Abstract and Applied Analysis

Abstract and Applied Analysis / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 161528 | 8 pages | https://doi.org/10.1155/2009/161528

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

Academic Editor: Simeon Reich
Received14 Dec 2008
Accepted23 Feb 2009
Published29 Apr 2009

Abstract

Here we introduce the 𝑛 th weighted space on the upper half-plane Π + = { 𝑧 ∈ ℂ ∶ I m 𝑧 > 0 } in the complex plane ℂ . For the case 𝑛 = 2 , we call it the Zygmund-type space, and denote it by 𝒵 ( Π + ) . The main result of the paper gives some necessary and sufficient conditions for the boundedness of the composition operator 𝐶 𝜑 𝑓 ( 𝑧 ) = 𝑓 ( 𝜑 ( 𝑧 ) ) from the Hardy space 𝐻 𝑝 ( Π + ) on the upper half-plane, to the Zygmund-type space, where 𝜑 is an analytic self-map of the upper half-plane.

1. Introduction

Let Π + be the upper half-plane, that is, the set { 𝑧 ∈ ℂ ∶ I m 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 𝑛 th weighted space on the upper half-plane. The 𝑛 th weighted space consists of all 𝑓 ∈ 𝐻 ( Π + ) such thatwhere 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 𝑓 ∈ 𝐻 ( Π + ) such that 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: if and only if

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 I m 𝑧 is the distance from the point 𝑧 ∈ Π + to the real axis in ℂ which is the boundary of Π + .

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 ∶ 𝐻 𝑝 ( Π + ) → 𝒜 ∞ ( Π + ) is bounded if and only if(b)The operator 𝐶 𝜑 ∶ 𝐻 𝑝 ( Π + ) → ℬ ∞ ( Π + ) is bounded if and only if

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 ğ‘Ž ⪯ 𝑏 means that there is a positive constant 𝐶 such that ğ‘Ž ≤ 𝐶 𝑏 . Moreover, if both ğ‘Ž ⪯ 𝑏 and 𝑏 ⪯ ğ‘Ž hold, then one says that ğ‘Ž ≍ 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 belongs to 𝐻 𝑝 ( Π + ) . Moreover

Proof. Let 𝑧 = 𝑥 + 𝑖 𝑦 and 𝑤 = 𝑢 + 𝑖 𝜐 . Then, we have 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 𝐶 𝜑 ∶ 𝐻 𝑝 ( Π + ) → 𝒵 ( Π + ) is bounded if and only if
Moreover, if the operator ‖ ‖ 𝐶 𝜑 ‖ ‖ 𝐻 𝑝 ( Π + ) → 𝒵 / ℙ 1 ( Π + ) ≍ s u p 𝑧 ∈ Π + I m 𝑧   I m 𝜑 ( 𝑧 ) 2 + 1 / 𝑝 | | 𝜑  | | ( 𝑧 ) 2 + s u p 𝑧 ∈ Π + I m 𝑧   I m 𝜑 ( 𝑧 ) 1 + 1 / 𝑝 | | 𝜑   | | ( 𝑧 ) . ( 3 . 3 ) is bounded, then

Proof. First assume that the operator 𝑤 ∈ Π + is bounded.
For 𝑓 𝑤 ( 𝑧 ) = ( I m 𝑤 ) 2 − 1 / 𝑝 𝜋 1 / 𝑝 ( 𝑧 − 𝑤 ) 2 . ( 3 . 4 ) , set
By Lemma 2.1 (case 𝑓 𝑤 ∈ 𝐻 𝑝 ( Π + ) ) we know that 𝑤 ∈ Π + for every s u p 𝑤 ∈ Π + ‖ ‖ 𝑓 𝑤 ‖ ‖ 𝐻 𝑝 ( Π + ) ≤ 1 . ( 3 . 5 ) . Moreover, we have that
From (3.5) and since the operator 𝑤 ∈ Π + is bounded, for every s u p 𝑧 ∈ Π + | | 𝑓 I m 𝑧 𝑤    𝜑 𝜑 ( 𝑧 )     ( 𝑧 ) 2 + 𝑓  𝑤   𝜑 𝜑 ( 𝑧 )   | | = ‖ ‖ 𝐶 ( 𝑧 ) 𝜑  𝑓 𝑤  ‖ ‖ 𝒵 ( Π + ) ≤ ‖ ‖ 𝐶 𝜑 ‖ ‖ 𝐻 𝑝 ( Π + ) → 𝒵 ( Π + ) . ( 3 . 6 ) , we obtain
We also have that
Replacing (3.7) in (3.6) and taking | | | | 3 I m 𝑧 8  𝜑   ( 𝑧 ) 2   I m 𝜑 ( 𝑧 ) 2 + 1 / 𝑝 − 𝑖 4 𝜑   ( 𝑧 )   I m 𝜑 ( 𝑧 ) 1 + 1 / 𝑝 | | | | ≤ 𝜋 1 / 𝑝 ‖ ‖ 𝐶 𝜑 ‖ ‖ 𝐻 𝑝 ( Π + ) → 𝒵 ( Π + ) , ( 3 . 8 ) , we obtain and consequently
Hence if we show that (3.1) holds then from the last inequality, condition (3.2) will follow.
For 𝑔 𝑤 3 ( 𝑧 ) = 𝑖 ( I m 𝑤 ) 2 − 1 / 𝑝 𝜋 1 / 𝑝 ( 𝑧 − 𝑤 ) 2 − 4 ( I m 𝑤 ) 3 − 1 / 𝑝 𝜋 1 / 𝑝 ( 𝑧 − 𝑤 ) 3 . ( 3 . 1 0 ) , set
Then it is easy to see that and by Lemma 2.1 (cases 𝑛 = 3 and s u p 𝑤 ∈ Π + ‖ ‖ 𝑔 𝑤 ‖ ‖ 𝐻 𝑝 < ∞ . ( 3 . 1 2 ) ) it is easy to see that
From this, since 𝑤 = 𝜑 ( 𝑧 ) is bounded and by taking 𝐶 I m 𝑧   I m 𝜑 ( 𝑧 ) 2 + 1 / 𝑝 | | 𝜑  | | ( 𝑧 ) 2 ≤ ‖ ‖ 𝐶 𝜑  𝑔 𝑤  ‖ ‖ 𝒵 ( Π + ) ‖ ‖ 𝐶 ≤ 𝐶 𝜑 ‖ ‖ 𝐻 𝑝 ( Π + ) → 𝒵 ( Π + ) , ( 3 . 1 3 ) , it follows that from which (3.1) follows, as desired.
Moreover, from (3.9) and (3.13) it follows that
Now assume that conditions (3.1) and (3.2) hold. By the Cauchy integral formula in 𝐻 𝑝 ( Π + ) for 𝑝 ≥ 1 functions (note that 1 𝑓 ( 𝑧 ) =  2 𝜋 𝑖 ∞ − ∞ 𝑓 ( 𝑡 ) 𝑡 − 𝑧 𝑑 𝑡 , 𝑧 ∈ Π + . ( 3 . 1 5 ) ), we have

By differentiating formula (3.15), we obtain for each | | 𝑓 ( 𝑛 ) | | ≤ ( 𝑧 ) 𝑛 !  2 𝜋 ∞ − ∞ | | | | 𝑓 ( 𝑡 )  ( 𝑡 − 𝑥 ) 2 + 𝑦 2  ( 𝑛 + 1 ) / 2 𝑑 𝑡 , 𝑧 ∈ Π + . ( 3 . 1 7 ) , from which it follows that

By using the change  ∞ − ∞ 𝑦 𝑛  ( 𝑡 − 𝑥 ) 2 + 𝑦 2  ( 𝑛 + 1 ) / 2  𝑑 𝑡 = ∞ − ∞ 𝑑 𝑠  𝑠 2  + 1 ( 𝑛 + 1 ) / 2 = ∶ 𝑐 𝑛 < ∞ , 𝑛 ∈ ℕ . ( 3 . 1 8 ) , we have that
From this, applying Jensen's inequality on (3.17) and an elementary inequality, we obtainwhere from which it follows that
Assume that ‖ ‖ 𝐶 𝜑 𝑓 ‖ ‖ 𝒵 ( Π + ) = | | 𝑓   | | + | |   𝜑 ( 𝑖 ) 𝑓 ∘ 𝜑  | | ( 𝑖 ) + s u p 𝑧 ∈ Π + | |  𝐶 I m 𝑧 𝜑 𝑓    | | = | | 𝑓   | | + | | 𝑓 ( 𝑧 ) 𝜑 ( 𝑖 )    | | | | 𝜑 𝜑 ( 𝑖 )  | | ( 𝑖 ) + s u p 𝑧 ∈ Π + | | 𝑓 I m 𝑧    𝜑 𝜑 ( 𝑧 )     ( 𝑧 ) 2 + 𝑓    𝜑 𝜑 ( 𝑧 )   | | ( 𝑧 ) ≤ 𝐶 ‖ 𝑓 ‖ 𝐻 𝑝 ( Π + )  1 + s u p 𝑧 ∈ Π + I m 𝑧   I m 𝜑 ( 𝑧 ) 2 + 1 / 𝑝 | | 𝜑  | | ( 𝑧 ) 2 + s u p 𝑧 ∈ Π + I m 𝑧   I m 𝜑 ( 𝑧 ) 1 + 1 / 𝑝 | | 𝜑   | |  . ( 𝑧 ) ( 3 . 2 2 ) . By applying (3.21) and Lemma 1 in [1, page 188], we have
From this and by conditions (3.1) and (3.2), it follows that the operator 𝒵 / ℙ 1 ( Π + ) is bounded. Moreover, if we consider the space ‖ ‖ 𝐶 𝜑 ‖ ‖ 𝐻 𝑝 ( Π + ) → 𝒵 / ℙ 1 ( Π + )  ≤ 𝐶 s u p 𝑧 ∈ Π + I m 𝑧   I m 𝜑 ( 𝑧 ) 2 + 1 / 𝑝 | | 𝜑  | | ( 𝑧 ) 2 + s u p 𝑧 ∈ Π + I m 𝑧   I m 𝜑 ( 𝑧 ) 1 + 1 / 𝑝 | | 𝜑   | |  ( 𝑧 ) . ( 3 . 2 3 ) , we have that
From (3.14) and (3.23), we obtain the asymptotic relation (3.3).

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Copyright © 2009 Stevo Stević. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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