About this Journal Submit a Manuscript Table of Contents
Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 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: RuhanΒ Zhao

Copyright Β© 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 ℬlog is the Banach space of analytic functions on the open unit disk 𝔻 whose elements 𝑓 satisfy the condition ‖𝑓‖=supπ‘§βˆˆπ”»(1βˆ’|𝑧|2)log(2/(1βˆ’|𝑧|2))|𝑓′(𝑧)|<∞. In this work we characterize the bounded and the compact weighted composition operators from the Hardy space 𝐻𝑝 (with 1β‰€π‘β‰€βˆž) 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 ℬlog consisting of the functions 𝑓 such that lim|𝑧|β†’1(1βˆ’|𝑧|2)log(2/(1βˆ’|𝑧|2))|𝑓′(𝑧)|=0.

1. Introduction

Let 𝑋 and π‘Œ be Banach spaces of analytic functions on a domain Ξ© in β„‚, πœ“ an analytic function on Ξ©, and let πœ‘ be an analytic function mapping Ξ© into itself. The weighted composition operator with symbols πœ“ and πœ‘ from 𝑋 to π‘Œ is the operator π‘Šπœ“,πœ‘ with range in π‘Œ defined byπ‘Šπœ“,πœ‘π‘“=π‘€πœ“πΆπœ‘π‘“=πœ“(π‘“βˆ˜πœ‘),forπ‘“βˆˆπ‘‹,(1.1) where π‘€πœ“ denotes the multiplication operator with symbol πœ“, and πΆπœ‘ denotes the composition operator with symbol πœ‘.

Let 𝐻(𝔻) be the set of analytic functions on 𝔻={π‘§βˆˆβ„‚βˆΆ|𝑧|<1}. For 0<𝑝<∞ the Hardy space 𝐻𝑝 is the space consisting of all π‘“βˆˆπ»(𝔻) such that‖𝑓‖𝑝𝐻𝑝=sup0<π‘Ÿ<1ξ€œ02πœ‹||π‘“ξ€·π‘Ÿπ‘’π‘–πœƒξ€Έ||π‘π‘‘πœƒ2πœ‹<∞.(1.2) Let 𝐻∞ denote the space of all π‘“βˆˆπ»(𝔻) for which β€–π‘“β€–βˆž=supπ‘§βˆˆπ”»|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=supπ‘§βˆˆπ”»ξ€·1βˆ’|𝑧|2ξ€Έ||π‘“ξ…ž||(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 β€–π‘“β€–π›½β‰€β€–π‘“β€–βˆž. The inclusion is proper, as the function 𝑓(𝑧)=log(1+𝑧)/(1βˆ’π‘§) shows.

The little Bloch space, denoted by ℬ0, is defined as the set of the analytic functions 𝑓 on 𝔻 such that lim|𝑧|β†’1(1βˆ’|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 ℬlog is defined as the set of functions 𝑓 on 𝔻 such that‖𝑓‖=supπ‘§βˆˆπ”»ξ€·1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||π‘“ξ…ž||(𝑧)<∞.(1.4) It is a Banach space under the norm defined by ‖𝑓‖ℬlog=|𝑓(0)|+‖𝑓‖. Clearly, if π‘“βˆˆβ„¬log, then lim|𝑧|β†’1(1βˆ’|𝑧|2)|𝑓′(𝑧)|=0, so ℬlog is a subset of the little Bloch space.

The little logarithmic Bloch space, denoted by ℬlog,0, is defined as the subspace of ℬlog whose elements 𝑓 satisfy the conditionlim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ2log||𝑓1βˆ’|𝑧|ξ…ž||(𝑧)=0.(1.5)

The space ℬlog arises in connection to the study of certain operators with symbol. Arazy [2] proved that the multiplication operator π‘€πœ“ is bounded on the Bloch space if and only if πœ“βˆˆβ„¬log∩𝐻∞. In [3], Brown and Shields extended this result to the little Bloch space.

The space ℬlog 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=(πΌβˆ’π‘ƒ)(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 π‘“βˆˆβ„¬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 ℬlog. He also introduced a class of Banach spaces 𝑄𝑝log (𝑝>0) closely related to ℬlog and studied the Taylor coefficients of the functions in ℬlog. In [7], Ye characterized the bounded and the compact weighted composition operators on the little logarithmic Bloch space ℬlog,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 1β‰€π‘β‰€βˆž) to the logarithmic Bloch space ℬlog as well as to its subspace ℬlog,0. The paper consists of five sections. Specifically, in Section 2, we consider the bounded weighted composition operators mapping 𝐻∞ into ℬlog and ℬlog,0. In particular, we show thatβ€–β€–π‘Šπœ“,πœ‘β€–β€–βˆΌsupπ‘›βˆˆβ„•βˆͺ{0}β€–πœ“πœ‘π‘›β€–β„¬log,(1.6) where the notation 𝐴∼𝐡 stands for 𝑐1𝐴≀𝐡≀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 𝐻𝑝 into ℬlog in the case when 1≀𝑝<∞. Finally, in Section 5, we study the operators mapping 𝐻𝑝 into ℬlog,0.

2. Boundedness of π‘Šπœ“,πœ‘ from 𝐻∞ into ℬlog and ℬlog,0

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

Theorem 2.1. Let πœ“ be an analytic function on 𝔻, and let πœ‘ be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator π‘Šπœ“,πœ‘βˆΆπ»βˆžβ†’β„¬log is bounded.(b)supπ‘›βˆˆβ„•βˆͺ{0}β€–πœ“πœ‘π‘›β€–β„¬log<∞.(c)πœ“βˆˆβ„¬log and πœŽπœ“,πœ‘=supπ‘§βˆˆπ”»((1βˆ’|𝑧|2)|πœ“(𝑧)πœ‘β€²(𝑧)|/1βˆ’|πœ‘(𝑧)|2)log(2/(1βˆ’|𝑧|2))<∞.

Proof. (a) β‡’ (b). For π‘›βˆˆβ„•, the function 𝑝𝑛(𝑧)=𝑧𝑛 is bounded and β€–π‘π‘›β€–βˆž=1. Therefore, if π‘Šπœ“,πœ‘ is bounded, then β€–πœ“πœ‘π‘›β€–β„¬logβ‰€β€–π‘Šπœ“,πœ‘β€–.
(b) β‡’ (c) Let 𝐢 be an upper bound for β€–πœ“πœ‘π‘›β€–β„¬log, 𝑛β‰₯0. Taking 𝑛=0, we deduce that β€–πœ“β€–β„¬log≀𝐢, so πœ“βˆˆβ„¬log.
For π‘βˆˆβ„• and 𝑛β‰₯2, define the sets 𝐸𝑁=||||1π‘§βˆˆπ”»βˆΆπœ‘(𝑧)≀1βˆ’π‘ξ‚‡,Δ𝑛=1π‘§βˆˆπ”»βˆΆ1βˆ’β‰€||||1π‘›βˆ’1πœ‘(𝑧)≀1βˆ’π‘›ξ‚‡.(2.1)
Fix an integer 𝑁>2, and π‘§βˆˆπ”». For π‘§βˆˆπΈπ‘, by the product rule, we have ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2≀1βˆ’|𝑧|2||ξ€Έξ€·(πœ“πœ‘)ξ…ž||+||πœ“(𝑧)ξ…ž||ξ€Έ(𝑧)πœ‘(𝑧)1βˆ’(1βˆ’1/𝑁)22log1βˆ’|𝑧|2≀11βˆ’(1βˆ’1/𝑁)2()β‰€β€–πœ“πœ‘β€–+β€–πœ“β€–2𝐢1βˆ’(1βˆ’1/𝑁)2.(2.2)
In the proof of Theorem 2 of [10], it was shown that infπ‘§βˆˆΞ”π‘›π‘›||||πœ‘(𝑧)π‘›βˆ’1ξ€·||||ξ€Έβ‰₯11βˆ’πœ‘(𝑧)𝑒.(2.3)
For |πœ‘(𝑧)|>1βˆ’1/𝑁, there exists 𝑛>𝑁 such that π‘§βˆˆΞ”π‘›. So ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2≀1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)π‘›πœ‘(𝑧)π‘›βˆ’1πœ‘ξ…ž||(𝑧)ξ€·||||𝑛||||1βˆ’πœ‘(𝑧)πœ‘(𝑧)π‘›βˆ’12log1βˆ’|𝑧|2≀𝑒1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||(πœ“πœ‘π‘›)ξ…ž||+ξ€·(𝑧)1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||πœ“ξ…ž(𝑧)πœ‘(𝑧)𝑛||≀2𝑒𝐢.(2.4) From (2.2) and (2.4), we deduce that πœŽπœ“,πœ‘ is finite.
(c) β‡’ (a) Let π‘“βˆˆπ»βˆž with β€–π‘“β€–βˆžβ‰€1 and pick π‘§βˆˆπ”». Then ξ€·1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||πœ“ξ…ž||≀(𝑧)𝑓(πœ‘(𝑧))β€–πœ“β€–,(2.5) and, since β€–π‘“β€–π›½β‰€β€–π‘“β€–βˆžβ‰€1, ξ€·1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||πœ“(𝑧)π‘“ξ…ž(πœ‘(𝑧))πœ‘ξ…ž||=ξ€·(𝑧)1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2ξ‚€||||1βˆ’πœ‘(𝑧)2||π‘“ξ…ž||≀(πœ‘(𝑧))1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2.(2.6) Thus, by (2.5) and (2.6), we deduce that β€–π‘Šπœ“,πœ‘π‘“β€–β„¬logβ‰€β€–πœ“β€–β„¬log+πœŽπœ“,πœ‘, completing the proof.

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

Theorem 2.2. Let πœ“ be an analytic function on 𝔻, and let πœ‘ be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator π‘Šπœ“,πœ‘βˆΆπ»βˆžβ†’β„¬log,0 is bounded.(b)For each integer 𝑛β‰₯0,πœ“πœ‘π‘›βˆˆβ„¬log,0 and supπ‘›βˆˆβ„•β€–πœ“πœ‘π‘›β€–β„¬log<∞.(c)πœ“βˆˆβ„¬log,0 and lim|𝑧|β†’1((1βˆ’|𝑧|2)|πœ“(𝑧)πœ‘ξ…ž(𝑧)|/(1βˆ’|πœ‘(𝑧)|2))log(2/(1βˆ’|𝑧|2))=0.

Proof. (a) β‡’ (b) is proved as in the case of the operator mapping into ℬlog.
(b) β‡’ (c) Suppose that (b) holds. If 𝐸𝑁=𝔻 for some integer 𝑁>1, then for all π‘§βˆˆπ”», we have ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2≀11βˆ’(1βˆ’1/𝑁)2ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||2(𝑧)log1βˆ’|𝑧|2≀11βˆ’(1βˆ’1/𝑁)2ξ€·1βˆ’|𝑧|2||ξ€Έξ€·(πœ“πœ‘)ξ…ž||+||πœ“(𝑧)ξ…ž||ξ€Έ2(𝑧)πœ‘(𝑧)log1βˆ’|𝑧|2≀11βˆ’(1βˆ’1/𝑁)2ξ€·1βˆ’|𝑧|2ξ€Έ||(πœ“πœ‘)ξ…ž(||2𝑧)log1βˆ’|𝑧|2+11βˆ’(1βˆ’1/𝑁)2ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||2(𝑧)log1βˆ’|𝑧|2⟢0,as|𝑧|⟢1.(2.7) If 𝐸𝑁 is properly contained in 𝔻, then for π‘§βˆˆπ”»β§΅πΈπ‘, arguing as in the proof of (b) β‡’ (c) in Theorem 2.1, there exists π‘˜β‰₯𝑁 such that π‘§βˆˆΞ”π‘˜, so that ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2≀1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)π‘˜πœ‘(𝑧)π‘˜βˆ’1πœ‘ξ…ž||(𝑧)ξ€·||||ξ€Έπ‘˜||||1βˆ’πœ‘(𝑧)πœ‘(𝑧)π‘˜βˆ’12log1βˆ’|𝑧|2≀𝑒1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2|||ξ€·πœ“πœ‘π‘˜ξ€Έξ…ž|||+ξ€·(𝑧)1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||πœ“ξ…ž||ξ‚Ή(𝑧)=𝐼+𝐼𝐼,(2.8) where 𝐼=𝑒(1βˆ’|𝑧|2)log(2/(1βˆ’|𝑧|2))|(πœ“πœ‘π‘˜)β€²(𝑧)| and 𝐼𝐼=𝑒(1βˆ’|𝑧|2)log(2/(1βˆ’|𝑧|2))|πœ“β€²(𝑧)|.
By the assumption of πœ“πœ‘π‘›βˆˆβ„¬log,0, we have 𝐼≀𝑒sup𝑛β‰₯π‘˜ξ€·1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||(πœ“πœ‘π‘›)ξ…ž||(𝑧)⟢0(2.9) as |𝑧|β†’1. On the other hand, since πœ“βˆˆβ„¬log,0, 𝐼𝐼→0 as |𝑧|β†’1. Therefore, lim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2=0.(2.10)
(c) β‡’ (a) Assume that (c) holds. To prove that π‘Šπœ“,πœ‘ is bounded, it suffices to show that π‘Šπœ“,πœ‘π‘“βˆˆβ„¬log,0 for each π‘“βˆˆπ»βˆž, since the boundedness of the operator can be shown as in the proof of Theorem 2.1. Since πœ“βˆˆβ„¬log,0, for π‘“βˆˆπ»βˆž and π‘§βˆˆπ”», we have ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||2(𝑧)𝑓(πœ‘(𝑧))log1βˆ’|𝑧|2≀1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||2(𝑧)log1βˆ’|𝑧|2β€–π‘“β€–βˆžβŸΆ0,(2.11) as |𝑧|β†’1. On the other hand, by (2.6), ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)π‘“ξ…ž(πœ‘(𝑧))πœ‘ξ…ž(||2𝑧)log1βˆ’|𝑧|2≀1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2β€–π‘“β€–βˆžβŸΆ0,(2.12) as |𝑧|β†’1. Hence, ξ€·1βˆ’|𝑧|2ξ€Έ|||ξ€·π‘Šπœ“,πœ‘π‘“ξ€Έξ…ž|||2(𝑧)log1βˆ’|𝑧|2⟢0(2.13) as |𝑧|β†’1, completing the proof.

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

3. Compactness of π‘Šπœ“,πœ‘ from 𝐻∞ into ℬlog and ℬlog,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 πœ“ be analytic on 𝔻, πœ‘ an analytic self-map of 𝔻, 1β‰€π‘β‰€βˆž. The operator π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log is compact if and only if for any bounded sequence {𝑓𝑛}π‘›βˆˆβ„• in 𝐻𝑝 which converges to zero uniformly on compact subsets of 𝔻, we have β€–π‘Šπœ“,πœ‘π‘“π‘›β€–β„¬logβ†’0 as π‘›β†’βˆž.

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 ℬlog,0 is compact if and only if it is bounded and satisfies the following: lim|𝑧|β†’1supπ‘“βˆˆπΎξ€·1βˆ’|𝑧|2ξ€Έ||π‘“ξ…ž||2(𝑧)log1βˆ’|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 π‘Žβˆˆπ”» and, for π‘§βˆˆπ”», defineπ‘“π‘Žξ‚΅(𝑧)=1βˆ’|π‘Ž|21βˆ’ξ‚Άπ‘Žπ‘§1/2,π‘”π‘Ž(𝑧)=1βˆ’|π‘Ž|21βˆ’π‘Žπ‘§.(3.2)

Theorem 3.3. Let πœ“ be analytic on 𝔻, πœ‘ an analytic self-map of 𝔻, and assume that π‘Šπœ“,πœ‘βˆΆπ»βˆžβ†’β„¬log is bounded. Then the following conditions are equivalent: (a)π‘Šπœ“,πœ‘βˆΆπ»βˆžβ†’β„¬log is compact.(b)lim|πœ‘(𝑀)|β†’1β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀)‖ℬlog=0 and lim|πœ‘(𝑀)|β†’1β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀)‖ℬlog=0.(c)lim|πœ‘(𝑀)|β†’1((1βˆ’|𝑀|2)|πœ“(𝑀)πœ‘β€²(𝑀)|/1βˆ’|πœ‘(𝑀)|2)log(2/(1βˆ’|𝑀|2))=0 and lim|πœ‘(𝑀)|β†’1(1βˆ’|𝑀|2)|πœ“β€²(𝑀)|log(2/(1βˆ’|𝑀|2))=0.(d)limπ‘›β†’βˆžβ€–πœ“πœ‘π‘›β€–β„¬log=0 and lim|πœ‘(𝑀)|β†’1(1βˆ’|𝑀|2)|πœ“ξ…ž(𝑀)|log(2/(1βˆ’|𝑀|2))=0.

Proof. We begin by showing that (a), (b), and (c) are equivalent.
(a)β‡’(b) Suppose that π‘Šπœ“,πœ‘ is compact and that {𝑀𝑛} is a sequence in 𝔻 such that |πœ‘(𝑀𝑛)|β†’1 as π‘›β†’βˆž. Since the sequences {π‘“πœ‘(𝑀𝑛)} and {π‘”πœ‘(𝑀𝑛)} are bounded in 𝐻∞ and converge to 0 uniformly on compact subsets of 𝔻, by Lemma 3.1, it follows that β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀𝑛)‖ℬlogβ†’0 and β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀𝑛)‖ℬlogβ†’0 as π‘›β†’βˆž.
(b)β‡’(c) Assume that (b) holds. Fix π‘€βˆˆπ”». A straightforward calculation shows that ξ€·πœ“ξ€·π‘“πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…ž(𝑀)=πœ“ξ…ž(𝑀)+πœ“(𝑀)πœ‘(𝑀)πœ‘ξ…ž(𝑀)2ξ‚€||||1βˆ’πœ‘(𝑀)2,ξ€·πœ“ξ€·π‘”πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…ž(𝑀)=πœ“ξ…ž(𝑀)+πœ“(𝑀)πœ‘(𝑀)πœ‘ξ…ž(𝑀)||||1βˆ’πœ‘(𝑀)2.(3.3) Eliminating πœ“β€²(𝑀), we obtain that πœ“(𝑀)πœ‘(𝑀)πœ‘ξ…ž(𝑀)2ξ‚€||||1βˆ’πœ‘(𝑀)2=ξ€·πœ“ξ€·π‘”πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…žξ€·πœ“ξ€·π‘“(𝑀)βˆ’πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…ž(𝑀).(3.4) Thus, for |πœ‘(𝑀)|>π‘Ÿβˆˆ(0,1), ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)22log1βˆ’|𝑀|2≀2π‘Ÿξ‚€β€–β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀)‖‖ℬlog+β€–β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀)‖‖ℬlog.(3.5) Taking the limit as |πœ‘(𝑀)|β†’1, we deduce that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)22log1βˆ’|𝑀|2⟢0.(3.6) On the other hand, using (3.3), we obtain that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||2(𝑀)log1βˆ’|𝑀|2β‰€β€–β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀)‖‖ℬlog+ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)22log1βˆ’|𝑀|2.(3.7) Taking the limit as |πœ‘(𝑀)|β†’1, we obtain that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||2(𝑀)log1βˆ’|𝑀|2⟢0,(3.8) proving (c).
(c)β‡’(a) Suppose that (c) holds. Let {𝑓𝑛} be a bounded sequence in 𝐻∞ converging to 0 uniformly on compact subsets of 𝔻. Set 𝐢=supπ‘›βˆˆβ„•β€–π‘“π‘›β€–βˆž. Then, given πœ€>0, there exists π‘Ÿβˆˆ(0,1) such that for |πœ‘(𝑀)|>π‘Ÿ, ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)22log1βˆ’|𝑀|2<πœ€,ξ€·2𝐢1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||2(𝑀)log1βˆ’|𝑀|2<πœ€.2𝐢(3.9) Then, for π‘€βˆˆπ”», noting that (1βˆ’|πœ‘(𝑀)|2)|π‘“ξ…žπ‘›(πœ‘(𝑀))|β‰€β€–π‘“π‘›β€–βˆžβ‰€πΆ, we obtain that ξ€·1βˆ’|𝑀|2ξ€Έ|||ξ€·πœ“ξ€·π‘“π‘›βˆ˜πœ‘ξ€Έξ€Έξ…ž|||2(𝑀)log1βˆ’|𝑀|2≀1βˆ’|𝑀|2||πœ“ξ€Έξ€·ξ…ž(𝑀)𝑓𝑛||+||πœ“(πœ‘(𝑀))(𝑀)π‘“ξ…žπ‘›(πœ‘(𝑀))πœ‘ξ…ž||ξ€Έ2(𝑀)log1βˆ’|𝑀|2≀1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||‖‖𝑓(𝑀)π‘›β€–β€–βˆž2log1βˆ’|𝑀|2+ξ€·1βˆ’|𝑀|2ξ€Έξ‚€||||1βˆ’πœ‘(𝑀)2||πœ“(𝑀)π‘“ξ…žπ‘›(πœ‘(𝑀))πœ‘ξ…ž||(𝑀)||πœ‘||1βˆ’(𝑀)22log1βˆ’|𝑀|2≀𝐢1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||+ξ€·(𝑀)1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)2ξƒ­2log1βˆ’|𝑀|2.(3.10) Thus, for |πœ‘(𝑀)|>π‘Ÿ, we have ξ€·1βˆ’|𝑀|2ξ€Έ|||ξ€·πœ“ξ€·π‘“π‘›βˆ˜πœ‘ξ€Έξ€Έξ…ž|||2(𝑀)log1βˆ’|𝑀|2<πœ€.(3.11) On the other hand, for |πœ‘(𝑀)|β‰€π‘Ÿ, ξ€·1βˆ’|𝑀|2ξ€Έ|||ξ€·πœ“ξ€·π‘“π‘›βˆ˜πœ‘ξ€Έξ€Έξ…ž|||2(𝑀)log1βˆ’|𝑀|2≀||π‘“β€–πœ“β€–π‘›||(πœ‘(𝑀))+πœŽπœ“,πœ‘||π‘“ξ…žπ‘›||(πœ‘(𝑀)).(3.12) Thus, by the uniform convergence to 0 of 𝑓𝑛 and π‘“ξ…žπ‘› on compact sets, we see that (3.11) holds also in this case for 𝑛 sufficiently large. Hence, β€–πœ“(π‘“π‘›βˆ˜πœ‘)β€–β‰€πœ€ for 𝑛 sufficiently large. Since |πœ“(0)𝑓𝑛(πœ‘(0))|β†’0, we deduce that β€–πœ“(π‘“π‘›βˆ˜πœ‘)‖ℬlogβ†’0 as π‘›β†’βˆž. Therefore, π‘Šπœ“,πœ‘ is compact.
(a) β‡’ (d) Suppose that π‘Šπœ“,πœ‘ is compact. Since the sequence {𝑝𝑛} defined by 𝑝𝑛(𝑧)=𝑧𝑛, π‘§βˆˆπ”» is bounded in 𝐻∞ and converges to 0 uniformly on compact subsets, by Lemma 3.1, it follows that β€–πœ“πœ‘π‘›β€–β„¬log=β€–π‘Šπœ“,πœ‘π‘π‘›β€–β„¬logβ†’0 as π‘›β†’βˆž. The second condition in (d) follows from the equivalence of (a) and (c).
(d) β‡’ (c) Fix πœ€>0 and choose 𝑁>2 such that β€–πœ“πœ‘π‘›β€–β„¬log<πœ€/2 for all 𝑛β‰₯𝑁 and (1βˆ’|𝑀|2)|πœ“β€²(𝑀)|log(2/(1βˆ’π‘€|2))<πœ€/2 for |πœ‘(𝑀)|>1βˆ’1/𝑁.
For |πœ‘(𝑀)|>1βˆ’1/𝑁, there exists 𝑛>𝑁 such that π‘§βˆˆΞ”π‘›. Using the product rule, we may write the following πœ“(𝑀)π‘›πœ‘(𝑀)π‘›βˆ’1πœ‘ξ…ž(𝑀)=(πœ“πœ‘π‘›)ξ…ž(𝑀)βˆ’πœ“ξ…ž(𝑀)πœ‘(𝑀)𝑛,(3.13) so that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)π‘›πœ‘(𝑀)π‘›βˆ’1πœ‘ξ…ž||2(𝑀)log1βˆ’|𝑀|2β‰€β€–πœ“πœ‘π‘›β€–β„¬log+ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž(𝑀)πœ‘(𝑀)𝑛||2log1βˆ’|𝑀|2.(3.14) The left-hand side of (3.14) can be written as ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)2ξ‚€||||1βˆ’πœ‘(𝑀)2𝑛||||πœ‘(𝑀)π‘›βˆ’12log1βˆ’|𝑀|2,(3.15) which, by (2.3), is bounded below by 1𝑒1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)22log1βˆ’|𝑀|2.(3.16) Thus, from (3.14), we deduce that 1𝑒1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)||||1βˆ’πœ‘(𝑀)22log1βˆ’|𝑀|2β‰€β€–πœ“πœ‘π‘›β€–β„¬log+ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||2(𝑀)log1βˆ’|𝑀|2<πœ€,(3.17) proving (c). The equivalence of statements (a)–(d) is now established.

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

Theorem 3.4. Let πœ“ be an analytic function on 𝔻, and let πœ‘ be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator π‘Šπœ“,πœ‘βˆΆπ»βˆžβ†’β„¬log,0 is compact.(b)For each integer 𝑛β‰₯0, πœ“πœ‘π‘›βˆˆβ„¬log,0 and limπ‘›β†’βˆžβ€–πœ“πœ‘π‘›β€–β„¬log=0.(c)πœ“βˆˆβ„¬log,0 andlim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2=0.(3.18)

Proof. (a) β‡’ (b) is immediate.
(b) β‡’ (c) It suffices to show that if (b) holds, then ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2⟢0(3.19) as |𝑧|β†’1. Fix πœ€>0 and let 𝑁>2 be an integer such that β€–πœ“πœ‘π‘›β€–β„¬log<πœ€/𝑒 for all 𝑛β‰₯𝑁. Observe that, since for π‘§βˆˆπ”», ||πœ“(𝑧)πœ‘ξ…ž(||≀||(𝑧)πœ“πœ‘)ξ…ž(||+||πœ“π‘§)ξ…ž(||≀||(𝑧)πœ‘(𝑧)πœ“πœ‘)ξ…ž(||+||πœ“π‘§)ξ…ž(||𝑧),(3.20) and, by assumption, the functions πœ“πœ‘ and πœ“ are in ℬlog,0, we have lim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||2(𝑧)log1βˆ’|𝑧|2=0.(3.21) Therefore, there is π‘Ÿβˆˆ(0,1), such that for |𝑧|>π‘Ÿ, ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||2(𝑧)log1βˆ’|𝑧|2<(2π‘βˆ’1)πœ€π‘2.(3.22) Thus, if π‘§βˆˆπΈπ‘, and |𝑧|>π‘Ÿ, then ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2≀𝑁2ξ€·2π‘βˆ’11βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž(||2𝑧)log1βˆ’|𝑧|2<πœ€.(3.23) On the other hand, if π‘§βˆ‰πΈπ‘, then there exists 𝑛>𝑁 such that π‘§βˆˆΞ”π‘›, so, as shown in the proof of (d) implies (c) of Theorem 3.3, we have ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2ξ‚Έβ‰€π‘’β€–πœ“πœ‘π‘›β€–β„¬log+ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž(||2𝑧)log1βˆ’|𝑧|2ξ‚Ήξ€·<πœ€+𝑒1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||2(𝑧)log1βˆ’|𝑧|2βŸΆπœ€,(3.24) as |𝑧|β†’1. Since πœ€ is arbitrary, the result follows.
(c) β‡’ (a) Let {𝑓𝑛} be a bounded sequence in 𝐻∞ converging to 0 uniformly on compact subsets, and let 𝐢=supπ‘›βˆˆβ„•β€–π‘“π‘›β€–βˆž. We wish to show that π‘Šπœ“,πœ‘π‘“π‘›βˆˆβ„¬log,0 and β€–π‘Šπœ“,πœ‘π‘“π‘›β€–β„¬logβ†’0 as π‘›β†’βˆž. As shown in the proof of (c) implies (a) of Theorem 3.3, for π‘§βˆˆπ”» and π‘›βˆˆβ„•, ξ€·1βˆ’|𝑧|2ξ€Έ|||ξ€·π‘Šπœ“,πœ‘π‘“π‘›ξ€Έξ…ž|||2(𝑧)log1βˆ’|𝑧|2≀𝐢1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||2(𝑧)log1βˆ’|𝑧|2ξ€·+𝐢1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)||||1βˆ’πœ‘(𝑧)22log1βˆ’|𝑧|2⟢0,(3.25) as |𝑧|β†’1. Thus, π‘Šπœ“,πœ‘π‘“π‘›βˆˆβ„¬log,0. The convergence to 0 of β€–π‘Šπœ“,πœ‘π‘“π‘›β€–β„¬log is proved as in the case of the operator mapping into ℬlog.

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

Corollary 3.5. Let πœ“ be an analytic function on 𝔻, and let πœ‘ be an analytic self-map of 𝔻. The following statements are equivalent. (a)The operator π‘Šπœ“,πœ‘βˆΆπ»βˆžβ†’β„¬log,0 is bounded.(b)The operator π‘Šπœ“,πœ‘βˆΆπ»βˆžβ†’β„¬log,0 is compact.(c)For each integer 𝑛β‰₯0, πœ“πœ‘π‘›βˆˆβ„¬log,0 and limπ‘›β†’βˆžβ€–πœ“πœ‘π‘›β€–β„¬log=0.(d)πœ“βˆˆβ„¬log,0 and lim|𝑧|β†’1((1βˆ’|𝑧|2)|πœ“(𝑧)πœ‘β€²(𝑧)|/(1βˆ’|πœ‘(𝑧)|2))log(2/(1βˆ’|𝑧|2))=0.

In the special cases when πœ‘ is the identity, respectively, πœ“ is identically 1, we obtain the following results.

Corollary 3.6. Let πœ“ be analytic on 𝔻. The following statements are equivalent: (a)π‘€πœ“βˆΆπ»βˆžβ†’β„¬log is bounded,(b)π‘€πœ“βˆΆπ»βˆžβ†’β„¬log,0 is bounded,(c)πœ“ is identically 0.

Corollary 3.7. Let πœ‘ be an analytic self map of 𝔻. Then the following statements are equivalent: (a)πΆπœ‘βˆΆπ»βˆžβ†’β„¬log is bounded,(b)supπ‘›βˆˆβ„•β€–πœ‘π‘›β€–β„¬log<∞,(c)supπ‘§βˆˆπ”»((1βˆ’|𝑧|2)|πœ‘β€²(𝑧)|/(1βˆ’|πœ‘(𝑧)|2))log(2/(1βˆ’|𝑧|2))<∞.

Corollary 3.8. Let πœ‘ be an analytic self map of 𝔻. Then the following statements are equivalent: (a)πΆπœ‘βˆΆπ»βˆžβ†’β„¬log is compact,(b)limπ‘›β†’βˆžβ€–πœ‘π‘›β€–β„¬log=0,(c)lim|πœ‘(𝑧)|β†’1((1βˆ’|𝑧|2)|πœ‘β€²(𝑧)|/(1βˆ’|πœ‘(𝑧)|2))log(2/(1βˆ’|𝑧|2))=0.

Corollary 3.9. Let πœ‘ be an analytic self map of 𝔻. Then the following statements are equivalent: (a)πΆπœ‘βˆΆπ»βˆžβ†’β„¬log,0 is bounded,(b)πΆπœ‘βˆΆπ»βˆžβ†’β„¬log,0 is compact,(c)πœ‘βˆˆβ„¬log,0 and limπ‘›β†’βˆžβ€–πœ‘π‘›β€–β„¬log=0,(d)lim|𝑧|β†’1((1βˆ’|𝑧|2)|πœ‘β€²(𝑧)|/(1βˆ’|πœ‘(𝑧)|2))log(2/(1βˆ’|𝑧|2))=0.

4. π‘Šπœ“,πœ‘ from 𝐻𝑝(1≀𝑝<∞) into ℬlog

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 π‘“βˆˆπ»π‘, π‘§βˆˆπ”», ||||≀𝑓(𝑧)‖𝑓‖𝐻𝑝1βˆ’|𝑧|2ξ€Έ1/𝑝.(4.1)

Lemma 4.2 (See [13]). Let 0<𝑝<∞. Then for any π‘“βˆˆπ»π‘, π‘§βˆˆπ”», ||π‘“ξ…ž||(𝑧)≀𝐢‖𝑓‖𝐻𝑝1βˆ’|𝑧|2ξ€Έ1+1/𝑝.(4.2)

Fix 1≀𝑝<∞ and π‘Žβˆˆπ”». For π‘§βˆˆπ”», define the functionsπ‘“π‘Žξ€·(𝑧)=1βˆ’|π‘Ž|2ξ€Έ2βˆ’1/𝑝1βˆ’ξ€Έπ‘Žπ‘§2,π‘”π‘Žξ€·(𝑧)=1βˆ’|π‘Ž|2ξ€Έ2ξ€·1βˆ’ξ€Έπ‘Žπ‘§2+1/𝑝.(4.3) Then π‘“π‘Ž,π‘”π‘Žβˆˆπ»π‘ and the norms β€–π‘“π‘Žβ€–π»π‘ and β€–π‘”π‘Žβ€–π»π‘ are bounded by constants only dependent of 𝑝. In addition, a straightforward calculation shows thatπ‘“π‘Ž(π‘Ž)=π‘”π‘Ž1(π‘Ž)=ξ€·1βˆ’|π‘Ž|2ξ€Έ1/𝑝,π‘“ξ…žπ‘Ž2(π‘Ž)=π‘Žξ€·1βˆ’|π‘Ž|2ξ€Έ1+1/𝑝,π‘”ξ…žπ‘Ž(π‘Ž)=(2+1/𝑝)π‘Žξ€·1βˆ’|π‘Ž|2ξ€Έ1+1/𝑝.(4.4)

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

Theorem 4.3. Let 1≀𝑝<∞, πœ“ analytic on 𝔻 and let πœ‘ be an analytic self-map of 𝔻. Then the following conditions are equivalent: (a)π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log is bounded,(b)πœ“πœ‘βˆˆβ„¬log, 𝐴=supπ‘€βˆˆπ”»β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀)‖ℬlog<∞ and 𝐡=supπ‘€βˆˆπ”»β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀)‖ℬlog<∞,(c)π‘₯πœ“,πœ‘=supπ‘§βˆˆπ”»ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21/𝑝2log1βˆ’|𝑧|2𝑦<∞,πœ“,πœ‘=supπ‘§βˆˆπ”»ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21+1/𝑝2log1βˆ’|𝑧|2<∞.(4.5)

Proof. (a) β‡’ (b) Assume that π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log is bounded. Then πœ“πœ‘βˆˆβ„¬log and for each π‘€βˆˆπ”», β€–β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀)‖‖ℬlogβ‰€β€–β€–π‘Šπœ“,πœ‘β€–β€–β€–β€–π‘“πœ‘(𝑀)β€–β€–π»π‘β€–β€–π‘Šβ‰€πΆπœ“,πœ‘β€–β€–,β€–β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀)‖‖ℬlogβ‰€β€–β€–π‘Šπœ“,πœ‘β€–β€–β€–β€–π‘”πœ‘(𝑀)β€–β€–π»π‘β€–β€–π‘Šβ‰€πΆπœ“,πœ‘β€–β€–,(4.6) for some constant 𝐢, so 𝐴 and 𝐡 are finite.
(b) β‡’ (c) Suppose that πœ“πœ‘βˆˆβ„¬log, and the quantities 𝐴 and 𝐡 are finite. From (4.4), for π‘€βˆˆπ”», we have ξ€·πœ“ξ€·π‘“πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…žπœ“(𝑀)=ξ…ž(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝+2πœ“(𝑀)πœ‘ξ…ž(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝,(4.7) whence ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝2log1βˆ’|𝑀|2≀1βˆ’|𝑀|2ξ€Έ2log1βˆ’|𝑀|2|||ξ€·πœ“ξ€·π‘“πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…ž|||ξ€·(𝑀)+21βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2β‰€β€–β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀)‖‖ℬlogξ€·+21βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2≀𝐴+21βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2.(4.8) Moreover, ξ€·πœ“ξ€·π‘”πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…žπœ“(𝑀)=ξ…ž(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝+(2+1/𝑝)πœ“(𝑀)πœ‘ξ…ž(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝.(4.9) Therefore, subtracting (4.7) from (4.9) and taking the modulus, we obtain 1𝑝||πœ“(𝑀)πœ‘ξ…ž||(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝≀|||ξ€·πœ“ξ€·π‘“πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…ž|||+|||ξ€·πœ“ξ€·π‘”(𝑀)πœ‘(𝑀)βˆ˜πœ‘ξ€Έξ€Έξ…ž|||(𝑀),(4.10) which yields that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2≀𝑝(𝐴+𝐡).(4.11) Consequently, from (4.8), we deduce that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝2log1βˆ’|𝑀|2≀(1+2𝑝)𝐴+2𝑝𝐡.(4.12) Taking the supremum over all π‘€βˆˆπ”», we see that π‘₯πœ“,πœ‘ is finite.
Fix π‘Ÿβˆˆ(0,1). If |πœ‘(𝑀)|>π‘Ÿ, then from (4.11) we have ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2<𝑝(𝐴+𝐡)π‘Ÿ.(4.13) On the other hand, since πœ“πœ‘βˆˆβ„¬log, if |πœ‘(𝑀)|β‰€π‘Ÿ, then, ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2≀1βˆ’|𝑀|2ξ€Έ||(πœ“πœ‘)ξ…ž||(𝑀)ξ€·1βˆ’π‘Ÿ2ξ€Έ1+1/𝑝2log1βˆ’|𝑀|2+ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||(𝑀)πœ‘(𝑀)ξ€·1βˆ’π‘Ÿ2ξ€Έξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝2log1βˆ’|𝑀|2β‰€β€–πœ“πœ‘β€–ξ€·1βˆ’π‘Ÿ2ξ€Έ1+1/𝑝+π‘₯πœ“,πœ‘1βˆ’π‘Ÿ2<∞.(4.14) Taking the supremum over all π‘€βˆˆπ”», it follows that π‘¦πœ“,πœ‘ is finite as well.
(c) β‡’ (a) Suppose that π‘₯πœ“,πœ‘ and π‘¦πœ“,πœ‘ are finite. For arbitrary 𝑧 in 𝔻 and π‘“βˆˆπ»π‘, by Lemmas 4.1 and 4.2, we have ξ€·1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2|||ξ€·π‘Šπœ“,πœ‘π‘“ξ€Έξ…ž|||≀(𝑧)1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2ξ€·||πœ“ξ…ž||||𝑓||+||𝑓(𝑧)(πœ‘(𝑧))ξ…ž||||πœ“(πœ‘(𝑧))(𝑧)πœ‘ξ…ž||ξ€Έβ‰€βŽ›βŽœβŽœβŽœβŽξ€·(𝑧)1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21/𝑝2log1βˆ’|𝑧|2ξ€·+𝐢1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21+1/𝑝2log1βˆ’|𝑧|2βŽžβŽŸβŽŸβŽŸβŽ β€–π‘“β€–π»π‘β‰€ξ€·π‘₯πœ“,πœ‘+πΆπ‘¦πœ“,πœ‘ξ€Έβ€–π‘“β€–π»π‘.(4.15)
Taking the supremum over all π‘§βˆˆπ”» and applying Lemma 4.1, we obtain that β€–β€–π‘Šπœ“,πœ‘π‘“β€–β€–β„¬log=||||+β€–β€–π‘Šπœ“(0)𝑓(πœ‘(0))πœ“,πœ‘π‘“β€–β€–β‰€βŽ›βŽœβŽœβŽœβŽ||||πœ“(0)ξ‚€||||1βˆ’πœ‘(0)21/𝑝+π‘₯πœ“,πœ‘+πΆπ‘¦πœ“,πœ‘βŽžβŽŸβŽŸβŽŸβŽ β€–π‘“β€–π»π‘.(4.16) The boundedness of the operator π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log follows by taking the supremum over all π‘“βˆˆπ»π‘.

Theorem 4.4. Let 1≀𝑝<∞, πœ“ analytic on 𝔻, πœ‘ an analytic self-map of 𝔻, and assume that π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log is bounded. Then the following conditions are equivalent: (a)π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log is compact,(b)lim|πœ‘(𝑀)|β†’1β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀)‖ℬlog=0 and lim|πœ‘(𝑀)|β†’1β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀)‖ℬlog=0,(c)lim||||πœ‘(𝑀)β†’1ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2=0,lim||||πœ‘(𝑀)β†’1ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝2log1βˆ’|𝑀|2=0.(4.17)

Proof. (a)β‡’(b) Suppose that π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log is compact. Let {𝑀𝑛} be a sequence in 𝔻 such that limπ‘›β†’βˆž|πœ‘(𝑀𝑛)|=1. Observe that the sequences {π‘“πœ‘(𝑀𝑛)} and {π‘”πœ‘(𝑀𝑛)} are bounded in 𝐻𝑝 and converge to 0 uniformly on compact subsets of 𝔻. By Lemma 3.1, it follows that β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀𝑛)‖ℬlogβ†’0 and β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀𝑛)‖ℬlogβ†’0 as π‘›β†’βˆž, proving (b).
(b)β‡’(c) Assume that the limits in (b) are 0. Using the inequality (4.10), we obtain that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2β‰€π‘ξ‚€β€–β€–π‘Šπœ“,πœ‘π‘“πœ‘(𝑀)‖‖ℬlog+β€–β€–π‘Šπœ“,πœ‘π‘”πœ‘(𝑀)‖‖ℬlog||||πœ‘(𝑀)⟢0(4.18) as |πœ‘(𝑀)|β†’1. Moreover, using (4.8), we deduce that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝2log1βˆ’|𝑀|2⟢0(4.19) as |πœ‘(𝑀)|β†’1.
(c)β‡’(a) Suppose that (c) holds. Let {𝑓𝑛} be a bounded sequence in 𝐻𝑝 converging to 0 uniformly on compact subsets of 𝔻. Set 𝐢=supπ‘›βˆˆβ„•β€–π‘“π‘›β€–π»π‘. Then, given πœ€>0, there exists π‘Ÿβˆˆ(0,1) such that ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2<πœ€,ξ€·2𝐢1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝2log1βˆ’|𝑀|2<πœ€,2𝐢(4.20) for |πœ‘(𝑀)|>π‘Ÿ. Therefore, again by Lemmas 4.1 and 4.2, and (4.20), for |πœ‘(𝑀)|>π‘Ÿ, we have ξ€·1βˆ’|𝑀|2ξ€Έ2log1βˆ’|𝑀|2|||ξ€·πœ“ξ€·π‘“π‘›βˆ˜πœ‘ξ€Έξ€Έξ…ž|||≀(𝑀)1βˆ’|𝑀|2ξ€Έ2log1βˆ’|𝑀|2ξ€·||πœ“ξ…ž(𝑀)𝑓𝑛||+||πœ“(πœ‘(𝑀))(𝑀)π‘“ξ…žπ‘›(πœ‘(𝑀))πœ‘ξ…ž||≀‖‖𝑓(𝑀)π‘›β€–β€–π»π‘βŽ›βŽœβŽœβŽœβŽξ€·1βˆ’|𝑀|2ξ€Έ||πœ“ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21/𝑝2log1βˆ’|𝑀|2+ξ€·1βˆ’|𝑀|2ξ€Έ||πœ“(𝑀)πœ‘ξ…ž||(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝2log1βˆ’|𝑀|2⎞⎟⎟⎟⎠<πœ€.(4.21) On the other hand, for |πœ‘(𝑀)|β‰€π‘Ÿ, by the uniform convergence to 0 of 𝑓𝑛 and π‘“ξ…žπ‘› on compact sets, we have ξ€·1βˆ’|𝑀|2ξ€Έ2log1βˆ’|𝑀|2|||ξ€·πœ“ξ€·π‘“π‘›βˆ˜πœ‘ξ€Έξ€Έξ…ž|||(𝑀)≀π‘₯πœ“,πœ‘||𝑓𝑛||(πœ‘(𝑀))+π‘¦πœ“,πœ‘||π‘“ξ…ž||(πœ‘(𝑀))⟢0,(4.22) as π‘›β†’βˆž. Since |πœ“(0)𝑓𝑛(πœ‘(0))|β†’0, we conclude that β€–π‘Šπœ“,πœ‘π‘“π‘›β€–β„¬logβ†’0 as π‘›β†’βˆž. Consequently, by Lemma 3.1, the operator π‘Šπœ“,πœ‘ is compact.

As a consequence of Theorems 4.3 and 4.4, noting that for π‘€βˆˆπ”»,ξ€·πΆπœ‘π‘”πœ‘(𝑀)ξ€Έξ…ž1(𝑀)=2ξ‚΅12+π‘ξ‚Άξ€·πΆπœ‘π‘“πœ‘(𝑀)ξ€Έξ…ž(𝑀)=(2+1/𝑝)πœ‘ξ…ž(𝑀)πœ‘(𝑀)ξ‚€||||1βˆ’πœ‘(𝑀)21+1/𝑝,(4.23) we obtain the following characterizations of the bounded and the compact composition operators from 𝐻𝑝 into ℬlog.

Corollary 4.5. Let πœ‘ be an analytic self-map of 𝔻, and 1≀𝑝<∞. The following statements are equivalent: (a)πΆπœ‘βˆΆπ»π‘β†’β„¬log is bounded,(b)supπ‘€βˆˆπ”»β€–πΆπœ‘π‘“πœ‘(𝑀)‖ℬlog<∞,(c)supπ‘€βˆˆπ”»β€–πΆπœ‘π‘”πœ‘(𝑀)‖ℬlog<∞,(d)supπ‘€βˆˆπ”»((1βˆ’|𝑀|2)|πœ‘β€²(𝑀)|/(1βˆ’|πœ‘(𝑀)|2)1+1/𝑝)log(2/(1βˆ’|𝑀|2))<∞.

Corollary 4.6. Let πœ‘ be an analytic self-map of 𝔻, and 1≀𝑝<∞. If πΆπœ‘βˆΆπ»π‘β†’β„¬log is bounded, then the following statements are equivalent: (a)πΆπœ‘βˆΆπ»π‘β†’β„¬log is compact,(b)lim|πœ‘(𝑀)|β†’1β€–πΆπœ‘π‘“πœ‘(𝑀)‖ℬlog=0,(c)lim|πœ‘(𝑀)|β†’1β€–πΆπœ‘π‘”πœ‘(𝑀)‖ℬlog=0,(d)lim|πœ‘(𝑀)|β†’1((1βˆ’|𝑀|2)|πœ‘β€²(𝑀)|/(1βˆ’|πœ‘(𝑀)|2)1+1/𝑝)log(2/(1βˆ’|𝑀|2))=0.

5. π‘Šπœ“,πœ‘ from 𝐻𝑝(1≀𝑝<∞) into ℬlog,0

In this section, we characterize the boundedness and the compactness of the weighted composition operators π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬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 πœ‘ is an analytic self-map of the unit disk, πœ“ analytic on 𝔻, and 1≀𝑝<∞. Then, lim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21/𝑝2log1βˆ’|𝑧|2=0(5.1) if and only if πœ“βˆˆβ„¬log,0 and lim||||πœ‘(𝑧)β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21/𝑝2log1βˆ’|𝑧|2=0.(5.2)

Lemma 5.2. Suppose that πœ‘ is an analytic self-map of the unit disk, πœ“ analytic on 𝔻, and 1≀𝑝<∞. Then, lim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21+1/𝑝2log1βˆ’|𝑧|2=0(5.3) if and only if lim|𝑧|β†’1(1βˆ’|𝑧|)2log(2/(1βˆ’|𝑧|2))|πœ“(𝑧)πœ‘β€²(𝑧)|=0 and lim||||πœ‘(𝑧)β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21+1/𝑝2log1βˆ’|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 πœ‘ be an analytic self-map of the unit disk, πœ“βˆˆπ»(𝔻), and 1≀𝑝<∞. Then, the operator π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log,0 is bounded if and only if π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log is bounded, πœ“βˆˆβ„¬log,0, and lim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ2log1βˆ’|𝑧|2||πœ“(𝑧)πœ‘ξ…ž||(𝑧)=0.(5.5)

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

Theorem 5.4. Suppose that πœ‘ is an analytic self-map of the unit disk, πœ“βˆˆπ»(𝔻), and 1≀𝑝<∞. Then, π‘Šπœ“,πœ‘βˆΆπ»π‘β†’β„¬log,0 is compact if and only if lim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21/𝑝2log1βˆ’|𝑧|2=0,lim|𝑧|β†’1ξ€·1βˆ’|𝑧|2ξ€Έ||πœ“(𝑧)πœ‘ξ…ž||(𝑧)ξ‚€||||1βˆ’πœ‘(𝑧)21+1/𝑝2log1βˆ’|𝑧|2=0.(5.6)

Proof. First, we assume that π‘Š