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

and Applied Analysis 3 Let β > 0. The weighted-type space or growth space on the upper half-planeA∞ 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 thatA∞ β Π is a Banach space with the norm defined above. For weightedtype 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 fromA α Π to theweighted-type spaceA∞ 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 Wφ,ψ : A α 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 Wφ,ψ : A α 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 Wφ,ψ : A α Π → A∞ β Π is bounded then the following asymptotic relationship holds: ∥Wφ,ψ ∥ A α Π →A∞ β Π M. 2.2 Proof. First suppose that 2.1 holds. Then for any z ∈ Π and f ∈ A α 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 ∥ A α Π , 2.3 and so by 2.1 ,Wφ,ψ : A α Π → A∞ β Π is bounded and moreover ∥Wφ,ψ ∥ A α Π →A∞ β Π M. 2.4 4 Abstract and Applied Analysis Conversely suppose Wφ,ψ : A α Π → A∞ β Π is bounded. Consider the function fw z Iw α 2 /p z −w 2α 4 /p , w ∈ Π . 2.5 Then fw ∈ A α Π and moreover supw∈Π ‖fw‖Ap α Π 1 see, e.g., Lemma 1 in 18 . Thus the boundedness of Wφ,ψ : A α Π → A∞ β Π implies that Iz β ∣ ∣ψ z ∣ ∣ ∣ ∣fw ( φ z )∣ ≤ ∥Wφ,ψfw ∥ ∥ A∞ β Π ∥ ∥Wφ,ψ ∥ ∥ A α 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 ∣ ∥Wφ,ψ ∥ A α Π →A∞ β Π . 2.7 Since z ∈ Π is arbitrary, 2.1 follows and moreover M ∥Wφ,ψ ∥ A α Π →A∞ β Π . 2.8 If Wφ,ψ : A α 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 Mψ : A α Π → A∞ β Π is bounded if and only if ψ ∈ X, where X ⎧ ⎨ ⎩ A∞ β− α 2 /p Π if α 2 < βp, H∞ Π if α 2 βp. 2.9 Example 2.3. Let 1 ≤ p < ∞, α > −1 and β > 0 be such that βp ≥ α 2 andw ∈ Π . Let ψw be a holomorphic map of Π defined as ψw z ⎧ ⎪ ⎪⎨ ⎪ ⎪⎩ 1 z −w β− α 2 /p if α 2 < βp, Iw z −w if α 2 βp. 2.10 Abstract and Applied Analysis 5 For z x iy and w u iv in Π , we haveand Applied Analysis 5 For z x iy and w u iv in Π , we have sup z∈Π Iz β− α 2 /p ∣ ∣ψw z ∣ ∣ sup z x iy∈Π yβ− α 2 /p ( x − u 2 y v2 ) βp− α 2 /2p ≤ sup z x iy∈Π yβ− α 2 /p ( y v )β− α 2 /p ≤ 1. 2.11 Thus ψw ∈ A∞ β− α 2 /p Π if α 2 < βp. Similarly ψw ∈ H∞ Π if α 2 βp. By Corollary 2.2, it follows that Mψw : A α Π → A∞ β Π is bounded. Corollary 2.4. Let 1 ≤ p < ∞, α > −1, β > 0, and let φ be a holomorphic self-map of Π . Then Cφ : A α 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 b cz d , a, b, c, d ∈ R, ad − bc > 0. 2.13 Then necessary and sufficient condition that Cφ : A α Π → A∞ β Π is bounded is that c 0 and α 2 βp. Proof. Assume that Cφ : A α 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 ad − bc α 2 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 ( d a )β < ∞. 2.15 Hence Cφ : A α 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′ . Then the weighted composition operator Wφ,ψ acts boundedly from A α Π toA∞ β Π . 6 Abstract and Applied Analysis Proof. By Theorem 2.1, Wφ,ψ : A α 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 ≤ 1 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 operatorWφ,ψ : A α 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 Cφ : A α Π → A∞ β Π nor Mψ : A p α Π → A∞ β Π is bounded, but Wφ,ψ : A p α 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, Cφ : A α 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, Mψ : A α Π → A∞ β Π is not bounded. However, by Corollary 2.6, we have that Wφ, φ′ β : A p α Π → A∞ β Π is bounded. The next Schwartz-type lemma characterizes compact weighted composition operators Wφ,ψ : A α 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 let φ be a holomorphic self-map of Π . Then Wφ,ψ : A α Π → A∞ β Π is compact if and only if, for any bounded sequence fn n∈N ⊂ A α Π converging to zero on compacts of Π , one has lim n→∞ ∥Wφ,ψfn ∥ 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 . IfWφ,ψ : A α Π → A∞ β Π is compact, then lim r→ 0 sup Iφ z 0 and a sequence zn n∈N ⊂ Π such that Iφ zn → 0 and Izn β ( Iφ zn ) 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 lim n→∞ ∥Wφ,ψfn ∥ A∞ β Π 0. 2.23 On the other hand, ∥Wφ,ψfn ∥ A∞ β Π ≥ Izn β ∣Wφ,ψfn ) zn ∣ Izn β ∣ψ zn ∣∣fn ( φ zn )∣ Izn β 2 2α 4 /p ( Iφ zn ) α 2 /p ∣ψ zn ∣ > δ 2 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 : Γa,b {z ∈ Π : a ≤ Iz ≤ b}. 2.25 Theorem 2.10. Let 1 ≤ p < ∞, α > −1, β > 0, ψ ∈ H Π , and let φ be a holomorphic self-map of Π and Wφ,ψ : A α Π → A∞ β Π be bounded. Suppose that ψ ∈ A∞ β Π and Iz |ψ z | → 0 as |Rφ z | → ∞ within Γa,b for all a and b, 0 < a < b < ∞. Then Wφ,ψ : A α 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, whenever Iφ z < M1. 2.26 8 Abstract and Applied Analysis Let fn n∈N be a sequence in A α Π such that supn∈N‖fn‖Ap α Π ≤ M and fn → 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 ∣ ∣ ∣ ∣fn ( φ z )∣ Iz β ( Iφ z ) α 2 /p ∣ ∣ψ z ∣ ∣ ∥ ∥fn ∥ ∥ A α Π < εM. 2.27 From estimate 1.6 we have ∣ ∣fn z ∣ ∣ ∥ ∥fn ∥ ∥ A α 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 ∈ Nwe have Iz β ∣ψ z ∣∣fn ( 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 |ψ z | < ε, whenever |Rφ z | > M3. Therefore, for each n ∈ Nwe have Iz β ∣ψ z ∣∣fn ( φ z )∣ ε ∥fn ∥ A α Π ( Iφ z ) α 2 /p ≤ ε M M α 2 /p 1 , 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 ∣ψ z ∣∣fn ( φ z )∣ < ε ∥ψ ∥∥ A∞ β Π . 2.32 Combining 2.27 – 2.32 , we have that ∥Wφ,ψfn ∥ 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 Wφ,ψ : A α Π → A∞ β Π is compact. Abstract and Applied Analysis 9 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 , thenRφ z x and Iφ z y 1. It is easy to see that ψ ∈ A∞ β Π . Beside this, for z ∈ Γa,b, we haveand Applied Analysis 9 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 , thenRφ z x and Iφ z y 1. It is easy to see that ψ ∈ A∞ β Π . Beside this, for z ∈ Γa,b, we have Iz β ∣ ∣ψ z ∣ ∣ y ( x2 ( y 1 )2)β/2 ≤ b β x2 a2 β/2 −→ 0 as Rφ z x −→ ∞. 2.34