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

Authors: Stevo Stević
Journal: Abstract and Applied Analysis
 (2009)

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## Abstract

Here we introduce thenth
weighted space on the upper half-plane Π+={z∈ℂ:Imz>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 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.


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## Body

## 1. Introduction

LetΠ+ be the upper half-plane, that is, the set {z∈ℂ:Imz>0} and H(Π+) the space of
all analytic functions on Π+. The Hardy space Hp(Π+)=Hp, p>0, consists of all f∈H(Π+) such that(1.1)∥f∥Hpp=supy>0∫−∞∞f|(x+iy)|pdx<∞. With this norm Hp(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∥Hpp, f,g∈Hp(Π+), [1].We introduce here thenth weighted space on the upper
half-plane. The nth weighted space consists of all f∈H(Π+) such that(1.2)supz∈Π+Imz|f(n)(z)|<∞,where n∈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 spaceℬ∞(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).Whenn=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(1.3)b𝒵(f)=supz∈Π+Imz|f′′(z)|<∞. The quantity is a seminorm on the Zygmund space or a norm on 𝒵/ℙ1, where ℙ1 is the set of
all linear polynomials. A natural norm on the Zygmund space can be introduced
as follows:(1.4)∥f∥𝒵=|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 disk𝔻={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:(1.5)suph>0,θ∈[0,2π]|f(ei(θ+h))+f(ei(θ−h))−2f(eiθ)|h<∞  if and only if(1.6)supz∈𝔻(1−|z|2)|f′′(z)|<∞.The family of all analytic functions on𝔻 satisfying
condition (1.6) is called the Zygmund class on the unit disk.With the norm(1.7)∥f∥=|f(0)|+|f′(0)|+supz∈𝔻(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, [15–19].Now note that1−|z| is the distance
from the point z∈𝔻 to the boundary
of the unit disc, that is, ∂𝔻, and that Imz 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.
Assumep≥1  and  φ  is a
holomorphic self-map of  Π+.  Then the following statements true hold.
(a)
The operatorCφ:Hp(Π+)→𝒜∞(Π+) is bounded if
and only if(1.8)supz∈Π+Imz(Imφ(z))1/p<∞.(b)
The operatorCφ:Hp(Π+)→ℬ∞(Π+) is bounded if
and only if(1.9)supz∈Π+Imz(Imφ(z))1+1/p|φ′(z)|<∞.Motivated by Theorem A, here we investigate the
boundedness of the operatorCφ: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, 21–27].Throughout this paper, constants are denoted byC, 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 thatp≥1, n∈ℕ,   and  w∈Π+.
Then the
function (2.1)fw,n(z)=(Imw)n−1/p(z−w¯)n,
belongs to Hp(Π+). Moreover(2.2)supw∈Π+∥fw,n∥Hp≤π1/p.Proof.
Letz=x+iy and w=u+iυ. Then, we have(2.3)∥fw,n∥Hpp=supy>0∫−∞∞|fw,n(x+iy)|pdx=(Imw)np−1supy>0∫−∞∞dx|z−w¯|np−2|z−w¯|2≤vnp−1supy>0∫−∞∞dx((y+v)2)(np−2)/2((x−u)2+(y+v)2)≤vnp−1supy>01(y+v)np−1∫−∞∞y+v(x−u)2+(y+v)2dx=supy>0vnp−1(y+v)np−1∫−∞∞dtt2+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.
Assumep≥1  and  φ  is a
holomorphic self-map of  Π+.  Then  Cφ:Hp(Π+)→𝒵(Π+)  is bounded if
and only if (3.1)supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2<∞,(3.2)supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|<∞.
Moreover, if the operatorCφ:Hp(Π+)→𝒵/ℙ1(Π+) is bounded,
then(3.3)∥Cφ∥Hp(Π+)→𝒵/ℙ1(Π+)≍supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|.Proof.
First assume that the operatorCφ:Hp(Π+)→𝒵(Π+) is bounded.
Forw∈Π+, set(3.4)fw(z)=(Imw)2−1/pπ1/p(z−w¯)2.
By Lemma2.1 (case n=2) we
know that fw∈Hp(Π+) for every w∈Π+. Moreover, we
have that(3.5)supw∈Π+∥fw∥Hp(Π+)≤1.
From (3.5) and since the operator Cφ:Hp(Π+)→𝒵(Π+) is bounded, for
every w∈Π+, we obtain(3.6)supz∈Π+Imz|fw′′(φ(z))(φ′(z))2+fw′(φ(z))φ′′(z)|=∥Cφ(fw)∥𝒵(Π+)≤∥Cφ∥Hp(Π+)→𝒵(Π+).
We also have that(3.7)fw′(z)=−2(Imw)2−1/pπ1/p(z−w¯)3,fw′′(z)=6(Imw)2−1/pπ1/p(z−w¯)4.
Replacing (3.7) in (3.6) and taking w=φ(z),  we obtain(3.8)Imz|38(φ′(z))2(Imφ(z))2+1/p−i4φ′′(z)(Imφ(z))1+1/p|≤π1/p∥Cφ∥Hp(Π+)→𝒵(Π+), and consequently(3.9)14Imz(Imφ(z))1+1/p|φ′′(z)|≤π1/p∥Cφ∥Hp(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.
Forw∈Π+, set(3.10)gw(z)=3i(Imw)2−1/pπ1/p(z−w¯)2−4(Imw)3−1/pπ1/p(z−w¯)3.
Then it is easy to see that(3.11)gw′(w)=0,gw′′(w)=Cw2+1/p, and by Lemma 2.1 (cases n=2 and n=3) it is
easy to see that(3.12)supw∈Π+∥gw∥Hp<∞.
From this, sinceCφ:Hp(Π+)→𝒵(Π+) is bounded and
by taking w=φ(z), it follows that(3.13)CImz(Imφ(z))2+1/p|φ′(z)|2≤∥Cφ(gw)∥𝒵(Π+)≤C∥Cφ∥Hp(Π+)→𝒵(Π+),
from which (3.1) follows, as desired.
Moreover, from (3.9) and (3.13) it follows that(3.14)supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+Imz(Imφ(z))1+1/p|φ′′(z)|≤C∥Cφ∥Hp(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 p≥1), we
have
(3.15)f(z)=12πi∫−∞∞f(t)t−zdt,z∈Π+.
By differentiating formula (3.15),  we obtain(3.16)f(n)(z)=n!2πi∫−∞∞f(t)(t−z)n+1dt,z∈Π+, for each n∈ℕ, from which it follows that
(3.17)|f(n)(z)|≤n!2π∫−∞∞|f(t)|[(t−x)2+y2](n+1)/2dt,z∈Π+.
By using the changet−x=sy, we have that(3.18)∫−∞∞yn[(t−x)2+y2](n+1)/2dt=∫−∞∞ds(s2+1)(n+1)/2=:cn<∞,n∈ℕ.
From this, applying Jensen's inequality on (3.17) and an
elementary inequality, we obtain(3.19)|f(n)(z)|p≤dn∫−∞∞|f(t)|pynpyn[(t−x)2+y2](n+1)/2dt≤dn∫−∞∞|f(t)|pynp+1dt≤dn∥f∥Hp(Π+)pynp+1,where(3.20)dn=(cnn!2π)p, from which it follows that(3.21)|f(n)(z)|≤C∥f∥Hp(Π+)yn+1/p.
Assume thatf∈Hp(Π+). By applying (3.21) and Lemma 1 in [1, page 188], we have (3.22)∥Cφf∥𝒵(Π+)=|f(φ(i))|+|(f∘φ)′(i)|+supz∈Π+Imz|(Cφf)′′(z)|=|f(φ(i))|+|f′(φ(i))||φ′(i)|+supz∈Π+Imz|f′′(φ(z))(φ′(z))2+f′(φ(z))φ′′(z)|≤C∥f∥Hp(Π+)(1+supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+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/ℙ1(Π+), we have that(3.23)∥Cφ∥Hp(Π+)→𝒵/ℙ1(Π+)≤C(supz∈Π+Imz(Imφ(z))2+1/p|φ′(z)|2+supz∈Π+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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*Source: 161528-2009-04-29.xml*