Composition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-Plane

and Applied Analysis 3 In two main theorems in 20 , the authors proved the following results, which we now incorporate in the next theorem. Theorem A. Assume p ≥ 1 and φ is a holomorphic self-map of Π . Then the following statements true hold. a The operator Cφ : H Π → A∞ Π is bounded if and only if sup z∈Π Im z ( Imφ z )1/p < ∞. 1.8 b The operator Cφ : H Π → B∞ Π is bounded if and only if sup z∈Π Im z ( Imφ z )1 1/p ∣∣φ′ z ∣∣ < ∞. 1.9 Motivated by Theorem A, here we investigate the boundedness of the operator Cφ : H Π → Z Π . Some recent results on composition and weighted composition operators can be found, for example, in 4, 6, 7, 10, 12, 18, 21–27 . 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. An Auxiliary Result In this section we prove an auxiliary result which will be used in the proof of the main result of the paper. Lemma 2.1. Assume that p ≥ 1, n ∈ N, and w ∈ Π . Then the function fw, n z Imw n−1/p z −w n , 2.1 belongs toH Π . Moreover sup w∈Π ∥∥fw, n∥Hp ≤ π1/p. 2.2 4 Abstract and Applied Analysis Proof. Let z x iy and w u iυ. Then, we have ∥∥fw, n∥pHp sup y>0 ∫∞ −∞ ∣∣fw, n x iy ∣∣pdx Imw np−1sup y>0 ∫∞ −∞ dx |z −w|np−2|z −w|2 ≤ vnp−1 sup y>0 ∫∞ −∞ dx ( y v 2 ) np−2 /2( x − u 2 y v 2) ≤ vnp−1 sup y>0 1 y v np−1 ∫∞ −∞ y v x − u 2 y v 2 dx sup y>0 vnp−1 y v np−1 ∫∞ −∞ dt t2 1 π, 2.3 where we have used the change of variables x u t y v . 3. Main Result Here we formulate and prove the main result of the paper. Theorem 3.1. Assume p ≥ 1 and φ is a holomorphic self-map of Π . Then Cφ : H Π → Z Π is bounded if and only if sup z∈Π Im z ( Imφ z )2 1/p ∣∣φ′ z ∣∣2 < ∞, 3.1 sup z∈Π Im z ( Imφ z )1 1/p ∣∣φ′′ z ∣∣ < ∞. 3.2 Moreover, if the operator Cφ : H Π → Z/P1 Π is bounded, then ∥Cφ∥Hp Π →Z/P1 Π sup z∈Π Im z ( Imφ z )2 1/p ∣∣φ′ z ∣∣2 sup z∈Π Im z ( Imφ z )1 1/p ∣∣φ′′ z ∣∣. 3.3 Proof. First assume that the operator Cφ : H Π → Z Π is bounded. For w ∈ Π , set fw z Imw 2−1/p π1/p z −w 2 . 3.4 Abstract and Applied Analysis 5 By Lemma 2.1 case n 2 we know that fw ∈ H Π for every w ∈ Π . Moreover, we have thatand Applied Analysis 5 By Lemma 2.1 case n 2 we know that fw ∈ H Π for every w ∈ Π . Moreover, we have that sup w∈Π ∥fw∥Hp Π ≤ 1. 3.5 From 3.5 and since the operator Cφ : H Π → Z Π is bounded, for every w ∈ Π , we obtain sup z∈Π Im z ∣∣f ′′ w(φ z )(φ′ z )2 f ′ w(φ z )φ′′ z ∣∣ ∥Cφ(fw)∥Z Π ≤ ∥Cφ∥Hp Π →Z Π . 3.6