim_{|𝜑(𝑤)|→1}((1−|𝑤|²)|𝜑′(𝑤)|/(1−|𝜑(𝑤)|²)^1+1/𝑝)log(2/(1−|𝑤|²))=0.

5. 𝑊_𝜓,𝜑 from 𝐻^𝑝(1≤𝑝<∞) into ℬ_log,0

In this section, we characterize the boundedness and the compactness of the weighted composition operators 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log,0. Arguing as in the proof of Lemma 4.2 of [14], we easily get the following two lemmas.

Lemma 5.1. Suppose that 𝜑 is an analytic self-map of the unit disk, 𝜓 analytic on 𝔻, and 1≤𝑝<∞. Then, lim_|𝑧|→11−|𝑧|²||𝜓^(𝑧)||1−||𝜑(𝑧)||²^1/𝑝log21−|𝑧|²=0(5.1) if and only if 𝜓∈ℬ_log,0 and lim_{||||𝜑(𝑧)→1}1−|𝑧|²||𝜓^(𝑧)||1−||𝜑(𝑧)||²^1/𝑝log21−|𝑧|²=0.(5.2)

Lemma 5.2. Suppose that 𝜑 is an analytic self-map of the unit disk, 𝜓 analytic on 𝔻, and 1≤𝑝<∞. Then, lim_|𝑧|→11−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²=0(5.3) if and only if lim_|𝑧|→1(1−|𝑧|)²log(2/(1−|𝑧|²))|𝜓(𝑧)𝜑′(𝑧)|=0 and lim_{||||𝜑(𝑧)→1}1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²=0.(5.4)

The proof of the following theorem is a straightforward adaptation of the proof of Theorem  4.4 in [14]. We omit the details.

Theorem 5.3. Let 𝜑 be an analytic self-map of the unit disk, 𝜓∈𝐻(𝔻), and 1≤𝑝<∞. Then, the operator 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log,0 is bounded if and only if 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is bounded, 𝜓∈ℬ_log,0, and lim_|𝑧|→11−|𝑧|²log21−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||=0.(5.5)

We are now ready to prove the main result of this section.

Theorem 5.4. Suppose that 𝜑 is an analytic self-map of the unit disk, 𝜓∈𝐻(𝔻), and 1≤𝑝<∞. Then, 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log,0 is compact if and only if lim_|𝑧|→11−|𝑧|²||𝜓^(𝑧)||1−||𝜑(𝑧)||²^1/𝑝log21−|𝑧|²lim_|𝑧|→1=0,1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²=0.(5.6)

Proof. First, we assume that 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log,0 is compact, and hence bounded. By Theorem 5.3, it follows that 𝜓∈ℬ_log,0 and lim_|𝑧|→1(1−|𝑧|)²log21−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||=0.(5.7) Since by assumption 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log is compact, from Theorem 4.4, we know that lim_{||||𝜑(𝑧)→1}1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²lim_{||||𝜑(𝑧)→1}=0,1−|𝑧|²||𝜓^(𝑧)||1−||𝜑(𝑧)||²^1/𝑝log21−|𝑧|²=0.(5.8) Applying Lemmas 5.1 and 5.2 yields the desired result.
Conversely, suppose that conditions (5.6) hold. From the proof of Theorem 4.3, we have that 1−|𝑧|²log21−|𝑧|²|||𝑊_𝜓,𝜑≤⎛⎜⎜⎜⎝1−|𝑧|₂𝑓^(𝑧)|||||𝜓^(𝑧)||1−||𝜑(𝑧)||²^1/𝑝log21−|𝑧|²+𝐶1−|𝑧|²||𝜓(𝑧)𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²⎞⎟⎟⎟⎠‖𝑓‖_𝐻𝑝.(5.9) Taking the supremum over all 𝑓∈𝐻^𝑝 such that ‖𝑓‖_𝐻^𝑝≤1, then letting |𝑧|→1, we obtain that lim_|𝑧|→1sup_{‖𝑓‖𝐻𝑝≤1}1−|𝑧|²log21−|𝑧|²|||𝑊_𝜓,𝜑(𝑓)^(𝑧)|||=0,(5.10) from which, by Lemma 3.2, we deduce that the operator 𝑊_𝜓,𝜑∶𝐻^𝑝→ℬ_log,0 is compact.

From Theorems 5.3 and 5.4, we obtain the following corollary.

Corollary 5.5. Suppose that 𝜑 is an analytic self-map of the unit disk and 1≤𝑝<∞. Then, the following statements hold. (i)𝐶_𝜑∶𝐻^𝑝→ℬ_log,0 is bounded if and only if 𝐶_𝜑∶𝐻^𝑝→ℬ_log is bounded and 𝜑∈ℬ_log,0.(ii)𝐶_𝜑∶𝐻^𝑝→ℬ_log,0 is compact if and only if lim_|𝑧|→11−|𝑧|²||𝜑^(𝑧)||1−||𝜑(𝑧)||²^1+1/𝑝log21−|𝑧|²=0.(5.11)

Acknowledgments

The authors wish to express their gratitude to the referees for their careful reading of the paper and for their helpful suggestions. S. Li is supported by the Guangdong Natural Science Foundation (no. 10451401501004305), The National Natural Science Foundation of China (no. 11001107), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (no. LYM11117).