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


Introduction
Let Π be the upper half-plane, that is, the set {z ∈ C : Im z > 0} and H Π the space of all analytic functions on Π .The Hardy space H p Π H p , p > 0, consists of all f ∈ H Π such that With this norm H p Π is 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 p H p , f, g ∈ H p Π , 1 .We introduce here the nth weighted space on the upper half-plane.The nth weighted space consists of all f ∈ H Π such that where n ∈ N 0 .For n 0 the space is called the growth space and is denoted by A ∞ Π A ∞ and for n 1 it is called the Bloch space B ∞ Π B ∞ for Bloch-type spaces on the unit disk, polydisk, or the unit ball and some operators on them, see, e.g., 2-14 and the references therein .
When n 2, we call the space the Zygmund-type space on the upper half-plane or simply the Zygmund space and denote it by Z Π Z. Recall that the space consists of all The quantity is a seminorm on the Zygmund space or a norm on Z/P 1 , where P 1 is the set of all linear polynomials.A natural norm on the Zygmund space can be introduced as follows: With this norm the Zygmund space becomes a Banach space.
To clarify the notation we have just introduced, we have to say that the main reason for this name is found in the fact that for the case of the unit disk D {z : |z| < 1} in the complex palne C, Zygmund see, e.g., 1, Theorem 5.3 proved that a holomorphic function on D continuous on the closed unit disk D satisfies the following condition: sup h>0, θ∈ 0,2π The family of all analytic functions on D satisfying condition 1.6 is called the Zygmund class on the unit disk.
With the norm the Zygmund class becomes a Banach space.Zygmund class with this norm is called the Zygmund space and is denoted by Z D .For some other information on this space and some operators on it, see, for example, 15-19 .Now note that 1 − |z| is the distance from the point z ∈ D to the boundary of the unit disc, that is, ∂D, and that Im z is the distance from the point z ∈ Π to the real axis in C which is the boundary of Π .

Abstract 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 C ϕ : Motivated by Theorem A, here we investigate the boundedness of the operator C ϕ : H p 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.

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.
Proof.Let z x iy and w u iυ.Then, we have where we have used the change of variables x u t y v .

Main Result
Here we formulate and prove the main result of the paper.

3.6
We also have that Replacing 3.7 in 3.6 and taking w ϕ z , we obtain Im z 3.9 Hence if we show that 3.1 holds then from the last inequality, condition 3.2 will follow.
For w ∈ Π , set Then it is easy to see that g w w 0 , g w w C w By differentiating formula 3.15 , we obtain for each n ∈ N, from which it follows that 3.17 By using the change t − x sy, we have that From this, applying Jensen's inequality on 3.17 and an elementary inequality, we obtain

3.22
From this and by conditions 3.1 and 3.2 , it follows that the operator C ϕ : H p Π → Z Π is bounded.Moreover, if we consider the space Z/P 1 Π , we have that

3.23
From 3.14 and 3.23 , we obtain the asymptotic relation 3.3 .

Lemma 2 . 1 .
Assume that p ≥ 1, n ∈ N, and w ∈ Π .Then the function C ϕ H p Π → Z/P 1 Π ≤ C sup z∈Π By Lemma 2.1 case n 2 we know that f w ∈ H p Π for every w ∈ Π .Moreover, Theorem 3.1.Assume p ≥ 1 and ϕ is a holomorphic self-map of Π .Then C ϕ :H p Π → Z Π is bounded if and only if sup z∈Π Im z ϕ : H p Π → Z/P 1 Π is bounded, then C ϕ H p Π → Z/P 1 Π sup z∈Π Im z ϕ : H p Π → Z Π is bounded.For w ∈ Π , set f w z Im w 2−1/p π 1/p z − w 2 .3.4 From this, since C ϕ : H p Π → Z Π is bounded and by taking w ϕ z , it follows that Now assume that conditions 3.1 and 3.2 hold.By the Cauchy integral formula in Π for H p Π functions note that p ≥ 1 , we have Assume that f ∈ H p Π .By applying 3.21 , and Lemma 1 in 1, page 188 , we have z∈Π Im z f ϕ z ϕ z 2 f ϕ z ϕ z ≤ C f H p Π 1 sup z∈Π Im z