Abstract

Let 𝜓 be a holomorphic mapping on the upper half-plane Π+={𝑧∈ℂ∶𝔍𝑧>0} and 𝜑 be a holomorphic self-map of Π+. 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 𝜓, we also characterize the compactness of these operators.

1. Introduction and Preliminaries

Let Π+={𝑧∈ℂ∶ℑ𝑧>0} be the upper half-plane, Ω a domain in ℂ or ℂ𝑛, and 𝐻(Ω) the space of all holomorphic functions on Ω. Let 𝜓∈𝐻(Ω), and let𝜑 be a holomorphic self-map of Ω. Then by 𝑊𝜑,𝜓(𝑓)(𝑧)=𝜓(𝑓∘𝜑)(𝑧),𝑧∈Ω,(1.1) is defined a linear operator on 𝐻(Ω) which is called weighted composition operator. If 𝜓(𝑧)=1, then 𝑊𝜑,𝜓 becomes composition operator and is denoted by 𝐶𝜑, and if 𝜑(𝑧)=𝑧, then 𝑊𝜑,𝜓 becomes multiplication operator and is denoted by 𝑀𝜓.

During the past few decades, composition operators and weighted composition operators have been studied extensively on spaces of holomorphic functions on various domains in ℂ 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 Π+ 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 𝜑 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 𝛼∈(−1,∞), let 𝔏𝑝(Π+,𝑑𝐴𝛼) denote the collection of all Lebesgue 𝑝-integrable functions 𝑓∶Π+→ℂ such that Π+||||𝑓(𝑧)𝑝𝑑𝐴𝛼(𝑧)<∞,(1.2) where 𝑑𝐴𝛼1(𝑧)=𝜋(𝛼+1)(2ℑ𝑧)𝛼𝑑𝐴(𝑧),(1.3)𝑑𝐴(𝑧)=𝑑𝑥𝑑𝑦, and 𝑧=𝑥+𝑖𝑦.

Let 𝒜𝑝𝛼(Π+)=𝔏𝑝(Π+,𝑑𝐴𝛼)∩𝐻(Π+). For 1≤𝑝<∞,𝒜𝑝𝛼(Π+) is a Banach space with the norm defined by ‖𝑓‖𝒜𝑝𝛼(Π+)=Π+||||𝑓(𝑧)𝑝𝑑𝐴𝛼(𝑧)1/𝑝<∞.(1.4) With this norm 𝒜𝑝𝛼(Π+) becomes a Banach space when 𝑝≥1, while for 𝑝∈(0,1) it is a Fréchet space with the translation invariant metric 𝑑(𝑓,𝑔)=‖𝑓−𝑔‖𝑝𝒜𝑝𝛼(Π+),𝑓,𝑔∈𝒜𝑝𝛼Π+.(1.5)

Recall that for every 𝑓∈𝒜𝑝𝛼(Π+) the following estimate holds: ||||𝑓(𝑥+𝑖𝑦)𝑝≤𝐶‖𝑓‖𝑝𝒜𝑝𝛼(Π+)𝑦𝛼+2,(1.6) where 𝐶 is a positive constant independent of 𝑓.

Let 𝛽>0. The weighted-type space (or growth space) on the upper half-plane ğ’œâˆžğ›½(Π+) consists of all 𝑓∈𝐻(Π+) such that â€–ğ‘“â€–ğ’œâˆžğ›½(Π+)=sup𝑧∈Π+(ℑ𝑧)𝛽||||𝑓(𝑧)<∞.(1.7) It is easy to check that ğ’œâˆžğ›½(Π+) 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 𝑌 and 𝑍, we recall that a linear map 𝑇∶𝑌→𝑍 is bounded if 𝑇(𝐸)⊂𝑍 is bounded for every bounded subset 𝐸 of 𝑌. In addition, we say that 𝑇 is compact if 𝑇(𝐸)⊂𝑍 is relatively compact for every bounded set 𝐸⊂𝑌.

In this paper, we consider the boundedness and compactness of weighted composition operators acting from 𝒜𝑝𝛼(Π+) to the weighted-type space ğ’œâˆžğ›½(Π+). 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 𝐶; they are positive and may differ from one occurrence to the other. The notation ğ‘Žâª¯ğ‘ means that there is a positive constant 𝐶 such that ğ‘Žâ‰¤ğ¶ğ‘. Moreover, if both ğ‘Žâª¯ğ‘ and ğ‘âª¯ğ‘Ž hold, then one says that ğ‘Žâ‰ğ‘.

2. Main Results

The boundedness and compactness of the weighted composition operator 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) are characterized in this section.

Theorem 2.1. Let 1≤𝑝<∞,𝛼>−1,𝛽>0,𝜓∈𝐻(Π+), and let 𝜑 be a holomorphic self-map of Π+. Then 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded if and only if 𝑀∶=sup𝑧∈Π+(ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝||||𝜓(𝑧)<∞.(2.1) Moreover, if the operator 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded then the following asymptotic relationship holds: ‖‖𝑊𝜑,𝜓‖‖𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+)≍𝑀.(2.2)

Proof. First suppose that (2.1) holds. Then for any 𝑧∈Π+ and 𝑓∈𝒜𝑝𝛼(Π+), by (1.6) we have (ℑ𝑧)𝛽||𝑊𝜑,𝜓𝑓||(𝑧)=(ℑ𝑧)𝛽||||||||⪯𝜓(𝑧)𝑓(𝜑(𝑧))(ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝||||𝜓(𝑧)‖𝑓‖𝒜𝑝𝛼(Π+),(2.3) and so by (2.1), 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded and moreover ‖‖𝑊𝜑,𝜓‖‖𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+)⪯𝑀.(2.4) Conversely suppose 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded. Consider the function 𝑓𝑤(𝑧)=(ℑ𝑤)(𝛼+2)/𝑝𝑧−𝑤(2𝛼+4)/𝑝,𝑤∈Π+.(2.5) Then 𝑓𝑤∈𝒜𝑝𝛼(Π+) and moreover sup𝑤∈Π+‖𝑓𝑤‖𝒜𝑝𝛼(Π+)⪯1 (see, e.g., Lemma 1 in [18]).
Thus the boundedness of 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) implies that (ℑ𝑧)𝛽||||||𝑓𝜓(𝑧)𝑤||≤‖‖𝑊(𝜑(𝑧))𝜑,ğœ“ğ‘“ğ‘¤â€–â€–ğ’œâˆžğ›½(Π+)≼‖‖𝑊𝜑,𝜓‖‖𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+),(2.6) for every 𝑧,𝑤∈Π+. In particular, if 𝑧∈Π+ is fixed then for 𝑤=𝜑(𝑧), we get (ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝||||⪯‖‖𝑊𝜓(𝑧)𝜑,𝜓‖‖𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+).(2.7) Since 𝑧∈Π+ is arbitrary, (2.1) follows and moreover ‖‖𝑊𝑀⪯𝜑,𝜓‖‖𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+).(2.8) If 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded then from (2.4) and (2.8) asymptotic relationship (2.2) follows.

Corollary 2.2. Let 1≤𝑝<∞,𝛼>−1, and 𝛽>0 be such that 𝛽𝑝≥𝛼+2 and 𝜓∈𝐻(Π+). Then 𝑀𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded if and only if 𝜓∈𝑋, where 𝒜𝑋=âˆžğ›½âˆ’((𝛼+2)/𝑝)Π+𝐻if𝛼+2<𝛽𝑝,∞Π+if𝛼+2=𝛽𝑝.(2.9)

Example 2.3. Let 1≤𝑝<∞,𝛼>−1 and 𝛽>0 be such that 𝛽𝑝≥𝛼+2 and 𝑤∈Π+. Let 𝜓𝑤 be a holomorphic map of Π+ defined as ğœ“ğ‘¤âŽ§âŽªâŽ¨âŽªâŽ©1(𝑧)=𝑧−𝑤𝛽−((𝛼+2)/𝑝)if𝛼+2<𝛽𝑝,ℑ𝑤𝑧−𝑤if𝛼+2=𝛽𝑝.(2.10) For 𝑧=𝑥+𝑖𝑦 and 𝑤=𝑢+𝑖𝑣 in Π+, we have sup𝑧∈Π+(ℑ𝑧)𝛽−(𝛼+2)/𝑝||𝜓𝑤||(𝑧)=sup𝑧=𝑥+𝑖𝑦∈Π+𝑦𝛽−(𝛼+2)/𝑝(𝑥−𝑢)2+(𝑦+𝑣)2(𝛽𝑝−(𝛼+2))/2𝑝≤sup𝑧=𝑥+𝑖𝑦∈Π+𝑦𝛽−(𝛼+2)/𝑝(𝑦+𝑣)𝛽−(𝛼+2)/𝑝≤1.(2.11) Thus ğœ“ğ‘¤âˆˆğ’œâˆžğ›½âˆ’(𝛼+2)/𝑝(Π+) if 𝛼+2<𝛽𝑝. Similarly ğœ“ğ‘¤âˆˆğ»âˆž(Π+) if 𝛼+2=𝛽𝑝. By Corollary 2.2, it follows that 𝑀𝜓𝑤∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded.

Corollary 2.4. Let 1≤𝑝<∞,𝛼>−1,𝛽>0, and let𝜑 be a holomorphic self-map of Π+. Then 𝐶𝜑∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded if and only if sup𝑧∈Π+(ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝<∞.(2.12)

Corollary 2.5. Let 𝜑 be the linear fractional map 𝜑(𝑧)=ğ‘Žğ‘§+𝑏𝑐𝑧+𝑑,ğ‘Ž,𝑏,𝑐,𝑑∈ℝ,ğ‘Žğ‘‘âˆ’ğ‘ğ‘>0.(2.13) Then necessary and sufficient condition that 𝐶𝜑∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded is that 𝑐=0 and 𝛼+2=𝛽𝑝.

Proof. Assume that 𝐶𝜑∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded. Then sup𝑧∈Π+(ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝=sup𝑧=𝑥+𝑖𝑦∈Π+(𝑐𝑥+𝑑)2+𝑐2𝑦2(𝛼+2)/𝑝𝑦𝛽(ğ‘Žğ‘‘âˆ’ğ‘ğ‘)(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 ğ‘Žâ‰ 0, and by some calculation sup𝑧∈Π+(ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝=î‚€ğ‘‘ğ‘Žî‚ğ›½<∞.(2.15) Hence 𝐶𝜑∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded.

Corollary 2.6. Let 1≤𝑝<∞,𝛼>−1, and 𝛽>0 be such that 𝛽𝑝=𝛼+2. Let 𝜑 be a holomorphic self-map of Π+ and 𝜓=(ğœ‘î…ž)𝛽. Then the weighted composition operator 𝑊𝜑,𝜓 acts boundedly from 𝒜𝑝𝛼(Π+) to ğ’œâˆžğ›½(Π+).

Proof. By Theorem 2.1, 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded if and only if sup𝑧∈Π+(ℑ𝑧)𝛽(ℑ𝜑(𝑧))𝛽||ğœ‘î…ž||(𝑧)𝛽<∞.(2.16) By the Schwarz-Pick theorem on the upper half-plane we have that for every holomorphic self-map 𝜑 of Π+ and all 𝑧∈Π+||ğœ‘î…ž||(𝑧)≤1ℑ𝜑(𝑧)ℑ𝑧,(2.17) where the equality holds when 𝜑 is a Möbius transformation given by (2.13). From (2.17), condition (2.16) follows and consequently the boundedness of the operator 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+).

Corollary 2.6 enables us to show that there exist 1≤𝑝<∞,𝛼>−1,𝛽>0, and holomorphic maps 𝜑 and 𝜓 of the upper half-plane Π+ such that neither 𝐶𝜑∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) nor 𝑀𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded, but 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded.

Example 2.7. Let 1≤𝑝<∞,𝛼>−1, and 𝛽>0 be such that 𝛽𝑝=𝛼+2. Let 𝜑(𝑧)=(ğ‘Žğ‘§+𝑏)/(𝑐𝑧+𝑑),ğ‘Ž,𝑏,𝑐,𝑑∈ℝ,ğ‘Žğ‘‘âˆ’ğ‘ğ‘>0, and 𝑐≠0. Then by Corollary 2.5, 𝐶𝜑∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is not bounded. On the other hand, if 𝜑𝜓(𝑧)=(𝑧)𝛽=î‚µğ‘Žğ‘‘âˆ’ğ‘ğ‘(𝑐𝑧+𝑑)2𝛽,(2.18) then ğœ“âˆ‰ğ»âˆž(Π+) and so by Corollary 2.2, 𝑀𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is not bounded. However, by Corollary 2.6, we have that 𝑊𝜑,(ğœ‘î…ž)𝛽∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is bounded.
The next Schwartz-type lemma characterizes compact weighted composition operators 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) and it follows from standard arguments ([4]).

Lemma 2.8. Let 1≤𝑝<∞,𝛼>−1,𝛽>0,𝜓∈𝐻(Π+), and let𝜑 be a holomorphic self-map of  Π+. Then 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is compact if and only if, for any bounded sequence (𝑓𝑛)𝑛∈ℕ⊂𝒜𝑝𝛼(Π+) converging to zero on compacts of  Π+, one has limğ‘›â†’âˆžâ€–â€–ğ‘Šğœ‘,ğœ“ğ‘“ğ‘›â€–â€–ğ’œâˆžğ›½(Π+)=0.(2.19)

Theorem 2.9. Let 1≤𝑝<∞,𝛼>−1,𝛽>0,𝜓∈𝐻(Π+) and 𝜑 be a holomorphic self-map of   Π+. If 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is compact, then lim𝑟→0supℑ𝜑(𝑧)<𝑟(ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝||||𝜓(𝑧)=0.(2.20)

Proof. Suppose 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is compact and (2.20) does not hold. Then there is a 𝛿>0 and a sequence (𝑧𝑛)𝑛∈ℕ⊂Π+ such that ℑ𝜑(𝑧𝑛)→0 and ℑ𝑧𝑛𝛽𝑧ℑ𝜑𝑛(𝛼+2)/𝑝||𝜓𝑧𝑛||>𝛿(2.21) for all 𝑛∈ℕ. Let 𝑤𝑛=𝜑(𝑧𝑛), 𝑛∈ℕ, and 𝑓𝑛(𝑧)=ℑ𝑤𝑛(𝛼+2)/𝑝𝑧−𝑤𝑛(2𝛼+4)/𝑝,𝑛∈ℕ.(2.22) Then 𝑓𝑛 is a norm bounded sequence and 𝑓𝑛→0 on compacts of Π+ as ℑ𝜑(𝑧𝑛)→0. By Lemma 2.8 it follows that limğ‘›â†’âˆžâ€–â€–ğ‘Šğœ‘,ğœ“ğ‘“ğ‘›â€–â€–ğ’œâˆžğ›½(Π+)=0.(2.23) On the other hand, ‖‖𝑊𝜑,ğœ“ğ‘“ğ‘›â€–â€–ğ’œâˆžğ›½(Π+)≥ℑ𝑧𝑛𝛽||𝑊𝜑,𝜓𝑓𝑛𝑧𝑛||=ℑ𝑧𝑛𝛽||𝜓𝑧𝑛||||𝑓𝑛𝜑𝑧𝑛||=ℑ𝑧𝑛𝛽2(2𝛼+4)/𝑝𝑧ℑ𝜑𝑛(𝛼+2)/𝑝||𝜓𝑧𝑛||>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 ğ‘Ž,𝑏∈(0,∞) such that ğ‘Ž<𝑏, the following subset of Π+: Î“ğ‘Ž,𝑏=𝑧∈Π+î€¾âˆ¶ğ‘Žâ‰¤â„‘ğ‘§â‰¤ğ‘.(2.25)

Theorem 2.10. Let 1≤𝑝<∞,𝛼>−1,𝛽>0,𝜓∈𝐻(Π+), and let𝜑 be a holomorphic self-map of Π+ and 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) be bounded. Suppose that ğœ“âˆˆğ’œâˆžğ›½(Π+) and (ℑ𝑧)𝛽|𝜓(𝑧)|→0 as |ℜ𝜑(𝑧)|→∞ within Î“ğ‘Ž,𝑏 for all ğ‘Ž and 𝑏,0<ğ‘Ž<𝑏<∞. Then 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is compact if condition (2.20) holds.

Proof. Assume (2.20) holds. Then for each 𝜀>0, there is an 𝑀1>0 such that (ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝||||𝜓(𝑧)<𝜀,wheneverℑ𝜑(𝑧)<𝑀1.(2.26) Let (𝑓𝑛)𝑛∈ℕ be a sequence in 𝒜𝑝𝛼(Π+) such that sup𝑛∈ℕ‖𝑓𝑛‖𝒜𝑝𝛼(Π+)≤𝑀 and 𝑓𝑛→0 uniformly on compact subsets of Π+ as ğ‘›â†’âˆž. Thus for 𝑧∈Π+ such that ℑ𝜑(𝑧)<𝑀1 and each 𝑛∈ℕ, we have (ℑ𝑧)𝛽||||||𝑓𝜓(𝑧)𝑛||⪯(𝜑(𝑧))(ℑ𝑧)𝛽(ℑ𝜑(𝑧))(𝛼+2)/𝑝||||‖‖𝑓𝜓(𝑧)𝑛‖‖𝒜𝑝𝛼(Π+)<𝜀𝑀.(2.27)
From estimate (1.6) we have ||𝑓𝑛(||⪯‖‖𝑓𝑧)𝑛‖‖𝒜𝑝𝛼(Π+)(ℑ𝑧)(𝛼+2)/𝑝⪯𝑀(ℑ𝑧)(𝛼+2)/𝑝.(2.28) Thus there is an 𝑀2>𝑀1 such that ||𝑓𝑛||(𝜑(𝑧))<𝜀,(2.29) whenever ℑ𝜑(𝑧)>𝑀2. Hence for 𝑧∈Π+ such that ℑ𝜑(𝑧)>𝑀2 and each 𝑛∈ℕ we have (ℑ𝑧)𝛽||||||𝑓𝜓(𝑧)𝑛||(𝜑(𝑧))<ğœ€â€–ğœ“â€–ğ’œâˆžğ›½(Π+).(2.30) If 𝑀1≤ℑ𝜑(𝑧)≤𝑀2, then by the assumption there is an 𝑀3>0 such that (ℑ𝑧)𝛽|𝜓(𝑧)|<𝜀, whenever |ℜ𝜑(𝑧)|>𝑀3. Therefore, for each 𝑛∈ℕ we have (ℑ𝑧)𝛽||||||𝑓𝜓(𝑧)𝑛(||‖‖𝑓𝜑(𝑧))⪯𝜀𝑛‖‖𝒜𝑝𝛼(Π+)(ℑ𝜑(𝑧))(𝛼+2)/𝑝𝑀≤𝜀𝑀1(𝛼+2)/𝑝,(2.31) whenever 𝑀1≤ℑ𝜑(𝑧)≤𝑀2 and |ℜ𝜑(𝑧)|>𝑀3.
If 𝑀1≤ℑ𝜑(𝑧)≤𝑀2 and |ℜ𝜑(𝑧)|≤𝑀3, then there exists some 𝑛0∈ℕ such that |𝑓𝑛(𝜑(𝑧))|<𝜀 for all 𝑛≥𝑛0, and so (ℑ𝑧)𝛽||||||𝑓𝜓(𝑧)𝑛||(𝜑(𝑧))<ğœ€â€–ğœ“â€–ğ’œâˆžğ›½(Π+).(2.32) Combining (2.27)–(2.32), we have that ‖‖𝑊𝜑,ğœ“ğ‘“ğ‘›â€–â€–ğ’œâˆžğ›½(Π+)<𝜀𝐶,(2.33) for 𝑛≥𝑛0 and some 𝐶>0 independent of 𝑛. Since 𝜀 is an arbitrary positive number, by Lemma 2.8, it follows that 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) is compact.

Example 2.11. Let 1≤𝑝<∞,𝛼>−1, and 𝛽>0 be such that 𝛼+2=𝛽𝑝. Let 𝜑(𝑧)=𝑧+𝑖 and 𝜓(𝑧)=1/(𝑧+𝑖)𝛽, then ℜ𝜑(𝑧)=𝑥 and ℑ𝜑(𝑧)=𝑦+1. It is easy to see that ğœ“âˆˆğ’œâˆžğ›½(Π+). Beside this, for ğ‘§âˆˆÎ“ğ‘Ž,𝑏, we have (ℑ𝑧)𝛽||||=𝑦𝜓(𝑧)𝛽𝑥2+(𝑦+1)2𝛽/2≤𝑏𝛽𝑥2+ğ‘Ž2𝛽/2⟶0asℜ𝜑(𝑧)=ğ‘¥âŸ¶âˆž.(2.34)
Also sup𝑧∈Π+(ℑ𝑧)𝛽(ℑ𝜑(𝑧))𝛽||||𝜓(𝑧)=sup𝑧=𝑥+𝑖𝑦∈Π+𝑦𝛽(𝑦+1)𝛽1𝑥2+(𝑦+1)2𝛽/2≤1<∞,(2.35) and the set {𝑧∶ℑ𝜑(𝑧)<1} is empty. Thus 𝜑 and 𝜓 satisfy all the assumptions of Theorem 2.10, and so 𝑊𝜑,𝜓∶𝒜𝑝𝛼(Π+)â†’ğ’œâˆžğ›½(Π+) 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).