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Journal of Function Spaces and Applications
Volume 2012 (2012), Article ID 454820, 20 pages
http://dx.doi.org/10.1155/2012/454820
Research Article

Weighted Composition Operators from Hardy Spaces into Logarithmic Bloch Spaces

1Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA
2Department of Mathematics, Jiaying University, Guangdong, Meizhou 514015, China

Received 2 February 2012; Accepted 18 March 2012

Academic Editor: Ruhan Zhao

Copyright © 2012 Flavia Colonna and Songxiao Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The logarithmic Bloch space log is the Banach space of analytic functions on the open unit disk 𝔻 whose elements 𝑓 satisfy the condition 𝑓=sup𝑧𝔻(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 𝐻𝑝 (with 1𝑝) into the logarithmic Bloch space. We also provide boundedness and compactness criteria for the weighted composition operator mapping 𝐻𝑝 into the little logarithmic Bloch space defined as the subspace of 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 , 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 𝑊𝜓,𝜑 with range in 𝑌 defined by𝑊𝜓,𝜑𝑓=𝑀𝜓𝐶𝜑𝑓=𝜓(𝑓𝜑),for𝑓𝑋,(1.1) where 𝑀𝜓 denotes the multiplication operator with symbol 𝜓, and 𝐶𝜑 denotes the composition operator with symbol 𝜑.

Let 𝐻(𝔻) be the set of analytic functions on 𝔻={𝑧|𝑧|<1}. For 0<𝑝< the Hardy space 𝐻𝑝 is the space consisting of all 𝑓𝐻(𝔻) such that𝑓𝑝𝐻𝑝=sup0<𝑟<102𝜋||𝑓𝑟𝑒𝑖𝜃||𝑝𝑑𝜃2𝜋<.(1.2) Let 𝐻 denote the space of all 𝑓𝐻(𝔻) for which 𝑓=sup𝑧𝔻|𝑓(𝑧)|<.

The Bloch space on the open unit disk 𝔻 is the Banach space consisting of the analytic functions 𝑓 on 𝔻 such that𝑓𝛽=sup𝑧𝔻1|𝑧|2||𝑓||(𝑧)<.(1.3) The Bloch norm is given by 𝑓=|𝑓(0)|+𝑓𝛽. Using the Schwarz-Pick lemma, it is easy to see that the Hardy space 𝐻 is contained in and 𝑓𝛽𝑓. The inclusion is proper, as the function 𝑓(𝑧)=log(1+𝑧)/(1𝑧) shows.

The little Bloch space, denoted by 0, 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 0 is a closed separable subspace of . The interested reader is referred to [1] for more information on the Bloch space.

The logarithmic Bloch space log is defined as the set of functions 𝑓 on 𝔻 such that𝑓=sup𝑧𝔻1|𝑧|22log1|𝑧|2||𝑓||(𝑧)<.(1.4) It is a Banach space under the norm defined by 𝑓log=|𝑓(0)|+𝑓. Clearly, if 𝑓log, then lim|𝑧|1(1|𝑧|2)|𝑓(𝑧)|=0, so log is a subset of the little Bloch space.

The little logarithmic Bloch space, denoted by log,0, is defined as the subspace of log whose elements 𝑓 satisfy the conditionlim|𝑧|11|𝑧|22log||𝑓1|𝑧|||(𝑧)=0.(1.5)

The space log arises in connection to the study of certain operators with symbol. Arazy [2] proved that the multiplication operator 𝑀𝜓 is bounded on the Bloch space if and only if 𝜓log𝐻. In [3], Brown and Shields extended this result to the little Bloch space.

The space 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 𝐻𝑓 on 𝐴1 is defined as 𝐻𝑓𝑔=(𝐼𝑃)(𝑓𝑔), where 𝐼 is the identity operator, and 𝑃 is the standard Bergman projection from 𝐿1 into 𝐴1. In [4], Attele showed that 𝐻𝑓 is bounded on 𝐴1 if and only if 𝑓log.

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 log. He also introduced a class of Banach spaces 𝑄𝑝log (𝑝>0) closely related to log and studied the Taylor coefficients of the functions in log. In [7], Ye characterized the bounded and the compact weighted composition operators on the little logarithmic Bloch space 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 𝐻𝑝 (with 1𝑝) to the logarithmic Bloch space log as well as to its subspace log,0. The paper consists of five sections. Specifically, in Section 2, we consider the bounded weighted composition operators mapping 𝐻 into log and log,0. In particular, we show that𝑊𝜓,𝜑sup𝑛{0}𝜓𝜑𝑛log,(1.6) where the notation 𝐴𝐵 stands for 𝑐1𝐴𝐵𝑐2𝐴, 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 𝐻𝑝 into log in the case when 1𝑝<. Finally, in Section 5, we study the operators mapping 𝐻𝑝 into log,0.

2. Boundedness of 𝑊𝜓,𝜑 from 𝐻 into log and log,0

In the following theorem, we give two characterizations of boundedness when the operator maps 𝐻 into 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 𝑊𝜓,𝜑𝐻log is bounded.(b)sup𝑛{0}𝜓𝜑𝑛log<.(c)𝜓log and 𝜎𝜓,𝜑=sup𝑧𝔻((1|𝑧|2)|𝜓(𝑧)𝜑(𝑧)|/1|𝜑(𝑧)|2)log(2/(1|𝑧|2))<.

Proof. (a) (b). For 𝑛, the function 𝑝𝑛(𝑧)=𝑧𝑛 is bounded and 𝑝𝑛=1. Therefore, if 𝑊𝜓,𝜑 is bounded, then 𝜓𝜑𝑛log𝑊𝜓,𝜑.
(b) (c) Let 𝐶 be an upper bound for 𝜓𝜑𝑛log, 𝑛0. Taking 𝑛=0, we deduce that 𝜓log𝐶, so 𝜓log.
For 𝑁 and 𝑛2, define the sets 𝐸𝑁=||||1𝑧𝔻𝜑(𝑧)1𝑁,Δ𝑛=1𝑧𝔻1||||1𝑛1𝜑(𝑧)1𝑛.(2.1)
Fix an integer 𝑁>2, and 𝑧𝔻. For 𝑧𝐸𝑁, by the product rule, we have 1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|21|𝑧|2||(𝜓𝜑)||+||𝜓(𝑧)||(𝑧)𝜑(𝑧)1(11/𝑁)22log1|𝑧|211(11/𝑁)2()𝜓𝜑+𝜓2𝐶1(11/𝑁)2.(2.2)
In the proof of Theorem 2 of [10], it was shown that inf𝑧Δ𝑛𝑛||||𝜑(𝑧)𝑛1||||11𝜑(𝑧)𝑒.(2.3)
For |𝜑(𝑧)|>11/𝑁, there exists 𝑛>𝑁 such that 𝑧Δ𝑛. So 1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|21|𝑧|2||𝜓(𝑧)𝑛𝜑(𝑧)𝑛1𝜑||(𝑧)||||𝑛||||1𝜑(𝑧)𝜑(𝑧)𝑛12log1|𝑧|2𝑒1|𝑧|22log1|𝑧|2||(𝜓𝜑𝑛)||+(𝑧)1|𝑧|22log1|𝑧|2||𝜓(𝑧)𝜑(𝑧)𝑛||2𝑒𝐶.(2.4) From (2.2) and (2.4), we deduce that 𝜎𝜓,𝜑 is finite.
(c) (a) Let 𝑓𝐻 with 𝑓1 and pick 𝑧𝔻. Then 1|𝑧|22log1|𝑧|2||𝜓||(𝑧)𝑓(𝜑(𝑧))𝜓,(2.5) and, since 𝑓𝛽𝑓1, 1|𝑧|22log1|𝑧|2||𝜓(𝑧)𝑓(𝜑(𝑧))𝜑||=(𝑧)1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|2||||1𝜑(𝑧)2||𝑓||(𝜑(𝑧))1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|2.(2.6) Thus, by (2.5) and (2.6), we deduce that 𝑊𝜓,𝜑𝑓log𝜓log+𝜎𝜓,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 𝑊𝜓,𝜑𝐻log,0 is bounded.(b)For each integer 𝑛0,𝜓𝜑𝑛log,0 and sup𝑛𝜓𝜑𝑛log<.(c)𝜓log,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 log.
(b) (c) Suppose that (b) holds. If 𝐸𝑁=𝔻 for some integer 𝑁>1, then for all 𝑧𝔻, we have 1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|211(11/𝑁)21|𝑧|2||𝜓(𝑧)𝜑||2(𝑧)log1|𝑧|211(11/𝑁)21|𝑧|2||(𝜓𝜑)||+||𝜓(𝑧)||2(𝑧)𝜑(𝑧)log1|𝑧|211(11/𝑁)21|𝑧|2||(𝜓𝜑)(||2𝑧)log1|𝑧|2+11(11/𝑁)21|𝑧|2||𝜓||2(𝑧)log1|𝑧|20,as|𝑧|1.(2.7) If 𝐸𝑁 is properly contained in 𝔻, then for 𝑧𝔻𝐸𝑁, arguing as in the proof of (b) (c) in Theorem 2.1, there exists 𝑘𝑁 such that 𝑧Δ𝑘, so that 1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|21|𝑧|2||𝜓(𝑧)𝑘𝜑(𝑧)𝑘1𝜑||(𝑧)||||𝑘||||1𝜑(𝑧)𝜑(𝑧)𝑘12log1|𝑧|2𝑒1|𝑧|22log1|𝑧|2|||𝜓𝜑𝑘|||+(𝑧)1|𝑧|22log1|𝑧|2||𝜓||(𝑧)=𝐼+𝐼𝐼,(2.8) where 𝐼=𝑒(1|𝑧|2)log(2/(1|𝑧|2))|(𝜓𝜑𝑘)(𝑧)| and 𝐼𝐼=𝑒(1|𝑧|2)log(2/(1|𝑧|2))|𝜓(𝑧)|.
By the assumption of 𝜓𝜑𝑛log,0, we have 𝐼𝑒sup𝑛𝑘1|𝑧|22log1|𝑧|2||(𝜓𝜑𝑛)||(𝑧)0(2.9) as |𝑧|1. On the other hand, since 𝜓log,0, 𝐼𝐼0 as |𝑧|1. Therefore, lim|𝑧|11|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|2=0.(2.10)
(c) (a) Assume that (c) holds. To prove that 𝑊𝜓,𝜑 is bounded, it suffices to show that 𝑊𝜓,𝜑𝑓log,0 for each 𝑓𝐻, since the boundedness of the operator can be shown as in the proof of Theorem 2.1. Since 𝜓log,0, for 𝑓𝐻 and 𝑧𝔻, we have 1|𝑧|2||𝜓||2(𝑧)𝑓(𝜑(𝑧))log1|𝑧|21|𝑧|2||𝜓||2(𝑧)log1|𝑧|2𝑓0,(2.11) as |𝑧|1. On the other hand, by (2.6), 1|𝑧|2||𝜓(𝑧)𝑓(𝜑(𝑧))𝜑(||2𝑧)log1|𝑧|21|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|2𝑓0,(2.12) as |𝑧|1. Hence, 1|𝑧|2|||𝑊𝜓,𝜑𝑓|||2(𝑧)log1|𝑧|20(2.13) as |𝑧|1, completing the proof.

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

3. Compactness of 𝑊𝜓,𝜑 from 𝐻 into log and 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, 1𝑝. The operator 𝑊𝜓,𝜑𝐻𝑝log is compact if and only if for any bounded sequence {𝑓𝑛}𝑛 in 𝐻𝑝 which converges to zero uniformly on compact subsets of 𝔻, we have 𝑊𝜓,𝜑𝑓𝑛log0 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 log,0 is compact if and only if it is bounded and satisfies the following: lim|𝑧|1sup𝑓𝐾1|𝑧|2||𝑓||2(𝑧)log1|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 𝑎𝔻 and, for 𝑧𝔻, define𝑓𝑎(𝑧)=1|𝑎|21𝑎𝑧1/2,𝑔𝑎(𝑧)=1|𝑎|21𝑎𝑧.(3.2)

Theorem 3.3. Let 𝜓 be analytic on 𝔻, 𝜑 an analytic self-map of 𝔻, and assume that 𝑊𝜓,𝜑𝐻log is bounded. Then the following conditions are equivalent: (a)𝑊𝜓,𝜑𝐻log is compact.(b)lim|𝜑(𝑤)|1𝑊𝜓,𝜑𝑓𝜑(𝑤)log=0 and lim|𝜑(𝑤)|1𝑊𝜓,𝜑𝑔𝜑(𝑤)log=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)lim𝑛𝜓𝜑𝑛log=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 𝑊𝜓,𝜑 is compact and that {𝑤𝑛} is a sequence in 𝔻 such that |𝜑(𝑤𝑛)|1 as 𝑛. Since the sequences {𝑓𝜑(𝑤𝑛)} and {𝑔𝜑(𝑤𝑛)} are bounded in 𝐻 and converge to 0 uniformly on compact subsets of 𝔻, by Lemma 3.1, it follows that 𝑊𝜓,𝜑𝑓𝜑(𝑤𝑛)log0 and 𝑊𝜓,𝜑𝑔𝜑(𝑤𝑛)log0 as 𝑛.
(b)(c) Assume that (b) holds. Fix 𝑤𝔻. A straightforward calculation shows that 𝜓𝑓𝜑(𝑤)𝜑(𝑤)=𝜓(𝑤)+𝜓(𝑤)𝜑(𝑤)𝜑(𝑤)2||||1𝜑(𝑤)2,𝜓𝑔𝜑(𝑤)𝜑(𝑤)=𝜓(𝑤)+𝜓(𝑤)𝜑(𝑤)𝜑(𝑤)||||1𝜑(𝑤)2.(3.3) Eliminating 𝜓(𝑤), we obtain that 𝜓(𝑤)𝜑(𝑤)𝜑(𝑤)2||||1𝜑(𝑤)2=𝜓𝑔𝜑(𝑤)𝜑𝜓𝑓(𝑤)𝜑(𝑤)𝜑(𝑤).(3.4) Thus, for |𝜑(𝑤)|>𝑟(0,1), 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)22log1|𝑤|22𝑟𝑊𝜓,𝜑𝑓𝜑(𝑤)log+𝑊𝜓,𝜑𝑔𝜑(𝑤)log.(3.5) Taking the limit as |𝜑(𝑤)|1, we deduce that 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)22log1|𝑤|20.(3.6) On the other hand, using (3.3), we obtain that 1|𝑤|2||𝜓||2(𝑤)log1|𝑤|2𝑊𝜓,𝜑𝑔𝜑(𝑤)log+1|𝑤|2||𝜓(𝑤)𝜑(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)22log1|𝑤|2.(3.7) Taking the limit as |𝜑(𝑤)|1, we obtain that 1|𝑤|2||𝜓||2(𝑤)log1|𝑤|20,(3.8) proving (c).
(c)(a) Suppose that (c) holds. Let {𝑓𝑛} be a bounded sequence in 𝐻 converging to 0 uniformly on compact subsets of 𝔻. Set 𝐶=sup𝑛𝑓𝑛. Then, given 𝜀>0, there exists 𝑟(0,1) such that for |𝜑(𝑤)|>𝑟, 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)22log1|𝑤|2<𝜀,2𝐶1|𝑤|2||𝜓||2(𝑤)log1|𝑤|2<𝜀.2𝐶(3.9) Then, for 𝑤𝔻, noting that (1|𝜑(𝑤)|2)|𝑓𝑛(𝜑(𝑤))|𝑓𝑛𝐶, we obtain that 1|𝑤|2|||𝜓𝑓𝑛𝜑|||2(𝑤)log1|𝑤|21|𝑤|2||𝜓(𝑤)𝑓𝑛||+||𝜓(𝜑(𝑤))(𝑤)𝑓𝑛(𝜑(𝑤))𝜑||2(𝑤)log1|𝑤|21|𝑤|2||𝜓||𝑓(𝑤)𝑛2log1|𝑤|2+1|𝑤|2||||1𝜑(𝑤)2||𝜓(𝑤)𝑓𝑛(𝜑(𝑤))𝜑||(𝑤)||𝜑||1(𝑤)22log1|𝑤|2𝐶1|𝑤|2||𝜓||+(𝑤)1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)22log1|𝑤|2.(3.10) Thus, for |𝜑(𝑤)|>𝑟, we have 1|𝑤|2|||𝜓𝑓𝑛𝜑|||2(𝑤)log1|𝑤|2<𝜀.(3.11) On the other hand, for |𝜑(𝑤)|𝑟, 1|𝑤|2|||𝜓𝑓𝑛𝜑|||2(𝑤)log1|𝑤|2||𝑓𝜓𝑛||(𝜑(𝑤))+𝜎𝜓,𝜑||𝑓𝑛||(𝜑(𝑤)).(3.12) Thus, by the uniform convergence to 0 of 𝑓𝑛 and 𝑓𝑛 on compact sets, we see that (3.11) holds also in this case for 𝑛 sufficiently large. Hence, 𝜓(𝑓𝑛𝜑)𝜀 for 𝑛 sufficiently large. Since |𝜓(0)𝑓𝑛(𝜑(0))|0, we deduce that 𝜓(𝑓𝑛𝜑)log0 as 𝑛. Therefore, 𝑊𝜓,𝜑 is compact.
(a) (d) Suppose that 𝑊𝜓,𝜑 is compact. Since the sequence {𝑝𝑛} defined by 𝑝𝑛(𝑧)=𝑧𝑛, 𝑧𝔻 is bounded in 𝐻 and converges to 0 uniformly on compact subsets, by Lemma 3.1, it follows that 𝜓𝜑𝑛log=𝑊𝜓,𝜑𝑝𝑛log0 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 𝜓𝜑𝑛log<𝜀/2 for all 𝑛𝑁 and (1|𝑤|2)|𝜓(𝑤)|log(2/(1𝑤|2))<𝜀/2 for |𝜑(𝑤)|>11/𝑁.
For |𝜑(𝑤)|>11/𝑁, there exists 𝑛>𝑁 such that 𝑧Δ𝑛. Using the product rule, we may write the following 𝜓(𝑤)𝑛𝜑(𝑤)𝑛1𝜑(𝑤)=(𝜓𝜑𝑛)(𝑤)𝜓(𝑤)𝜑(𝑤)𝑛,(3.13) so that 1|𝑤|2||𝜓(𝑤)𝑛𝜑(𝑤)𝑛1𝜑||2(𝑤)log1|𝑤|2𝜓𝜑𝑛log+1|𝑤|2||𝜓(𝑤)𝜑(𝑤)𝑛||2log1|𝑤|2.(3.14) The left-hand side of (3.14) can be written as 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)2||||1𝜑(𝑤)2𝑛||||𝜑(𝑤)𝑛12log1|𝑤|2,(3.15) which, by (2.3), is bounded below by 1𝑒1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)22log1|𝑤|2.(3.16) Thus, from (3.14), we deduce that 1𝑒1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)22log1|𝑤|2𝜓𝜑𝑛log+1|𝑤|2||𝜓||2(𝑤)log1|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 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 𝑊𝜓,𝜑𝐻log,0 is compact.(b)For each integer 𝑛0, 𝜓𝜑𝑛log,0 and lim𝑛𝜓𝜑𝑛log=0.(c)𝜓log,0 andlim|𝑧|11|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|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)||||1𝜑(𝑧)22log1|𝑧|20(3.19) as |𝑧|1. Fix 𝜀>0 and let 𝑁>2 be an integer such that 𝜓𝜑𝑛log<𝜀/𝑒 for all 𝑛𝑁. Observe that, since for 𝑧𝔻, ||𝜓(𝑧)𝜑(||||(𝑧)𝜓𝜑)(||+||𝜓𝑧)(||||(𝑧)𝜑(𝑧)𝜓𝜑)(||+||𝜓𝑧)(||𝑧),(3.20) and, by assumption, the functions 𝜓𝜑 and 𝜓 are in log,0, we have lim|𝑧|11|𝑧|2||𝜓(𝑧)𝜑||2(𝑧)log1|𝑧|2=0.(3.21) Therefore, there is 𝑟(0,1), such that for |𝑧|>𝑟, 1|𝑧|2||𝜓(𝑧)𝜑||2(𝑧)log1|𝑧|2<(2𝑁1)𝜀𝑁2.(3.22) Thus, if 𝑧𝐸𝑁, and |𝑧|>𝑟, then 1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|2𝑁22𝑁11|𝑧|2||𝜓(𝑧)𝜑(||2𝑧)log1|𝑧|2<𝜀.(3.23) On the other hand, if 𝑧𝐸𝑁, then there exists 𝑛>𝑁 such that 𝑧Δ𝑛, so, as shown in the proof of (d) implies (c) of Theorem 3.3, we have 1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|2𝑒𝜓𝜑𝑛log+1|𝑧|2||𝜓(||2𝑧)log1|𝑧|2<𝜀+𝑒1|𝑧|2||𝜓||2(𝑧)log1|𝑧|2𝜀,(3.24) as |𝑧|1. Since 𝜀 is arbitrary, the result follows.
(c) (a) Let {𝑓𝑛} be a bounded sequence in 𝐻 converging to 0 uniformly on compact subsets, and let 𝐶=sup𝑛𝑓𝑛. We wish to show that 𝑊𝜓,𝜑𝑓𝑛log,0 and 𝑊𝜓,𝜑𝑓𝑛log0 as 𝑛. As shown in the proof of (c) implies (a) of Theorem 3.3, for 𝑧𝔻 and 𝑛, 1|𝑧|2|||𝑊𝜓,𝜑𝑓𝑛|||2(𝑧)log1|𝑧|2𝐶1|𝑧|2||𝜓||2(𝑧)log1|𝑧|2+𝐶1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)22log1|𝑧|20,(3.25) as |𝑧|1. Thus, 𝑊𝜓,𝜑𝑓𝑛log,0. The convergence to 0 of 𝑊𝜓,𝜑𝑓𝑛log is proved as in the case of the operator mapping into 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 𝑊𝜓,𝜑𝐻log,0 is bounded.(b)The operator 𝑊𝜓,𝜑𝐻log,0 is compact.(c)For each integer 𝑛0, 𝜓𝜑𝑛log,0 and lim𝑛𝜓𝜑𝑛log=0.(d)𝜓log,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)𝑀𝜓𝐻log is bounded,(b)𝑀𝜓𝐻log,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)𝐶𝜑𝐻log is bounded,(b)sup𝑛𝜑𝑛log<,(c)sup𝑧𝔻((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)𝐶𝜑𝐻log is compact,(b)lim𝑛𝜑𝑛log=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)𝐶𝜑𝐻log,0 is bounded,(b)𝐶𝜑𝐻log,0 is compact,(c)𝜑log,0 and lim𝑛𝜑𝑛log=0,(d)lim|𝑧|1((1|𝑧|2)|𝜑(𝑧)|/(1|𝜑(𝑧)|2))log(2/(1|𝑧|2))=0.

4. 𝑊𝜓,𝜑 from 𝐻𝑝(1𝑝<) into 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 𝑓𝐻𝑝, 𝑧𝔻, ||||𝑓(𝑧)𝑓𝐻𝑝1|𝑧|21/𝑝.(4.1)

Lemma 4.2 (See [13]). Let 0<𝑝<. Then for any 𝑓𝐻𝑝, 𝑧𝔻, ||𝑓||(𝑧)𝐶𝑓𝐻𝑝1|𝑧|21+1/𝑝.(4.2)

Fix 1𝑝< and 𝑎𝔻. For 𝑧𝔻, define the functions𝑓𝑎(𝑧)=1|𝑎|221/𝑝1𝑎𝑧2,𝑔𝑎(𝑧)=1|𝑎|221𝑎𝑧2+1/𝑝.(4.3) Then 𝑓𝑎,𝑔𝑎𝐻𝑝 and the norms 𝑓𝑎𝐻𝑝 and 𝑔𝑎𝐻𝑝 are bounded by constants only dependent of 𝑝. In addition, a straightforward calculation shows that𝑓𝑎(𝑎)=𝑔𝑎1(𝑎)=1|𝑎|21/𝑝,𝑓𝑎2(𝑎)=𝑎1|𝑎|21+1/𝑝,𝑔𝑎(𝑎)=(2+1/𝑝)𝑎1|𝑎|21+1/𝑝.(4.4)

We use these two families of functions to characterize the bounded and the compact weighted composition operators from 𝐻𝑝 to log.

Theorem 4.3. Let 1𝑝<, 𝜓 analytic on 𝔻 and let 𝜑 be an analytic self-map of 𝔻. Then the following conditions are equivalent: (a)𝑊𝜓,𝜑𝐻𝑝log is bounded,(b)𝜓𝜑log, 𝐴=sup𝑤𝔻𝑊𝜓,𝜑𝑓𝜑(𝑤)log< and 𝐵=sup𝑤𝔻𝑊𝜓,𝜑𝑔𝜑(𝑤)log<,(c)𝑥𝜓,𝜑=sup𝑧𝔻1|𝑧|2||𝜓||(𝑧)||||1𝜑(𝑧)21/𝑝2log1|𝑧|2𝑦<,𝜓,𝜑=sup𝑧𝔻1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)21+1/𝑝2log1|𝑧|2<.(4.5)

Proof. (a) (b) Assume that 𝑊𝜓,𝜑𝐻𝑝log is bounded. Then 𝜓𝜑log and for each 𝑤𝔻, 𝑊𝜓,𝜑𝑓𝜑(𝑤)log𝑊𝜓,𝜑𝑓𝜑(𝑤)𝐻𝑝𝑊𝐶𝜓,𝜑,𝑊𝜓,𝜑𝑔𝜑(𝑤)log𝑊𝜓,𝜑𝑔𝜑(𝑤)𝐻𝑝𝑊𝐶𝜓,𝜑,(4.6) for some constant 𝐶, so 𝐴 and 𝐵 are finite.
(b) (c) Suppose that 𝜓𝜑log, and the quantities 𝐴 and 𝐵 are finite. From (4.4), for 𝑤𝔻, we have 𝜓𝑓𝜑(𝑤)𝜑𝜓(𝑤)=(𝑤)||||1𝜑(𝑤)21/𝑝+2𝜓(𝑤)𝜑(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝,(4.7) whence 1|𝑤|2||𝜓||(𝑤)||||1𝜑(𝑤)21/𝑝2log1|𝑤|21|𝑤|22log1|𝑤|2|||𝜓𝑓𝜑(𝑤)𝜑|||(𝑤)+21|𝑤|2||𝜓(𝑤)𝜑||(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2𝑊𝜓,𝜑𝑓𝜑(𝑤)log+21|𝑤|2||𝜓(𝑤)𝜑||(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2𝐴+21|𝑤|2||𝜓(𝑤)𝜑||(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2.(4.8) Moreover, 𝜓𝑔𝜑(𝑤)𝜑𝜓(𝑤)=(𝑤)||||1𝜑(𝑤)21/𝑝+(2+1/𝑝)𝜓(𝑤)𝜑(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝.(4.9) Therefore, subtracting (4.7) from (4.9) and taking the modulus, we obtain 1𝑝||𝜓(𝑤)𝜑||(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝|||𝜓𝑓𝜑(𝑤)𝜑|||+|||𝜓𝑔(𝑤)𝜑(𝑤)𝜑|||(𝑤),(4.10) which yields that 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2𝑝(𝐴+𝐵).(4.11) Consequently, from (4.8), we deduce that 1|𝑤|2||𝜓||(𝑤)||||1𝜑(𝑤)21/𝑝2log1|𝑤|2(1+2𝑝)𝐴+2𝑝𝐵.(4.12) Taking the supremum over all 𝑤𝔻, we see that 𝑥𝜓,𝜑 is finite.
Fix 𝑟(0,1). If |𝜑(𝑤)|>𝑟, then from (4.11) we have 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2<𝑝(𝐴+𝐵)𝑟.(4.13) On the other hand, since 𝜓𝜑log, if |𝜑(𝑤)|𝑟, then, 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|21|𝑤|2||(𝜓𝜑)||(𝑤)1𝑟21+1/𝑝2log1|𝑤|2+1|𝑤|2||𝜓||(𝑤)𝜑(𝑤)1𝑟2||||1𝜑(𝑤)21/𝑝2log1|𝑤|2𝜓𝜑1𝑟21+1/𝑝+𝑥𝜓,𝜑1𝑟2<.(4.14) Taking the supremum over all 𝑤𝔻, it follows that 𝑦𝜓,𝜑 is finite as well.
(c) (a) Suppose that 𝑥𝜓,𝜑 and 𝑦𝜓,𝜑 are finite. For arbitrary 𝑧 in 𝔻 and 𝑓𝐻𝑝, by Lemmas 4.1 and 4.2, we have 1|𝑧|22log1|𝑧|2|||𝑊𝜓,𝜑𝑓|||(𝑧)1|𝑧|22log1|𝑧|2||𝜓||||𝑓||+||𝑓(𝑧)(𝜑(𝑧))||||𝜓(𝜑(𝑧))(𝑧)𝜑||(𝑧)1|𝑧|2||𝜓||(𝑧)||||1𝜑(𝑧)21/𝑝2log1|𝑧|2+𝐶1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)21+1/𝑝2log1|𝑧|2𝑓𝐻𝑝𝑥𝜓,𝜑+𝐶𝑦𝜓,𝜑𝑓𝐻𝑝.(4.15)
Taking the supremum over all 𝑧𝔻 and applying Lemma 4.1, we obtain that 𝑊𝜓,𝜑𝑓log=||||+𝑊𝜓(0)𝑓(𝜑(0))𝜓,𝜑𝑓||||𝜓(0)||||1𝜑(0)21/𝑝+𝑥𝜓,𝜑+𝐶𝑦𝜓,𝜑𝑓𝐻𝑝.(4.16) The boundedness of the operator 𝑊𝜓,𝜑𝐻𝑝log follows by taking the supremum over all 𝑓𝐻𝑝.

Theorem 4.4. Let 1𝑝<, 𝜓 analytic on 𝔻, 𝜑 an analytic self-map of 𝔻, and assume that 𝑊𝜓,𝜑𝐻𝑝log is bounded. Then the following conditions are equivalent: (a)𝑊𝜓,𝜑𝐻𝑝log is compact,(b)lim|𝜑(𝑤)|1𝑊𝜓,𝜑𝑓𝜑(𝑤)log=0 and lim|𝜑(𝑤)|1𝑊𝜓,𝜑𝑔𝜑(𝑤)log=0,(c)lim||||𝜑(𝑤)11|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2=0,lim||||𝜑(𝑤)11|𝑤|2||𝜓||(𝑤)||||1𝜑(𝑤)21/𝑝2log1|𝑤|2=0.(4.17)

Proof. (a)(b) Suppose that 𝑊𝜓,𝜑𝐻𝑝log is compact. Let {𝑤𝑛} be a sequence in 𝔻 such that lim𝑛|𝜑(𝑤𝑛)|=1. Observe that the sequences {𝑓𝜑(𝑤𝑛)} and {𝑔𝜑(𝑤𝑛)} are bounded in 𝐻𝑝 and converge to 0 uniformly on compact subsets of 𝔻. By Lemma 3.1, it follows that 𝑊𝜓,𝜑𝑓𝜑(𝑤𝑛)log0 and 𝑊𝜓,𝜑𝑔𝜑(𝑤𝑛)log0 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)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2𝑝𝑊𝜓,𝜑𝑓𝜑(𝑤)log+𝑊𝜓,𝜑𝑔𝜑(𝑤)log||||𝜑(𝑤)0(4.18) as |𝜑(𝑤)|1. Moreover, using (4.8), we deduce that 1|𝑤|2||𝜓||(𝑤)||||1𝜑(𝑤)21/𝑝2log1|𝑤|20(4.19) as |𝜑(𝑤)|1.
(c)(a) Suppose that (c) holds. Let {𝑓𝑛} be a bounded sequence in 𝐻𝑝 converging to 0 uniformly on compact subsets of 𝔻. Set 𝐶=sup𝑛𝑓𝑛𝐻𝑝. Then, given 𝜀>0, there exists 𝑟(0,1) such that 1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2<𝜀,2𝐶1|𝑤|2||𝜓||(𝑤)||||1𝜑(𝑤)21/𝑝2log1|𝑤|2<𝜀,2𝐶(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|22log1|𝑤|2|||𝜓𝑓𝑛𝜑|||(𝑤)1|𝑤|22log1|𝑤|2||𝜓(𝑤)𝑓𝑛||+||𝜓(𝜑(𝑤))(𝑤)𝑓𝑛(𝜑(𝑤))𝜑||𝑓(𝑤)𝑛𝐻𝑝1|𝑤|2||𝜓||(𝑤)||||1𝜑(𝑤)21/𝑝2log1|𝑤|2+1|𝑤|2||𝜓(𝑤)𝜑||(𝑤)||||1𝜑(𝑤)21+1/𝑝2log1|𝑤|2<𝜀.(4.21) On the other hand, for |𝜑(𝑤)|𝑟, by the uniform convergence to 0 of 𝑓𝑛 and 𝑓𝑛 on compact sets, we have 1|𝑤|22log1|𝑤|2|||𝜓𝑓𝑛𝜑|||(𝑤)𝑥𝜓,𝜑||𝑓𝑛||(𝜑(𝑤))+𝑦𝜓,𝜑||𝑓||(𝜑(𝑤))0,(4.22) as 𝑛. Since |𝜓(0)𝑓𝑛(𝜑(0))|0, we conclude that 𝑊𝜓,𝜑𝑓𝑛log0 as 𝑛. Consequently, by Lemma 3.1, the operator 𝑊𝜓,𝜑 is compact.

As a consequence of Theorems 4.3 and 4.4, noting that for 𝑤𝔻,𝐶𝜑𝑔𝜑(𝑤)1(𝑤)=212+𝑝𝐶𝜑𝑓𝜑(𝑤)(𝑤)=(2+1/𝑝)𝜑(𝑤)𝜑(𝑤)||||1𝜑(𝑤)21+1/𝑝,(4.23) we obtain the following characterizations of the bounded and the compact composition operators from 𝐻𝑝 into log.

Corollary 4.5. Let 𝜑 be an analytic self-map of 𝔻, and 1𝑝<. The following statements are equivalent: (a)𝐶𝜑𝐻𝑝log is bounded,(b)sup𝑤𝔻𝐶𝜑𝑓𝜑(𝑤)log<,(c)sup𝑤𝔻𝐶𝜑𝑔𝜑(𝑤)log<,(d)sup𝑤𝔻((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 1𝑝<. If 𝐶𝜑𝐻𝑝log is bounded, then the following statements are equivalent: (a)𝐶𝜑𝐻𝑝log is compact,(b)lim|𝜑(𝑤)|1𝐶𝜑𝑓𝜑(𝑤)log=0,(c)lim|𝜑(𝑤)|1𝐶𝜑𝑔𝜑(𝑤)log=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. 𝑊𝜓,𝜑 from 𝐻𝑝(1𝑝<) into log,0

In this section, we characterize the boundedness and the compactness of the weighted composition operators 𝑊𝜓,𝜑𝐻𝑝log,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 1𝑝<. Then, lim|𝑧|11|𝑧|2||𝜓||(𝑧)||||1𝜑(𝑧)21/𝑝2log1|𝑧|2=0(5.1) if and only if 𝜓log,0 and lim||||𝜑(𝑧)11|𝑧|2||𝜓||(𝑧)||||1𝜑(𝑧)21/𝑝2log1|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 1𝑝<. Then, lim|𝑧|11|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)21+1/𝑝2log1|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)11|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)21+1/𝑝2log1|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, 𝜓𝐻(𝔻), and 1𝑝<. Then, the operator 𝑊𝜓,𝜑𝐻𝑝log,0 is bounded if and only if 𝑊𝜓,𝜑𝐻𝑝log is bounded, 𝜓log,0, and lim|𝑧|11|𝑧|22log1|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, 𝜓𝐻(𝔻), and 1𝑝<. Then, 𝑊𝜓,𝜑𝐻𝑝log,0 is compact if and only if lim|𝑧|11|𝑧|2||𝜓||(𝑧)||||1𝜑(𝑧)21/𝑝2log1|𝑧|2=0,lim|𝑧|11|𝑧|2||𝜓(𝑧)𝜑||(𝑧)||||1𝜑(𝑧)21+1/𝑝2log1|𝑧|2=0.(5.6)

Proof. First, we assume that 𝑊𝜓,𝜑𝐻𝑝log,0 is compact, and hence bounded. By Theorem 5.3, it follows that 𝜓log,0 and lim|𝑧|1(1|𝑧|)22log1|𝑧|2||𝜓(𝑧)𝜑||(𝑧)=0.(5.7) Since by assumption