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

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 ℂ^𝑛, 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 𝑊_𝜑,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 𝑊_𝜑,𝜓 becomes composition operator and is denoted by 𝐶_𝜑, and if 𝜑(𝑧)=𝑧, then 𝑊_𝜑,𝜓 becomes multiplication operator and is denoted by 𝑀_𝜓.

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 ℂ^𝑛 (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 𝔏^𝑝(Π⁺,𝑑𝐴_𝛼) denote the collection of all Lebesgue 𝑝-integrable functions 𝑓∶Π⁺→ℂ such that _Π+||𝑓(𝑧)||^𝑝𝑑𝐴_𝛼(𝑧)<∞,(1.2) where 𝑑𝐴_𝛼(𝑧)=1𝜋(𝛼+1)(2ℑ𝑧)^𝛼𝑑𝐴(𝑧),(1.3)𝑑𝐴(𝑧)=𝑑𝑥𝑑𝑦, and 𝑧=𝑥+𝑖𝑦.

Let 𝒜^𝑝𝛼(Π⁺)=𝔏^𝑝(Π⁺,𝑑𝐴_𝛼)∩𝐻(Π⁺). For 1≤𝑝<∞,𝒜^𝑝𝛼(Π⁺) is a Banach space with the norm defined by ‖𝑓‖_𝒜^𝑝𝛼_(Π⁺₎=_Π+||𝑓(𝑧)||^𝑝𝑑𝐴_𝛼(𝑧)^1/𝑝<∞.(1.4) With this norm 𝒜^𝑝𝛼(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‖^𝑝𝒜_{𝑝𝛼(Π+)},𝑓,𝑔∈𝒜^𝑝𝛼Π⁺.(1.5)

Recall that for every 𝑓∈𝒜^𝑝𝛼(Π⁺) the following estimate holds: ||𝑓(𝑥+𝑖𝑦)||^𝑝≤𝐶‖𝑓‖^{𝑝𝒜𝑝𝛼(Π+)}𝑦^𝛼+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 𝒜^∞𝛽(Π⁺) consists of all 𝑓∈𝐻(Π⁺) such that ‖𝑓‖_𝒜^∞𝛽_(Π⁺₎=sup_𝑧∈Π+(ℑ𝑧)^𝛽||𝑓(𝑧)||<∞.(1.7) It is easy to check that 𝒜^∞𝛽(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 𝒜^𝑝𝛼(Π⁺) to the weighted-type space 𝒜^∞𝛽(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 𝑎⪯𝑏 means that there is a positive constant 𝐶 such that 𝑎≤𝐶𝑏. Moreover, if both 𝑎⪯𝑏 and 𝑏⪯𝑎 hold, then one says that 𝑎≍𝑏.

2. Main Results

The boundedness and compactness of the weighted composition operator 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded if and only if 𝑀∶=sup_𝑧∈Π+(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^{(𝛼+2)/𝑝}||𝜓(𝑧)||<∞.(2.1) Moreover, if the operator 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded then the following asymptotic relationship holds: ‖‖𝑊_𝜑,𝜓‖‖_{𝒜𝑝𝛼(Π+)→𝒜∞𝛽(Π+)}≍𝑀.(2.2)

Proof. First suppose that (2.1) holds. Then for any 𝑧∈Π⁺ and 𝑓∈𝒜^𝑝𝛼(Π⁺), by (1.6) we have (ℑ𝑧)^𝛽||𝑊_𝜑,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), 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded and moreover ‖‖𝑊_𝜑,𝜓‖‖_{𝒜𝑝𝛼(Π+)→𝒜∞𝛽(Π+)}⪯𝑀.(2.4) Conversely suppose 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded. Consider the function 𝑓_𝑤(𝑧)=(ℑ𝑤)^{(𝛼+2)/𝑝}𝑧−𝑤^{(2𝛼+4)/𝑝},𝑤∈Π⁺.(2.5) Then 𝑓_𝑤∈𝒜^𝑝𝛼(Π⁺) and moreover sup_𝑤∈Π⁺‖𝑓_𝑤‖_𝒜^𝑝𝛼_(Π⁺₎⪯1 (see, e.g., Lemma 1 in [18]).
Thus the boundedness of 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) implies that (ℑ𝑧)^𝛽||𝜓(𝑧)||||𝑓_𝑤(𝜑(𝑧))||≤‖‖𝑊_𝜑,𝜓𝑓_𝑤‖‖_{𝒜∞𝛽(Π+)}≼‖‖𝑊_𝜑,𝜓‖‖_{𝒜𝑝𝛼(Π+)→𝒜∞𝛽(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}||𝜓(𝑧)||⪯‖‖𝑊_𝜑,𝜓‖‖_{𝒜𝑝𝛼(Π+)→𝒜∞𝛽(Π+)}.(2.7) Since 𝑧∈Π⁺ is arbitrary, (2.1) follows and moreover 𝑀⪯‖‖𝑊_𝜑,𝜓‖‖_{𝒜𝑝𝛼(Π+)→𝒜∞𝛽(Π+)}.(2.8) If 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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 𝑀_𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded if and only if 𝜓∈𝑋, where 𝑋=𝒜^{∞𝛽−((𝛼+2)/𝑝)}𝐻Π⁺if𝛼+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 𝜓_𝑤 be a holomorphic map of Π⁺ defined as 𝜓_𝑤(𝑧)=⎧⎪⎨⎪⎩𝑧−𝑤1^{𝛽−((𝛼+2)/𝑝)}ℑ𝑤if𝛼+2<𝛽𝑝,𝑧−𝑤if𝛼+2=𝛽𝑝.(2.10) For 𝑧=𝑥+𝑖𝑦 and 𝑤=𝑢+𝑖𝑣 in Π⁺, we have sup_𝑧∈Π+(ℑ𝑧)^{𝛽−(𝛼+2)/𝑝}||𝜓_𝑤(𝑧)||=sup_{𝑧=𝑥+𝑖𝑦∈Π+}𝑦^{𝛽−(𝛼+2)/𝑝}(𝑥−𝑢)²+(𝑦+𝑣)²^{(𝛽𝑝−(𝛼+2))/2𝑝}≤sup_{𝑧=𝑥+𝑖𝑦∈Π+}𝑦^{𝛽−(𝛼+2)/𝑝}(𝑦+𝑣)^{𝛽−(𝛼+2)/𝑝}≤1.(2.11) Thus 𝜓_𝑤∈𝒜^{∞𝛽−(𝛼+2)/𝑝}(Π⁺) if 𝛼+2<𝛽𝑝. Similarly 𝜓_𝑤∈𝐻^∞(Π⁺) if 𝛼+2=𝛽𝑝. By Corollary 2.2, it follows that 𝑀_𝜓𝑤∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded.

Corollary 2.4. Let 1≤𝑝<∞,𝛼>−1,𝛽>0, and let𝜑 be a holomorphic self-map of Π⁺. Then 𝐶_𝜑∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded if and only if sup_𝑧∈Π+(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^{(𝛼+2)/𝑝}<∞.(2.12)

Corollary 2.5. Let 𝜑 be the linear fractional map 𝜑(𝑧)=𝑎𝑧+𝑏𝑐𝑧+𝑑,𝑎,𝑏,𝑐,𝑑∈ℝ,𝑎𝑑−𝑏𝑐>0.(2.13) Then necessary and sufficient condition that 𝐶_𝜑∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded is that 𝑐=0 and 𝛼+2=𝛽𝑝.

Proof. Assume that 𝐶_𝜑∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded. Then sup_𝑧∈Π+(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^{(𝛼+2)/𝑝}=sup_{𝑧=𝑥+𝑖𝑦∈Π+}(𝑐𝑥+𝑑)²+𝑐²𝑦²^{(𝛼+2)/𝑝}𝑦^𝛽(𝑎𝑑−𝑏𝑐)^{(𝛼+2)/𝑝}𝑦^{(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 𝑎≠0, and by some calculation sup_𝑧∈Π+(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^{(𝛼+2)/𝑝}=𝑑𝑎^𝛽<∞.(2.15) Hence 𝐶_𝜑∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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=(𝜑^)^𝛽. Then the weighted composition operator 𝑊_𝜑,𝜓 acts boundedly from 𝒜^𝑝𝛼(Π⁺) to 𝒜^∞𝛽(Π⁺).

Proof. By Theorem 2.1, 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded if and only if sup_𝑧∈Π+(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^𝛽||𝜑^(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⁺||𝜑^(𝑧)||ℑ𝜑(𝑧)≤ℑ1𝑧,(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 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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 𝐶_𝜑∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) nor 𝑀_𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded, but 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded.

Example 2.7. Let 1≤𝑝<∞,𝛼>−1, and 𝛽>0 be such that 𝛽𝑝=𝛼+2. Let 𝜑(𝑧)=(𝑎𝑧+𝑏)/(𝑐𝑧+𝑑),𝑎,𝑏,𝑐,𝑑∈ℝ,𝑎𝑑−𝑏𝑐>0, and 𝑐≠0. Then by Corollary 2.5, 𝐶_𝜑∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is not bounded. On the other hand, if 𝜓(𝑧)=𝜑^(𝑧)^𝛽=𝑎𝑑−𝑏𝑐(𝑐𝑧+𝑑)²^𝛽,(2.18) then 𝜓∉𝐻^∞(Π⁺) and so by Corollary 2.2, 𝑀_𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is not bounded. However, by Corollary 2.6, we have that 𝑊_{𝜑,(𝜑)}^𝛽∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is bounded.
The next Schwartz-type lemma characterizes compact weighted composition operators 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is compact if and only if, for any bounded sequence (𝑓_𝑛)_𝑛∈ℕ⊂𝒜^𝑝𝛼(Π⁺) converging to zero on compacts of  Π⁺, one has lim_𝑛→∞‖‖𝑊_𝜑,𝜓𝑓_𝑛‖‖_{𝒜∞𝛽(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 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is compact, then lim_𝑟→0sup_{ℑ𝜑(𝑧)<𝑟}(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^{(𝛼+2)/𝑝}||𝜓(𝑧)||=0.(2.20)

Proof. Suppose 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) is compact and (2.20) does not hold. Then there is a 𝛿>0 and a sequence (𝑧_𝑛)_𝑛∈ℕ⊂Π⁺ such that ℑ𝜑(𝑧_𝑛)→0 and ℑ𝑧_𝑛^𝛽ℑ𝜑𝑧_𝑛^{(𝛼+2)/𝑝}||𝜓𝑧_𝑛||>𝛿(2.21) for all 𝑛∈ℕ. Let 𝑤_𝑛=𝜑(𝑧_𝑛), 𝑛∈ℕ, and 𝑓_𝑛(𝑧)=ℑ𝑤^𝑛^{(𝛼+2)/𝑝}𝑧−𝑤_𝑛^{(2𝛼+4)/𝑝},𝑛∈ℕ.(2.22) Then 𝑓_𝑛 is a norm bounded sequence and 𝑓_𝑛→0 on compacts of Π⁺ as ℑ𝜑(𝑧_𝑛)→0. By Lemma 2.8 it follows that lim_𝑛→∞‖‖𝑊_𝜑,𝜓𝑓_𝑛‖‖_{𝒜∞𝛽(Π+)}=0.(2.23) On the other hand, ‖‖𝑊_𝜑,𝜓𝑓_𝑛‖‖_{𝒜∞𝛽(Π+)}≥=ℑℑ𝑧𝑧_𝑛𝑛^𝛽||𝑊_𝜑,𝜓𝑓_𝑛𝑧_𝑛||=^𝛽||𝜓ℑ𝑧𝑧_𝑛𝑛||||𝑓_𝑛𝜑𝑧_𝑛||^𝛽2^{(2𝛼+4)/𝑝}ℑ𝜑𝑧_𝑛^{(𝛼+2)/𝑝}||𝜓𝑧_𝑛||>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 𝑎,𝑏∈(0,∞) such that 𝑎<𝑏, the following subset of Π⁺: Γ_𝑎,𝑏=𝑧∈Π⁺∶𝑎≤ℑ𝑧≤𝑏.(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 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(Π⁺) be bounded. Suppose that 𝜓∈𝒜^∞𝛽(Π⁺) and (ℑ𝑧)^𝛽|𝜓(𝑧)|→0 as |ℜ𝜑(𝑧)|→∞ within Γ_𝑎,𝑏 for all 𝑎 and 𝑏,0<𝑎<𝑏<∞. Then 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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 (𝑓_𝑛)_𝑛∈ℕ be a sequence in 𝒜^𝑝𝛼(Π⁺) such that sup_𝑛∈ℕ‖𝑓_𝑛‖_𝒜^𝑝𝛼_(Π⁺₎≤𝑀 and 𝑓_𝑛→0 uniformly on compact subsets of Π⁺ as 𝑛→∞. Thus for 𝑧∈Π⁺ such that ℑ𝜑(𝑧)<𝑀₁ and each 𝑛∈ℕ, we have (ℑ𝑧)^𝛽||𝜓(𝑧)||||𝑓_𝑛(𝜑(𝑧))||⪯(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^{(𝛼+2)/𝑝}||𝜓(𝑧)||‖‖𝑓_𝑛‖‖_𝒜^𝑝𝛼_(Π+)<𝜀𝑀.(2.27)
From estimate (1.6) we have ||𝑓_𝑛(𝑧)||⪯‖‖𝑓^𝑛‖‖_𝒜^𝑝𝛼_(Π+)(ℑ𝑧)^{(𝛼+2)/𝑝}⪯𝑀(ℑ𝑧)^{(𝛼+2)/𝑝}.(2.28) Thus there is an 𝑀₂>𝑀₁ such that ||𝑓_𝑛(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||𝜓(𝑧)||||𝑓_𝑛(𝜑(𝑧))||<𝜀‖𝜓‖_𝒜^∞𝛽_(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||𝜓(𝑧)||||𝑓_𝑛(𝜑(𝑧))||⪯𝜀‖‖𝑓^𝑛‖‖_𝒜^𝑝𝛼_(Π+)(ℑ𝜑(𝑧))^{(𝛼+2)/𝑝}≤𝜀𝑀𝑀^{1(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 |𝑓_𝑛(𝜑(𝑧))|<𝜀 for all 𝑛≥𝑛₀, and so (ℑ𝑧)^𝛽||𝜓(𝑧)||||𝑓_𝑛(𝜑(𝑧))||<𝜀‖𝜓‖_𝒜^∞𝛽_(Π⁺₎.(2.32) Combining (2.27)–(2.32), we have that ‖‖𝑊_𝜑,𝜓𝑓_𝑛‖‖_{𝒜∞𝛽(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 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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 𝜓∈𝒜^∞𝛽(Π⁺). Beside this, for 𝑧∈Γ_𝑎,𝑏, we have (ℑ𝑧)^𝛽||𝜓(𝑧)||=𝑦^𝛽𝑥²+(𝑦+1)²^𝛽/2≤𝑏^𝛽𝑥²+𝑎²^𝛽/2⟶0asℜ𝜑(𝑧)=𝑥⟶∞.(2.34)
Also sup_𝑧∈Π+(ℑ𝑧)^𝛽(ℑ𝜑(𝑧))^𝛽||𝜓(𝑧)||=sup_{𝑧=𝑥+𝑖𝑦∈Π+}𝑦^𝛽(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 𝑊_𝜑,𝜓∶𝒜^𝑝𝛼(Π⁺)→𝒜^∞𝛽(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).