AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation16152810.1155/2009/161528161528Research ArticleComposition Operators from the Hardy Space to the Zygmund-Type Space on the Upper Half-PlaneStevićStevoReichSimeonMathematical Institute of the Serbian Academy of SciencesKnez Mihailova 36/III11001 BeogradSerbiasanu.ac.yu200908032009200914122008230220092009Copyright © 2009This 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.

Here we introduce the nth weighted space on the upper half-plane Π+={z:Imz>0} in the complex plane . 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 Cφf(z)=f(φ(z)) from the Hardy space Hp(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:Imz>0} and H(Π+) the space of all analytic functions on Π+. The Hardy space Hp(Π+)=Hp, p>0, consists of all fH(Π+) such thatfHpp=supy>0f|(x+iy)|pdx<. With this norm Hp(Π+) is a Banach space when p1, while for p(0,1) it is a Fréchet space with the translation invariant metric d(f,g)=fgHpp, f,gHp(Π+), .

We introduce here the nth weighted space on the upper half-plane. The nth weighted space consists of all fH(Π+) such thatsupzΠ+  Imz|f(n)(z)|<,where n0. 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(Π+)= (for Bloch-type spaces on the unit disk, polydisk, or the unit ball and some operators on them, see, e.g.,  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 fH(Π+) such thatb𝒵(f)=supzΠ+Imz|f(z)|<. The quantity is a seminorm on the Zygmund space or a norm on 𝒵/1, where 1 is the set of all linear polynomials. A natural norm on the Zygmund space can be introduced as follows:f𝒵=|f(i)|+|f(i)|+b𝒵(f). 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 , 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:suph>0,θ[0,2π]|f(ei(θ+h))+f(ei(θh))2f(eiθ)|h< if and only ifsupz𝔻(1|z|2)|f(z)|<.

The family of all analytic functions on 𝔻 satisfying condition (1.6) is called the Zygmund class on the unit disk.

With the normf=|f(0)|+|f(0)|+supz𝔻(1|z|2)|f(z)|, 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, .

Now note that 1|z| is the distance from the point z𝔻 to the boundary of the unit disc, that is, 𝔻, and that Imz is the distance from the point zΠ+ to the real axis in which is the boundary of Π+.

In two main theorems in , the authors proved the following results, which we now incorporate in the next theorem.

Theorem A.

Assume p1 and φ is a holomorphic self-map of Π+. Then the following statements true hold.

The operator Cφ:Hp(Π+)𝒜(Π+) is bounded if and only ifsupzΠ+Imz(Imφ(z))1/p<.

The operator Cφ:Hp(Π+)(Π+) is bounded if and only ifsupzΠ+Imz(Imφ(z))1+1/p|φ(z)|<.

Motivated by Theorem A, here we investigate the boundedness of the operator Cφ:Hp(Π+)𝒵(Π+). Some recent results on composition and weighted composition operators can be found, for example, in [4, 6, 7, 10, 12, 18, 2127].

Throughout this paper, constants are denoted by C, they are positive and may differ from one occurrence to the other. The notation ab means that there is a positive constant C such that aCb. Moreover, if both ab and ba hold, then one says that ab.

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 p1, n, and wΠ+. Then the function fw,n(z)=(Imw)n1/p(zw¯)n, belongs to Hp(Π+). MoreoversupwΠ+fw,nHpπ1/p.

Proof.

Let z=x+iy and w=u+iυ. Then, we havefw,nHpp=supy>0|fw,n(x+iy)|pdx=(Imw)np1supy>0dx|zw¯|np2|zw¯|2vnp1supy>0dx((y+v)2)(np2)/2((xu)2+(y+v)2)vnp1supy>01(y+v)np1y+v(xu)2+(y+v)2dx=supy>0vnp1(y+v)np1dtt2+1=π, 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 p1 and φ is a holomorphic self-map of Π+. Then Cφ:Hp(Π+)𝒵(Π+) is bounded if and only if supzΠ+Imz(Imφ(z))2+1/p|φ(z)|2<,supzΠ+Imz(Imφ(z))1+1/p|φ(z)|<.

Moreover, if the operator Cφ:Hp(Π+)𝒵/1(Π+) is bounded, thenCφHp(Π+)𝒵/1(Π+)supzΠ+Imz(Imφ(z))2+1/p|φ(z)|2+supzΠ+Imz(Imφ(z))1+1/p|φ(z)|.

Proof.

First assume that the operator Cφ:Hp(Π+)𝒵(Π+) is bounded.

For wΠ+, setfw(z)=(Imw)21/pπ1/p(zw¯)2.

By Lemma 2.1 (case n=2) we know that fwHp(Π+) for every wΠ+. Moreover, we have thatsupwΠ+fwHp(Π+)1.

From (3.5) and since the operator Cφ:Hp(Π+)𝒵(Π+) is bounded, for every wΠ+, we obtainsupzΠ+Imz|fw(φ(z))(φ(z))2+fw(φ(z))φ(z)|=Cφ(fw)𝒵(Π+)CφHp(Π+)𝒵(Π+).

We also have thatfw(z)=2(Imw)21/pπ1/p(zw¯)3 ,fw(z)=6(Imw)21/pπ1/p(zw¯)4.

Replacing (3.7) in (3.6) and taking w=φ(z), we obtainImz|38(φ(z))2(Imφ(z))2+1/pi4φ(z)(Imφ(z))1+1/p|π1/pCφHp(Π+)𝒵(Π+), and consequently14Imz(Imφ(z))1+1/p|φ(z)|π1/pCφHp(Π+)𝒵(Π+)+38Imz(Imφ(z))2+1/p|φ(z)|2.

Hence if we show that (3.1) holds then from the last inequality, condition (3.2) will follow.

For wΠ+, setgw(z)=3i(Imw)21/pπ1/p(zw¯)24(Imw)31/pπ1/p(zw¯)3.

Then it is easy to see thatgw(w)=0 ,gw(w)=Cw2+1/p, and by Lemma 2.1 (cases n=2 and n=3) it is easy to see thatsupwΠ+gwHp<.

From this, since Cφ:Hp(Π+)𝒵(Π+) is bounded and by taking w=φ(z), it follows thatCImz(Imφ(z))2+1/p|φ(z)|2Cφ(gw)𝒵(Π+)CCφHp(Π+)𝒵(Π+), from which (3.1) follows, as desired.

Moreover, from (3.9) and (3.13) it follows thatsupzΠ+Imz(Imφ(z))2+1/p|φ(z)|2+supzΠ+Imz(Imφ(z))1+1/p|φ(z)|CCφHp(Π+)𝒵(Π+).

Now assume that conditions (3.1) and (3.2) hold. By the Cauchy integral formula in Π+ for Hp(Π+) functions (note that p1), we have

f(z)=12πif(t)tzdt,zΠ+.

By differentiating formula (3.15), we obtainf(n)(z)=n!2πif(t)(tz)n+1dt,zΠ+, for each n, from which it follows that

|f(n)(z)|n!2π|f(t)|[(tx)2+y2](n+1)/2dt,zΠ+.

By using the change tx=sy, we have thatyn[(tx)2+y2](n+1)/2dt=ds(s2+1)(n+1)/2=:cn<,n.

From this, applying Jensen's inequality on (3.17) and an elementary inequality, we obtain|f(n)(z)|pdn|f(t)|pynpyn[(tx)2+y2](n+1)/2dtdn|f(t)|pynp+1dtdnfHp(Π+)pynp+1,wheredn=(cnn!2π)p, from which it follows that|f(n)(z)|CfHp(Π+)yn+1/p.

Assume that fHp(Π+). By applying (3.21) and Lemma 1 in [1, page 188], we have Cφf𝒵(Π+)=|f(φ(i))|+|(fφ)(i)|+supzΠ+Imz|(Cφf)(z)|=|f(φ(i))|+|f(φ(i))||φ(i)|+supzΠ+Imz|f(φ(z))(φ(z))2+f(φ(z))φ(z)|CfHp(Π+)(1+supzΠ+Imz(Imφ(z))2+1/p|φ(z)|2+supzΠ+Imz(Imφ(z))1+1/p|φ(z)|).

From this and by conditions (3.1) and (3.2), it follows that the operator Cφ:Hp(Π+)𝒵(Π+) is bounded. Moreover, if we consider the space 𝒵/1(Π+), we have thatCφHp(Π+)𝒵/1(Π+)C(supzΠ+Imz(Imφ(z))2+1/p|φ(z)|2+supzΠ+Imz(Imφ(z))1+1/p|φ(z)|).

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

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