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Abstract and Applied Analysis
VolumeΒ 2011, Article IDΒ 989625, 10 pages
http://dx.doi.org/10.1155/2011/989625
Research Article

Weighted Composition Operators from Weighted Bergman Spaces to Weighted-Type Spaces on the Upper Half-Plane

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11000 Beograd, Serbia
2School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, J & K Katra 182320, India
3Department of Mathematics, University of Jammu, Jammu 180006, India

Received 25 January 2011; Accepted 3 May 2011

Academic Editor: Narcisa C.Β Apreutesei

Copyright © 2011 Stevo Stević et al. 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

Let πœ“ be a holomorphic mapping on the upper half-plane Ξ +={π‘§βˆˆβ„‚βˆΆπ”π‘§>0} and πœ‘ be a holomorphic self-map of Ξ +. We characterize bounded weighted composition operators acting from the weighted Bergman space to the weighted-type space on the upper half-plane. Under a mild condition on πœ“, we also characterize the compactness of these operators.

1. Introduction and Preliminaries

Let Ξ +={π‘§βˆˆβ„‚βˆΆβ„‘π‘§>0} be the upper half-plane, Ξ© a domain in β„‚ or ℂ𝑛, and 𝐻(Ξ©) the space of all holomorphic functions on Ξ©. Let πœ“βˆˆπ»(Ξ©), and letπœ‘ be a holomorphic self-map of Ξ©. Then by π‘Šπœ‘,πœ“(𝑓)(𝑧)=πœ“(π‘“βˆ˜πœ‘)(𝑧),π‘§βˆˆΞ©,(1.1) is defined a linear operator on 𝐻(Ξ©) which is called weighted composition operator. If πœ“(𝑧)=1, then π‘Šπœ‘,πœ“ becomes composition operator and is denoted by πΆπœ‘, and if πœ‘(𝑧)=𝑧, then π‘Šπœ‘,πœ“ becomes multiplication operator and is denoted by π‘€πœ“.

During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in β„‚ or ℂ𝑛 (see, e.g., [1–22] and the references therein). There are many reasons for this interest, for example, it is well known that the surjective isometries of Hardy and Bergman spaces are certain weighted composition operators (see [23, 24]). For some other operators related to weighted composition operators, see [25–30] and the references therein.

While there is a vast literature on composition and weighted composition operators between spaces of holomorphic functions on the unit disk 𝔻, there are few papers on these and related operators on spaces of functions holomorphic in the upper half-plane (see, e.g., [2, 3, 5, 7–9, 11, 12, 16–18, 31] and the references therein). For related results in the setting of the complex plane see also papers [19–21].

The behaviour of composition operators on spaces of functions holomorphic in the upper half-plane is considerably different from the behaviour of composition operators on spaces of functions holomorphic in the unit disk 𝔻. For example, there are holomorphic self-maps of Ξ + which do not induce composition operators on Hardy and Bergman spaces on the upper half-plane, whereas it is a well-known consequence of the Littlewood subordination principle that every holomorphic self-map πœ‘ of 𝔻 induces a bounded composition operator on the Hardy and weighted Bergman spaces on 𝔻. Also, Hardy and Bergman spaces on the upper half-plane do not support compact composition operators (see [3, 5]).

For 0<𝑝<∞ and π›Όβˆˆ(βˆ’1,∞), let 𝔏𝑝(Ξ +,𝑑𝐴𝛼) denote the collection of all Lebesgue 𝑝-integrable functions π‘“βˆΆΞ +β†’β„‚ such that ξ€œΞ +||||𝑓(𝑧)𝑝𝑑𝐴𝛼(𝑧)<∞,(1.2) where 𝑑𝐴𝛼1(𝑧)=πœ‹(𝛼+1)(2ℑ𝑧)𝛼𝑑𝐴(𝑧),(1.3)𝑑𝐴(𝑧)=𝑑π‘₯𝑑𝑦, and 𝑧=π‘₯+𝑖𝑦.

Let π’œπ‘π›Ό(Ξ +)=𝔏𝑝(Ξ +,𝑑𝐴𝛼)∩𝐻(Ξ +). For 1≀𝑝<∞,π’œπ‘π›Ό(Ξ +) is a Banach space with the norm defined by β€–π‘“β€–π’œπ‘π›Ό(Ξ +)=ξ‚΅ξ€œΞ +||||𝑓(𝑧)𝑝𝑑𝐴𝛼(𝑧)1/𝑝<∞.(1.4) With this norm π’œπ‘π›Ό(Ξ +) becomes a Banach space when 𝑝β‰₯1, while for π‘βˆˆ(0,1) it is a FrΓ©chet space with the translation invariant metric 𝑑(𝑓,𝑔)=β€–π‘“βˆ’π‘”β€–π‘π’œπ‘π›Ό(Ξ +),𝑓,π‘”βˆˆπ’œπ‘π›Όξ€·Ξ +ξ€Έ.(1.5)

Recall that for every π‘“βˆˆπ’œπ‘π›Ό(Ξ +) the following estimate holds: ||||𝑓(π‘₯+𝑖𝑦)π‘β‰€πΆβ€–π‘“β€–π‘π’œπ‘π›Ό(Ξ +)𝑦𝛼+2,(1.6) where 𝐢 is a positive constant independent of 𝑓.

Let 𝛽>0. The weighted-type space (or growth space) on the upper half-plane π’œβˆžπ›½(Ξ +) consists of all π‘“βˆˆπ»(Ξ +) such that β€–π‘“β€–π’œβˆžπ›½(Ξ +)=supπ‘§βˆˆΞ +(ℑ𝑧)𝛽||||𝑓(𝑧)<∞.(1.7) It is easy to check that π’œβˆžπ›½(Ξ +) is a Banach space with the norm defined above. For weighted-type spaces on the unit disk, polydisk, or the unit ball see, for example, papers [10, 32, 33] and the references therein.

Given two Banach spaces π‘Œ and 𝑍, we recall that a linear map π‘‡βˆΆπ‘Œβ†’π‘ is bounded if 𝑇(𝐸)βŠ‚π‘ is bounded for every bounded subset 𝐸 of π‘Œ. In addition, we say that 𝑇 is compact if 𝑇(𝐸)βŠ‚π‘ is relatively compact for every bounded set πΈβŠ‚π‘Œ.

In this paper, we consider the boundedness and compactness of weighted composition operators acting from π’œπ‘π›Ό(Ξ +) to the weighted-type space π’œβˆžπ›½(Ξ +). Related results on the unit disk and the unit ball can be found, for example, in [6, 13, 15].

Throughout this paper, constants are denoted by 𝐢; they are positive and may differ from one occurrence to the other. The notation π‘Žβͺ―𝑏 means that there is a positive constant 𝐢 such that π‘Žβ‰€πΆπ‘. Moreover, if both π‘Žβͺ―𝑏 and 𝑏βͺ―π‘Ž hold, then one says that π‘Žβ‰π‘.

2. Main Results

The boundedness and compactness of the weighted composition operator π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) are characterized in this section.

Theorem 2.1. Let 1≀𝑝<∞,𝛼>βˆ’1,𝛽>0,πœ“βˆˆπ»(Ξ +), and let πœ‘ be a holomorphic self-map of Ξ +. Then π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded if and only if π‘€βˆΆ=supπ‘§βˆˆΞ +(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝||||πœ“(𝑧)<∞.(2.1) Moreover, if the operator π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded then the following asymptotic relationship holds: β€–β€–π‘Šπœ‘,πœ“β€–β€–π’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +)≍𝑀.(2.2)

Proof. First suppose that (2.1) holds. Then for any π‘§βˆˆΞ + and π‘“βˆˆπ’œπ‘π›Ό(Ξ +), by (1.6) we have (ℑ𝑧)𝛽||ξ€·π‘Šπœ‘,πœ“π‘“ξ€Έ||(𝑧)=(ℑ𝑧)𝛽||||||||βͺ―πœ“(𝑧)𝑓(πœ‘(𝑧))(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝||||πœ“(𝑧)β€–π‘“β€–π’œπ‘π›Ό(Ξ +),(2.3) and so by (2.1), π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded and moreover β€–β€–π‘Šπœ‘,πœ“β€–β€–π’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +)βͺ―𝑀.(2.4) Conversely suppose π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded. Consider the function 𝑓𝑀(𝑧)=(ℑ𝑀)(𝛼+2)/π‘ξ€·π‘§βˆ’π‘€ξ€Έ(2𝛼+4)/𝑝,π‘€βˆˆΞ +.(2.5) Then π‘“π‘€βˆˆπ’œπ‘π›Ό(Ξ +) and moreover supπ‘€βˆˆΞ +β€–π‘“π‘€β€–π’œπ‘π›Ό(Ξ +)βͺ―1 (see, e.g., Lemma 1 in [18]).
Thus the boundedness of π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) implies that (ℑ𝑧)𝛽||||||π‘“πœ“(𝑧)𝑀||β‰€β€–β€–π‘Š(πœ‘(𝑧))πœ‘,πœ“π‘“π‘€β€–β€–π’œβˆžπ›½(Ξ +)β‰Όβ€–β€–π‘Šπœ‘,πœ“β€–β€–π’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +),(2.6) for every 𝑧,π‘€βˆˆΞ +. In particular, if π‘§βˆˆΞ + is fixed then for 𝑀=πœ‘(𝑧), we get (ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝||||βͺ―β€–β€–π‘Šπœ“(𝑧)πœ‘,πœ“β€–β€–π’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +).(2.7) Since π‘§βˆˆΞ + is arbitrary, (2.1) follows and moreover β€–β€–π‘Šπ‘€βͺ―πœ‘,πœ“β€–β€–π’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +).(2.8) If π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded then from (2.4) and (2.8) asymptotic relationship (2.2) follows.

Corollary 2.2. Let 1≀𝑝<∞,𝛼>βˆ’1, and 𝛽>0 be such that 𝛽𝑝β‰₯𝛼+2 and πœ“βˆˆπ»(Ξ +). Then π‘€πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded if and only if πœ“βˆˆπ‘‹, where ξƒ―π’œπ‘‹=βˆžπ›½βˆ’((𝛼+2)/𝑝)ξ€·Ξ +𝐻if𝛼+2<𝛽𝑝,βˆžξ€·Ξ +ξ€Έif𝛼+2=𝛽𝑝.(2.9)

Example 2.3. Let 1≀𝑝<∞,𝛼>βˆ’1 and 𝛽>0 be such that 𝛽𝑝β‰₯𝛼+2 and π‘€βˆˆΞ +. Let πœ“π‘€ be a holomorphic map of Ξ + defined as πœ“π‘€βŽ§βŽͺ⎨βŽͺ⎩1(𝑧)=ξ€·π‘§βˆ’π‘€ξ€Έπ›½βˆ’((𝛼+2)/𝑝)if𝛼+2<𝛽𝑝,β„‘π‘€π‘§βˆ’π‘€if𝛼+2=𝛽𝑝.(2.10) For 𝑧=π‘₯+𝑖𝑦 and 𝑀=𝑒+𝑖𝑣 in Ξ +, we have supπ‘§βˆˆΞ +(ℑ𝑧)π›½βˆ’(𝛼+2)/𝑝||πœ“π‘€||(𝑧)=sup𝑧=π‘₯+π‘–π‘¦βˆˆΞ +π‘¦π›½βˆ’(𝛼+2)/𝑝(π‘₯βˆ’π‘’)2+(𝑦+𝑣)2ξ€Έ(π›½π‘βˆ’(𝛼+2))/2𝑝≀sup𝑧=π‘₯+π‘–π‘¦βˆˆΞ +π‘¦π›½βˆ’(𝛼+2)/𝑝(𝑦+𝑣)π›½βˆ’(𝛼+2)/𝑝≀1.(2.11) Thus πœ“π‘€βˆˆπ’œβˆžπ›½βˆ’(𝛼+2)/𝑝(Ξ +) if 𝛼+2<𝛽𝑝. Similarly πœ“π‘€βˆˆπ»βˆž(Ξ +) if 𝛼+2=𝛽𝑝. By Corollary 2.2, it follows that π‘€πœ“π‘€βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded.

Corollary 2.4. Let 1≀𝑝<∞,𝛼>βˆ’1,𝛽>0, and letπœ‘ be a holomorphic self-map of Ξ +. Then πΆπœ‘βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded if and only if supπ‘§βˆˆΞ +(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝<∞.(2.12)

Corollary 2.5. Let πœ‘ be the linear fractional map πœ‘(𝑧)=π‘Žπ‘§+𝑏𝑐𝑧+𝑑,π‘Ž,𝑏,𝑐,π‘‘βˆˆβ„,π‘Žπ‘‘βˆ’π‘π‘>0.(2.13) Then necessary and sufficient condition that πΆπœ‘βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded is that 𝑐=0 and 𝛼+2=𝛽𝑝.

Proof. Assume that πΆπœ‘βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded. Then supπ‘§βˆˆΞ +(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝=sup𝑧=π‘₯+π‘–π‘¦βˆˆΞ +ξ€·(𝑐π‘₯+𝑑)2+𝑐2𝑦2ξ€Έ(𝛼+2)/𝑝𝑦𝛽(π‘Žπ‘‘βˆ’π‘π‘)(𝛼+2)/𝑝𝑦(𝛼+2)/𝑝,(2.14) which is finite only if 𝑐=0 and 𝛼+2=𝛽𝑝.
Conversely, if 𝑐=0 and 𝛼+2=𝛽𝑝, then from (2.13) we get π‘Žβ‰ 0, and by some calculation supπ‘§βˆˆΞ +(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝=ξ‚€π‘‘π‘Žξ‚π›½<∞.(2.15) Hence πΆπœ‘βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded.

Corollary 2.6. Let 1≀𝑝<∞,𝛼>βˆ’1, and 𝛽>0 be such that 𝛽𝑝=𝛼+2. Let πœ‘ be a holomorphic self-map of Ξ + and πœ“=(πœ‘ξ…ž)𝛽. Then the weighted composition operator π‘Šπœ‘,πœ“ acts boundedly from π’œπ‘π›Ό(Ξ +) to π’œβˆžπ›½(Ξ +).

Proof. By Theorem 2.1, π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded if and only if supπ‘§βˆˆΞ +(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))𝛽||πœ‘ξ…ž||(𝑧)𝛽<∞.(2.16) By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map πœ‘ of Ξ + and all π‘§βˆˆΞ +||πœ‘ξ…ž||(𝑧)≀1β„‘πœ‘(𝑧)ℑ𝑧,(2.17) where the equality holds when πœ‘ is a MΓΆbius transformation given by (2.13). From (2.17), condition (2.16) follows and consequently the boundedness of the operator π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +).

Corollary 2.6 enables us to show that there exist 1≀𝑝<∞,𝛼>βˆ’1,𝛽>0, and holomorphic maps πœ‘ and πœ“ of the upper half-plane Ξ + such that neither πΆπœ‘βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) nor π‘€πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded, but π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded.

Example 2.7. Let 1≀𝑝<∞,𝛼>βˆ’1, and 𝛽>0 be such that 𝛽𝑝=𝛼+2. Let πœ‘(𝑧)=(π‘Žπ‘§+𝑏)/(𝑐𝑧+𝑑),π‘Ž,𝑏,𝑐,π‘‘βˆˆβ„,π‘Žπ‘‘βˆ’π‘π‘>0, and 𝑐≠0. Then by Corollary 2.5, πΆπœ‘βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is not bounded. On the other hand, if ξ€·πœ‘πœ“(𝑧)=ξ…žξ€Έ(𝑧)𝛽=ξ‚΅π‘Žπ‘‘βˆ’π‘π‘(𝑐𝑧+𝑑)2𝛽,(2.18) then πœ“βˆ‰π»βˆž(Ξ +) and so by Corollary 2.2, π‘€πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is not bounded. However, by Corollary 2.6, we have that π‘Šπœ‘,(πœ‘ξ…ž)π›½βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is bounded.
The next Schwartz-type lemma characterizes compact weighted composition operators π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) and it follows from standard arguments ([4]).

Lemma 2.8. Let 1≀𝑝<∞,𝛼>βˆ’1,𝛽>0,πœ“βˆˆπ»(Ξ +), and letπœ‘ be a holomorphic self-map of  Π+. Then π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is compact if and only if, for any bounded sequence (𝑓𝑛)π‘›βˆˆβ„•βŠ‚π’œπ‘π›Ό(Ξ +) converging to zero on compacts of  Π+, one has limπ‘›β†’βˆžβ€–β€–π‘Šπœ‘,πœ“π‘“π‘›β€–β€–π’œβˆžπ›½(Ξ +)=0.(2.19)

Theorem 2.9. Let 1≀𝑝<∞,𝛼>βˆ’1,𝛽>0,πœ“βˆˆπ»(Ξ +) and πœ‘ be a holomorphic self-map of   Ξ +. If π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is compact, then limπ‘Ÿβ†’0supβ„‘πœ‘(𝑧)<π‘Ÿ(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝||||πœ“(𝑧)=0.(2.20)

Proof. Suppose π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is compact and (2.20) does not hold. Then there is a 𝛿>0 and a sequence (𝑧𝑛)π‘›βˆˆβ„•βŠ‚Ξ + such that β„‘πœ‘(𝑧𝑛)β†’0 and ξ€·β„‘π‘§π‘›ξ€Έπ›½ξ€·ξ€·π‘§β„‘πœ‘π‘›ξ€Έξ€Έ(𝛼+2)/𝑝||πœ“ξ€·π‘§π‘›ξ€Έ||>𝛿(2.21) for all π‘›βˆˆβ„•. Let 𝑀𝑛=πœ‘(𝑧𝑛), π‘›βˆˆβ„•, and 𝑓𝑛(𝑧)=ℑ𝑀𝑛(𝛼+2)/π‘ξ€·π‘§βˆ’π‘€π‘›ξ€Έ(2𝛼+4)/𝑝,π‘›βˆˆβ„•.(2.22) Then 𝑓𝑛 is a norm bounded sequence and 𝑓𝑛→0 on compacts of Ξ + as β„‘πœ‘(𝑧𝑛)β†’0. By Lemma 2.8 it follows that limπ‘›β†’βˆžβ€–β€–π‘Šπœ‘,πœ“π‘“π‘›β€–β€–π’œβˆžπ›½(Ξ +)=0.(2.23) On the other hand, β€–β€–π‘Šπœ‘,πœ“π‘“π‘›β€–β€–π’œβˆžπ›½(Ξ +)β‰₯ℑ𝑧𝑛𝛽||ξ€·π‘Šπœ‘,πœ“π‘“π‘›π‘§ξ€Έξ€·π‘›ξ€Έ||=ℑ𝑧𝑛𝛽||πœ“ξ€·π‘§π‘›ξ€Έ||||π‘“π‘›ξ€·πœ‘ξ€·π‘§π‘›||=ℑ𝑧𝑛𝛽2(2𝛼+4)/π‘ξ€·ξ€·π‘§β„‘πœ‘π‘›ξ€Έξ€Έ(𝛼+2)/𝑝||πœ“ξ€·π‘§π‘›ξ€Έ||>𝛿2(2𝛼+4)/𝑝,(2.24) which is a contradiction. Hence (2.20) must hold, as claimed.

Before we formulate and prove a converse of Theorem 2.9, we define, for every π‘Ž,π‘βˆˆ(0,∞) such that π‘Ž<𝑏, the following subset of Ξ +: Ξ“π‘Ž,𝑏=ξ€½π‘§βˆˆΞ +ξ€ΎβˆΆπ‘Žβ‰€β„‘π‘§β‰€π‘.(2.25)

Theorem 2.10. Let 1≀𝑝<∞,𝛼>βˆ’1,𝛽>0,πœ“βˆˆπ»(Ξ +), and letπœ‘ be a holomorphic self-map of Ξ + and π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) be bounded. Suppose that πœ“βˆˆπ’œβˆžπ›½(Ξ +) and (ℑ𝑧)𝛽|πœ“(𝑧)|β†’0 as |β„œπœ‘(𝑧)|β†’βˆž within Ξ“π‘Ž,𝑏 for all π‘Ž and 𝑏,0<π‘Ž<𝑏<∞. Then π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is compact if condition (2.20) holds.

Proof. Assume (2.20) holds. Then for each πœ€>0, there is an 𝑀1>0 such that (ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝||||πœ“(𝑧)<πœ€,wheneverβ„‘πœ‘(𝑧)<𝑀1.(2.26) Let (𝑓𝑛)π‘›βˆˆβ„• be a sequence in π’œπ‘π›Ό(Ξ +) such that supπ‘›βˆˆβ„•β€–π‘“π‘›β€–π’œπ‘π›Ό(Ξ +)≀𝑀 and 𝑓𝑛→0 uniformly on compact subsets of Ξ + as π‘›β†’βˆž. Thus for π‘§βˆˆΞ + such that β„‘πœ‘(𝑧)<𝑀1 and each π‘›βˆˆβ„•, we have (ℑ𝑧)𝛽||||||π‘“πœ“(𝑧)𝑛||βͺ―(πœ‘(𝑧))(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))(𝛼+2)/𝑝||||β€–β€–π‘“πœ“(𝑧)π‘›β€–β€–π’œπ‘π›Ό(Ξ +)<πœ€π‘€.(2.27)
From estimate (1.6) we have ||𝑓𝑛(||βͺ―‖‖𝑓𝑧)π‘›β€–β€–π’œπ‘π›Ό(Ξ +)(ℑ𝑧)(𝛼+2)/𝑝βͺ―𝑀(ℑ𝑧)(𝛼+2)/𝑝.(2.28) Thus there is an 𝑀2>𝑀1 such that ||𝑓𝑛||(πœ‘(𝑧))<πœ€,(2.29) whenever β„‘πœ‘(𝑧)>𝑀2. Hence for π‘§βˆˆΞ + such that β„‘πœ‘(𝑧)>𝑀2 and each π‘›βˆˆβ„• we have (ℑ𝑧)𝛽||||||π‘“πœ“(𝑧)𝑛||(πœ‘(𝑧))<πœ€β€–πœ“β€–π’œβˆžπ›½(Ξ +).(2.30) If 𝑀1β‰€β„‘πœ‘(𝑧)≀𝑀2, then by the assumption there is an 𝑀3>0 such that (ℑ𝑧)𝛽|πœ“(𝑧)|<πœ€, whenever |β„œπœ‘(𝑧)|>𝑀3. Therefore, for each π‘›βˆˆβ„• we have (ℑ𝑧)𝛽||||||π‘“πœ“(𝑧)𝑛(||β€–β€–π‘“πœ‘(𝑧))βͺ―πœ€π‘›β€–β€–π’œπ‘π›Ό(Ξ +)(β„‘πœ‘(𝑧))(𝛼+2)/π‘π‘€β‰€πœ€π‘€1(𝛼+2)/𝑝,(2.31) whenever 𝑀1β‰€β„‘πœ‘(𝑧)≀𝑀2 and |β„œπœ‘(𝑧)|>𝑀3.
If 𝑀1β‰€β„‘πœ‘(𝑧)≀𝑀2 and |β„œπœ‘(𝑧)|≀𝑀3, then there exists some 𝑛0βˆˆβ„• such that |𝑓𝑛(πœ‘(𝑧))|<πœ€ for all 𝑛β‰₯𝑛0, and so (ℑ𝑧)𝛽||||||π‘“πœ“(𝑧)𝑛||(πœ‘(𝑧))<πœ€β€–πœ“β€–π’œβˆžπ›½(Ξ +).(2.32) Combining (2.27)–(2.32), we have that β€–β€–π‘Šπœ‘,πœ“π‘“π‘›β€–β€–π’œβˆžπ›½(Ξ +)<πœ€πΆ,(2.33) for 𝑛β‰₯𝑛0 and some 𝐢>0 independent of 𝑛. Since πœ€ is an arbitrary positive number, by Lemma 2.8, it follows that π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is compact.

Example 2.11. Let 1≀𝑝<∞,𝛼>βˆ’1, and 𝛽>0 be such that 𝛼+2=𝛽𝑝. Let πœ‘(𝑧)=𝑧+𝑖 and πœ“(𝑧)=1/(𝑧+𝑖)𝛽, then β„œπœ‘(𝑧)=π‘₯ and β„‘πœ‘(𝑧)=𝑦+1. It is easy to see that πœ“βˆˆπ’œβˆžπ›½(Ξ +). Beside this, for π‘§βˆˆΞ“π‘Ž,𝑏, we have (ℑ𝑧)𝛽||||=π‘¦πœ“(𝑧)𝛽π‘₯2+(𝑦+1)2𝛽/2≀𝑏𝛽π‘₯2+π‘Ž2𝛽/2⟢0asβ„œπœ‘(𝑧)=π‘₯⟢∞.(2.34)
Also supπ‘§βˆˆΞ +(ℑ𝑧)𝛽(β„‘πœ‘(𝑧))𝛽||||πœ“(𝑧)=sup𝑧=π‘₯+π‘–π‘¦βˆˆΞ +𝑦𝛽(𝑦+1)𝛽1ξ€·π‘₯2+(𝑦+1)2𝛽/2≀1<∞,(2.35) and the set {π‘§βˆΆβ„‘πœ‘(𝑧)<1} is empty. Thus πœ‘ and πœ“ satisfy all the assumptions of Theorem 2.10, and so π‘Šπœ‘,πœ“βˆΆπ’œπ‘π›Ό(Ξ +)β†’π’œβˆžπ›½(Ξ +) is compact.

Acknowledgments

The work is partially supported by the Serbian Ministry of Science, projects III 41025 and III 44006. The work of the second author is a part of the research project sponsored by National Board of Higher Mathematics (NBHM)/DAE, India (Grant no. 48/4/2009/R&D-II/426).

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