Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 161528, 8 pages
doi:10.1155/2009/161528
Research Article
Composition Operators from the Hardy Space to
the Zygmund-Type Space on the Upper Half-Plane
Stevo Stević
Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, 11001 Beograd, Serbia
Correspondence should be addressed to Stevo Stević, sstevic@ptt.rs
Received 14 December 2008; Accepted 23 February 2009
Recommended by Simeon Reich
Here we introduce the nth weighted space on the upper half-plane Π {z ∈ C : Im z > 0} in the
complex plane C. For the case n 2, we call it the Zygmund-type space, and denote it by ZΠ .
The main result of the paper gives some necessary and sufficient conditions for the boundedness
of the composition operator Cϕ fz fϕz from the Hardy space H p Π on the upper halfplane, to the Zygmund-type space, where ϕ is an analytic self-map of the upper half-plane.
Copyright q 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.
1. Introduction
Let Π be the upper half-plane, that is, the set {z ∈ C : Im z > 0} and HΠ the space of all
analytic functions on Π . The Hardy space H p Π H p , p > 0, consists of all f ∈ HΠ such that
p
fH p
sup
y>0
∞
−∞
p
f x iy dx < ∞.
1.1
With this norm H p Π is a Banach space when p ≥ 1, while for p ∈ 0, 1 it is a Fréchet space
p
with the translation invariant metric df, g f − gH 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 ∈ HΠ such that
sup Im zf n z < ∞,
z∈Π
1.2
2
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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 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 ZΠ Z. Recall that the space consists of all
f ∈ HΠ such that
bZ f sup Im zf z < ∞.
1.3
z∈Π
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:
fZ fi f i bZ f.
1.4
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
iθh f e
f eiθ−h − 2f eiθ h
h>0, θ∈0,2π
<∞
1.5
if and only if
sup 1 − |z|2 f z < ∞.
1.6
z∈D
The family of all analytic functions on D satisfying condition 1.6 is called the
Zygmund class on the unit disk.
With the norm
f f0 f 0 sup 1 − |z|2 f z,
1.7
z∈D
the Zygmund class becomes a Banach space. Zygmund class with this norm is called the
Zygmund space and is denoted by ZD. 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 Cϕ : H p Π → A∞ Π is bounded if and only if
Im z
sup z∈Π
Im ϕz
1/p < ∞.
1.8
b The operator Cϕ : H p Π → B∞ Π is bounded if and only if
Im z
11/p ϕ z < ∞.
Im ϕ z
sup z∈Π
1.9
Motivated by Theorem A, here we investigate the boundedness of the operator Cϕ :
H p Π → ZΠ . 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
Im wn−1/p
,
z − wn
2.1
sup fw, n H p ≤ π 1/p .
2.2
fw, n z belongs to H p Π . Moreover
w∈Π
4
Abstract and Applied Analysis
Proof. Let z x iy and w u iυ. Then, we have
fw, n p p sup
H
∞
−∞
y>0
fw, n x iyp dx
Im wnp−1 sup
∞
y>0
≤ vnp−1 sup
y>0
∞
−∞
≤ vnp−1 sup
dx
np−2 |z − w|2
|z
−
w|
−∞
y>0
v
2 np−2/2 np−1
y v
∞
np−1
y v
x − u2 y v2
y v
∞
1
y>0
sup
dx
np−1
−∞
−∞ x
yv
− u2 y v2
2.3
dx
dt
π,
1
t2
where we have used the change of variables x u ty 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 p Π → ZΠ is bounded if and only if
2
Im z
21/p ϕ z < ∞,
Im ϕz
3.1
Im z
11/p ϕ z < ∞.
Im ϕz
3.2
sup z∈Π
sup z∈Π
Moreover, if the operator Cϕ : H p Π → Z/P1 Π is bounded, then
Cϕ H p Π → Z/P1 Π 2
Im z
Im z
21/p ϕ z sup 11/p ϕ z.
z∈Π Im ϕz
Im ϕz
sup z∈Π
3.3
Proof. First assume that the operator Cϕ : H p Π → ZΠ is bounded.
For w ∈ Π , set
fw z Im w2−1/p
π 1/p z − w2
.
3.4
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5
By Lemma 2.1 case n 2 we know that fw ∈ H p Π for every w ∈ Π . Moreover,
we have that
sup fw H p Π ≤ 1.
3.5
w∈Π
From 3.5 and since the operator Cϕ : H p Π → ZΠ is bounded, for every w ∈
Π , we obtain
2
sup Im zfw ϕz ϕ z fw ϕz ϕ z Cϕ fw ZΠ ≤ Cϕ H p Π → ZΠ .
z∈Π
3.6
We also have that
fw z −2
Im w2−1/p
π 1/p z − w3
,
fw z 6
Im w2−1/p
π 1/p z − w4
.
3.7
Replacing 3.7 in 3.6 and taking w ϕz, we obtain
2
3
ϕ z
ϕ z
i
Im z −
≤ π 1/p Cϕ H p Π → ZΠ ,
8 Im ϕz 21/p 4 Im ϕz 11/p 3.8
and consequently
2
1
Im z
Im z
3
ϕ z ≤ π 1/p Cϕ p
ϕ z .
H Π → ZΠ 4 Im ϕz11/p
8 Im ϕz21/p
3.9
Hence if we show that 3.1 holds then from the last inequality, condition 3.2 will
follow.
For w ∈ Π , set
gw z 3 Im w2−1/p
Im w3−1/p
−
4
.
i π 1/p z − w2
π 1/p z − w3
3.10
Then it is easy to see that
gw
w 0 ,
gw
w C
w21/p
,
3.11
and by Lemma 2.1 cases n 2 and n 3 it is easy to see that
sup gw H p < ∞.
w∈Π
3.12
6
Abstract and Applied Analysis
From this, since Cϕ : H p Π → ZΠ is bounded and by taking w ϕz, it follows
that
C
2 Im z
21/p ϕ z ≤ Cϕ gw ZΠ ≤ CCϕ H p Π → ZΠ ,
Im ϕz
3.13
from which 3.1 follows, as desired.
Moreover, from 3.9 and 3.13 it follows that
2
Im z
Im z
21/p ϕ z sup 11/p ϕ z ≤ CCϕ H p Π → ZΠ .
z∈Π Im ϕz
Im ϕz
sup z∈Π
3.14
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
1
fz 2πi
∞
ft
dt,
−∞ t − z
z ∈ Π .
3.15
By differentiating formula 3.15, we obtain
f n z n!
2πi
∞
ft
−∞ t
− zn1
dt,
z ∈ Π ,
3.16
for each n ∈ N, from which it follows that
n n!
f z ≤
2π
ft
∞
−∞
t − x2 y2
n1/2 dt,
z ∈ Π .
3.17
By using the change t − x sy, we have that
∞
−∞
yn
2
t − x y
n1/2 dt 2
∞
−∞ s2
ds
n1/2 : cn < ∞,
1
n ∈ N.
3.18
From this, applying Jensen’s inequality on 3.17 and an elementary inequality, we
obtain
n p
f z ≤ dn
≤ dn
p
∞ ft
−∞
ynp
p
∞ ft
−∞
ynp1
yn
n1/2 dt
t − x2 y2
p
dt ≤ dn
fH p Π ynp1
,
3.19
Abstract and Applied Analysis
7
where
dn cn n!
2π
p
,
3.20
from which it follows that
p
n f z ≤ C fH Π .
n1/p
y
3.21
Assume that f ∈ H p Π . By applying 3.21, and Lemma 1 in 1, page 188, we have
Cϕ f ZΠ f ϕi f ◦ ϕ i sup Im z Cϕ f z
z∈Π
2
f ϕi f ϕi ϕ i sup Im zf ϕz ϕ z f ϕz ϕ z
z∈Π
z∈Π
2
Im z
Im z
.
ϕ
ϕ
z
sup
z
21/p
11/p
z∈Π Im ϕz
Im ϕz
3.22
≤ CfH p Π 1 sup From this and by conditions 3.1 and 3.2, it follows that the operator Cϕ : H p Π →
ZΠ is bounded. Moreover, if we consider the space Z/P1 Π , we have that
Cϕ p
≤
C
sup H Π → Z/P1 Π z∈Π
2
Im z
Im z
ϕ
ϕ
z
sup
z
.
21/p
11/p
z∈Π Im ϕz
Im ϕz
3.23
From 3.14 and 3.23, we obtain the asymptotic relation 3.3.
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