Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces
The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is quite difficult. In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for the choice of a fairly simple symbol. We compute the operator norm...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507013692260352 |
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| author | Sharma, Ajay K. Sharma, Ajay K. Sharma, Ajay K. |
| author_facet | Sharma, Ajay K. Sharma, Ajay K. Sharma, Ajay K. |
| author_sort | Sharma, Ajay K. |
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| description | The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is quite difficult. In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for the choice of a fairly simple symbol. We compute the operator norm of composition operator acting between the space of Cauchy transforms and Zygmund-type spaces. We also characterize bounded and compact composition operators acting between these spaces.
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| first_indexed | 2026-03-24T02:02:34Z |
| format | Article |
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UDC 517.5
Ajay K. Sharma (Central Univ. Jammu, (Bagla), Raya-Suchani, Samba, India)
NORM OF A COMPOSITION OPERATOR FROM THE SPACE
OF CAUCHY TRANSFORMS INTO ZYGMUND-TYPE SPACES
НОРМА ОПЕРАТОРА КОМПОЗИЦIЇ З ПРОСТОРУ
ПЕРЕТВОРЕНЬ КОШI У ПРОСТОРИ ТИПУ ЗИГМУНДА
The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is
quite difficult. In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for
the choice of a fairly simple symbol. We compute the operator norm of composition operator acting between the space
of Cauchy transforms and Zygmund-type spaces. We also characterize bounded and compact composition operators acting
between these spaces.
Проблема визначення норми оператора композицiї, що дiє мiж просторами голоморфних функцiй, є дуже складною.
В просторах Гардi та зважених просторах Бергмана норма оператора композицiї є невiдомою навiть у випадку, коли
вибрано досить простий символ. Ми знаходимо операторну норму оператора композицiї, що дiє мiж просторами
перетворень Кошi та просторами типу Зигмунда. Крiм того, наведено характеристику обмежених та компактних
операторiв композицiї, що дiють мiж цими просторами.
1. Introduction. Let \BbbD be the open unit disk in the complex plane \BbbC , \BbbT the unit circle, H(\BbbD ) the
class of all holomorphic functions on \BbbD , H\infty the space of all bounded holomorphic functions on \BbbD
with the norm \| f\| \infty = \mathrm{s}\mathrm{u}\mathrm{p}z\in \BbbD | f(z)| , dA(w) the normalized area measure on \BbbD (i.e., A(\BbbD ) = 1)
and \scrM , the space of all complex Borel measures on \BbbT . Let
\eta z(w) =
z - w
1 - \=zw
, z, w \in \BbbD ,
that is, \eta z is the involutive automorphism of \BbbD interchanging points z and 0. The space of Cauchy
transforms \scrF is the collection of functions f \in H(\BbbD ) which admit a representation of the form
f(w) =
\int
\BbbT
d\mu (x)
1 - xw
.
The space \scrF becomes a Banach space under the norm
\| f\| \scrF = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \| \mu \| : f(w) =
\int
\BbbT
d\mu (x)
1 - xw
\right\} ,
where \| \mu \| denotes the total variation of the measure \mu . It is well known that
| f(w)| \leq \| f\| \scrF
1 - | w|
(1.1)
for every w \in \BbbD and f \in \scrF . For more about these spaces see [2 – 5, 8 – 12].
c\bigcirc AJAY K. SHARMA, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12 1699
1700 AJAY K. SHARMA
A strictly positive continuous function \nu on \BbbD is called weight. A weight \nu is called radial
if \nu (z) = \nu (| z| ) for all z \in \BbbD . A weight \nu is normal if there exist positive numbers \eta and \tau ,
0 < \eta < \tau , and \delta \in [0, 1) such that
\nu (r)
(1 - r)\eta
is decreasing on [\delta , 1) and \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 1
\nu (r)
(1 - r)\eta
= 0,
\nu (r)
(1 - r)\tau
is increasing on [\delta , 1) and \mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 1
\nu (r)
(1 - r)\tau
= \infty .
If we say that a function \nu : \BbbD \rightarrow [0,\infty ) is a normal weight function, then we also assume that it is
radial. It is well known that classical weights \sigma \alpha (w) = (1 - | w| 2)\alpha , \alpha > - 1 are normal weights.
For a normal weight \nu , the Zygmund-type class \scrZ \nu = \scrZ \nu (\BbbD ) consists of all f \in H(\BbbD ) such that
b\nu (f) := \mathrm{s}\mathrm{u}\mathrm{p}
w\in \BbbD
\nu (w)| f \prime \prime (w)| <\infty
with the norm
\| f\| \scrZ \nu = | f(0)| + | f \prime (0)| + b\nu (f)
the Zygmund-type class becomes a Banach space, called the Zygmund-type space.
The little Zygmund-type space, denoted by \scrZ \nu ,0 = \scrZ \nu ,0(\BbbD ) is the closed subspace of \scrZ \nu con-
sisting of all functions f such that
\mathrm{l}\mathrm{i}\mathrm{m}
| w| \rightarrow 1
\nu (w)| f \prime \prime (w)| = 0.
If \nu (w) = 1 - | w| 2, then we get the Zygmund space and the little Zygmund space. To characterize
composition operators, we need an equivalent norm for \scrZ \nu . The following lemma is possibly a
known result, but we have not managed to find it in the literature, so we give a proof.
Lemma 1. Let \nu : \BbbD \rightarrow [0,\infty ) be a normal weight function and d\lambda (w) = dA(w)/(1 - | w| 2)2.
Then f \in \scrZ \nu if and only if
\| f\| 2\scrZ \nu
\asymp | f(0)| 2 + | f \prime (0)| 2 + \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| f \prime \prime (w)| 2\nu 2(w)(1 - | \eta z(w)| 2)2d\lambda (w) <\infty ,
where the notation A \asymp B means that B \lesssim A \lesssim B and A \lesssim B means that there is some positive
constant C such that A \leq CB.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
NORM OF A COMPOSITION OPERATOR FROM THE SPACE OF CAUCHY TRANSFORMS . . . 1701
Proof. Let D(z, 1/2) =
\bigl\{
w \in \BbbD : | \eta z(w)| < 1/2
\bigr\}
. Then
| f \prime \prime (z)| 2 = | f \prime \prime (\eta z(0))| 2 \leq 4
\int
| z| <1/2
| f \prime \prime (\eta z(w))| 2dA(w) =
= 4
\int
D(z,1/2)
| f \prime \prime (w)| 2| \eta \prime z(w)| 2dA(w). (1.2)
Using the identity
(1 - | w| 2)| \eta \prime z(w)| = 1 - | \eta z(w)| 2 =
(1 - | z| 2)(1 - | w| 2)
| 1 - \=zw| 2
and the fact that \nu (z) \asymp \nu (w) for w \in D(z, 1/2) in (1.2), we have
| f \prime \prime (z)| 2 \lesssim 1
\nu 2(z)
\int
D(z,1/2)
| f \prime \prime (w)| 2\nu 2(w)(1 - | \eta z(w)| 2)2d\lambda (w).
Thus we obtain
b2\nu (f) \lesssim \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| f \prime \prime (w)| 2\nu 2(| w| )(1 - | \eta z(w)| 2)2d\lambda (w).
Therefore,
\| f\| 2\scrZ \nu
\lesssim | f(0)| 2 + | f \prime (0)| 2 + \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| f \prime \prime (w)| 2\nu 2(w)(1 - | \eta z(w)| 2)2d\lambda (w). (1.3)
Again
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| f \prime \prime (w)| 2\nu 2(w)(1 - | \eta z(w)| 2)2d\lambda (w) \leq
\leq b2\nu (f) \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
(1 - | z| 2)2
| 1 - \=zw| 4
dA(z) \lesssim b2\nu (f).
Thus,
| f(0)| 2 + | f \prime (0)| 2 + \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| f \prime \prime (w)| 2\nu 2(w)(1 - | \eta z(w)| 2)2d\lambda (w) \lesssim \| f\| 2\scrZ \nu
. (1.4)
Combining (1.3) and (1.4), we get the desired result.
The compactness of a closed subset L \subset \scrZ \nu ,0 can be characterized as follows.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1702 AJAY K. SHARMA
Lemma 2. A closed set L in \scrZ \nu ,0 is compact if and only if it is bounded with respect to the
norm \| \cdot \| \scrZ \nu and satisfies
\mathrm{l}\mathrm{i}\mathrm{m}
| w| \rightarrow 1
\mathrm{s}\mathrm{u}\mathrm{p}
f\in L
\nu (w)| f \prime \prime (w)| = 0.
This result for the for little Bloch space with \nu (w) = (1 - | w| 2) was proved by Madigan and
Matheson [13]. The above lemma can be proved by a slight modification in their proof (see also
Lemma 4.4 in [14]).
The composition operator C\varphi induced by \varphi is defined by C\varphi f = f \circ \varphi for f \in H(\BbbD ). Recently,
there has been some interest in computing the exact norm of a composition operator acting between
two holomorphic function spaces. Of course the problem of computing the operator norm of a
composition operator is quite difficult. On Hardy and weighted Bergman spaces, the norm of a
composition operator even for a choice of fairly simple symbol is unknown, see [1, 7]. In this
paper, we compute the operator norm of composition operator acting between Cauchy transforms
and Zygmund spaces. We also characterize bounded and compact composition operators acting
between these spaces, thereby, continuing the line of research in the papers [3, 15]. For more about
composition operators on the space of Cauchy integral transforms, we refer [3, 8, 9, 15].
2. Boundedness and compactness of \bfitC \bfitvarphi : \bfscrF \rightarrow \bfscrZ \bfitnu . In this section, we give the operator
norm of composition operator acting from the space of Cauchy transforms \scrF to Zygmund spaces \scrZ \nu .
Theorem 1. Let \nu be a normal weight and \varphi be a holomorphic self-map of \BbbD . Then the follo-
wing conditions are equivalent:
(a) C\varphi : \scrF \rightarrow \scrZ \nu is bounded,
(b) L := \mathrm{s}\mathrm{u}\mathrm{p}x\in \BbbT \mathrm{s}\mathrm{u}\mathrm{p}w\in \BbbD \nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| <\infty ,
(c) M := \mathrm{s}\mathrm{u}\mathrm{p}x\in \BbbT \mathrm{s}\mathrm{u}\mathrm{p}w\in \BbbD
\int
\BbbD
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2 \nu 2(w)(1 - | \eta z(w)| 2)2d\lambda (w)<\infty .
Moreover, if C\varphi : \scrF \rightarrow \scrZ \nu is bounded, then
\| C\varphi \| \scrF \rightarrow \scrZ \nu = \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| + \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| + L \asymp (2.1)
\asymp \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| + \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| +M1/2. (2.2)
Proof. (a) \leftrightarrow (b). First suppose that (a) holds. Consider the family of functions
fx(w) =
1
1 - xw
, x \in \BbbT . (2.3)
Then \| fx\| \scrF = 1, for each x \in \BbbT (see, e.g., [2, p. 468]). Thus, by the boundedness of C\varphi :
\scrF \rightarrow \scrZ \nu we have
| fx(\varphi (0))| + | f \prime x(\varphi (0))| + \mathrm{s}\mathrm{u}\mathrm{p}
w\in \BbbD
\nu (w)| (fx \circ \varphi )\prime \prime (w)| = \| C\varphi fx\| \scrZ \nu \leq \| C\varphi \| \scrF \rightarrow \scrZ \nu
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
NORM OF A COMPOSITION OPERATOR FROM THE SPACE OF CAUCHY TRANSFORMS . . . 1703
for every x \in \BbbT . Therefore, \bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| + \bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| +
+ \mathrm{s}\mathrm{u}\mathrm{p}
w\in \BbbD
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| \leq \| C\varphi - C\psi \| \scrF \rightarrow \scrZ \nu .
Taking supremum over x \in \BbbT , we see that (b) holds and
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| + \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| + L \leq \| C\varphi - C\psi \| \scrF \rightarrow \scrZ \nu . (2.4)
Conversely, suppose that (b) holds. Let f \in \scrF . Then there is a \mu \in \scrM such that \| \mu \| = \| f\| \scrF and
f(w) =
\int
\BbbT
d\mu (x)
1 - xw
. (2.5)
Differentiating (2.5) with respect to w, we get
f \prime (w) =
\int
\BbbT
x
(1 - xw)2
d\mu (x). (2.6)
Again differentiating (2.6) with respect to w, we obtain
f \prime (w) =
\int
\BbbT
2(x)2
(1 - xw)3
d\mu (x). (2.7)
Replacing w in (2.6) and in (2.7) by \varphi (w), multiplying such obtained inequalities, respectively, by
\varphi \prime \prime (w) and 2(\varphi \prime (w))2 and adding the obtained equations, we have
f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2 =
=
\int
\BbbT
\biggl(
x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
\biggr)
d\mu (x). (2.8)
By using (2.8) and an elementary inequality, we obtain
\nu (w)| (f \circ \varphi )\prime \prime (w)| = \nu (w)| f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2| \leq
\leq \nu (w)
\int
\BbbT
\bigm| \bigm| \bigm| \bigm| x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| d| \mu | (x) \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
w\in \BbbD
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| \int
\BbbT
d| \mu | (x) \leq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
1704 AJAY K. SHARMA
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
w\in \BbbD
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| \| f\| \scrF . (2.9)
Thus, it follows that
b\nu (C\varphi f) \leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
w\in \BbbD
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| \| f\| \scrF . (2.10)
Replacing w in (2.5) by \varphi (0) and using an elementary inequality, we get
| (C\varphi f)(0)| \leq
\int
\BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| d| \mu | (x) \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| \int
\BbbT
d| \mu | (x) \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| \| f\| \scrF . (2.11)
Again, replacing w in (2.6) by \varphi (0) and using an elementary inequality, we obtain
| (C\varphi f)\prime (0)| \leq
\int
\BbbT
\bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| d| \mu | (x) \leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| \| f\| \scrF . (2.12)
Combining (2.10), (2.11) and (2.12), we see that
\| C\varphi f\| \scrZ \nu \leq
\biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| + \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| + L
\biggr\}
\| f\| \scrF .
Thus, C\varphi : \scrF \rightarrow \scrZ \nu is bounded and
\| C\varphi \| \scrF \rightarrow \scrZ \nu \leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| 1
1 - x\varphi (0)
\bigm| \bigm| \bigm| \bigm| + \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\bigm| \bigm| \bigm| \bigm| \varphi \prime (0)
(1 - x\varphi (0))2
\bigm| \bigm| \bigm| \bigm| + L. (2.13)
Also from (2.6) and (2.13), (2.2) follows.
(b) \leftrightarrow (c). Assume that (b) holds. Since \nu is normal, \nu (z) \asymp \nu (w) when w \in D(z, (1 -
- | z| )/2) = \{ | w - z| < (1 - | z| )/2\} . Also it is known that | 1 - \=zw| \asymp 1 - | z| 2, for w \in
\in \Omega (z, (1 - | z| )/2). Using these facts and the subharmonicity of the function\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2,
we have
M = \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\Omega (z,(1 - | z| )/2)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2\nu 2(w)d\lambda z(w) \geq
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
NORM OF A COMPOSITION OPERATOR FROM THE SPACE OF CAUCHY TRANSFORMS . . . 1705
\geq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\nu (z)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (z)
(1 - x\varphi (z))2
+
2x(\varphi \prime (z))2
(1 - x\varphi (z))3
\bigm| \bigm| \bigm| \bigm| 2\times
\times \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\Omega (z,(1 - | z| )/2)
(1 - | z| 2)4
| 1 - zw| 4
dA(w) \geq
\geq C \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\nu (z)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2 = CL2. (2.14)
Next assume that (c) holds. Then
M \leq L2 \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
(1 - | z| 2)2
| 1 - \=zw| 4
dA(w) = L2. (2.15)
The asymptotic relation M \asymp L2 follows from (2.14) and (2.15). Moreover, the equivalence of (2.1)
and (2.2) follows.
Corollary 1. Let \nu be a normal weight, \scrA \nu is the space of all f \in H(\BbbD ) such that \| f\| \scrA \nu :=
:= \mathrm{s}\mathrm{u}\mathrm{p}w\in \BbbD \nu (w)| f(w)| <\infty and \varphi be a holomorphic self-map of \BbbD . For x \in \BbbT , let
fx(w) =
\bigl\{
\varphi \prime \prime (w)/(1 - x\varphi (w))2
\bigr\}
+
\bigl\{
2x(\varphi \prime (w))2/(1 - x\varphi (w))3
\bigr\}
.
Then C\varphi : \scrF \rightarrow \scrZ \nu is bounded if and only if the family \{ fx : x \in \BbbT \} is norm bounded in \scrA \nu .
By (1.1) it is easy to see that the unit ball of \scrF is a normal family of holomorphic functions. A
standard normal family argument then yields the proof of the following lemma (see, e.g., Proposi-
tion 3.11 of [6]).
Lemma 3. Let \nu be a normal weight and \varphi be a holomorphic self-map of \BbbD . Then C\varphi : \scrF \rightarrow \scrZ \nu
is compact if and only if for any sequence \{ fn\} n\in \BbbN in \scrF with \mathrm{s}\mathrm{u}\mathrm{p}n\in \BbbN \| fn\| \scrF \leq N which converges
to zero on compact subsets of \BbbD , we have \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| C\varphi fn\| \scrZ \nu = 0.
Theorem 2. Let \nu be a normal weight, \varphi be a holomorphic self-map of \BbbD and d\lambda z(w) = \{ (1 -
- | \eta z(w)| 2)2/(1 - | w| 2)2\} dA(w) and C\varphi : \scrF \rightarrow \scrZ \nu are bounded. Then C\varphi : \scrF \rightarrow \scrZ \nu is compact if
and only if
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 1
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (w)| >r
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2\nu 2(w)d\lambda z(w) = 0. (2.16)
Proof. First suppose that \{ fj\} j\in \BbbN is a norm bounded sequence in \scrF and \mathrm{s}\mathrm{u}\mathrm{p}j \| fj\| \scrF \leq N and
fj converges to 0 uniformly on compact subsets of \BbbD as j \rightarrow \infty . Then by the Weierstrass theorem,
f \prime j and f \prime \prime j also converges to 0 uniformly on compact subsets of \BbbD for each j \in \BbbN . By Lemma 3,
we need to show that \| C\varphi fj\| \scrZ \nu \rightarrow 0 as j \rightarrow \infty . For each j \in \BbbN , we can find a \nu j \in \scrM with
\| \nu j\| = \| fj\| \scrF such that
fj(w) =
\int
\BbbT
d\mu j(x)
1 - xw
.
Then proceeding as in (2.8), we have
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1706 AJAY K. SHARMA
f \prime j(\varphi (w))\varphi
\prime \prime (w) + f \prime \prime j (\varphi (w))(\varphi
\prime (w))2 =
\int
\BbbT
\biggl(
x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
\biggr)
d\mu j(x).
Applying Jensen’s inequality, as well as the norm boundedness of the sequence \{ fj\} j\in \BbbN , we obtain
| f \prime j(\varphi (w))\varphi \prime \prime (w) + f \prime \prime j (\varphi (w))(\varphi
\prime (w))2| 2 \leq
\leq \| \mu j\| 2
\int
\BbbT
\bigm| \bigm| \bigm| \bigm| x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2d| \mu j | (x)\| \mu j\|
\leq
\leq N
\int
\BbbT
\bigm| \bigm| \bigm| \bigm| x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2d| \mu j | (x). (2.17)
By (2.16) we have that, for every \varepsilon > 0, there is an r1 \in (0, 1) such that, for r \in (r1, 1),
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
\bigm| \bigm| \bigm| \bigm| x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2\nu 2(w)d\lambda z(w) < \varepsilon . (2.18)
Now
\| C\varphi fj\| 2\scrZ \nu
\lesssim | fj(\varphi (0))| 2 + | f \prime j(\psi (0))| 2+
+\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| \leq r
| f \prime j(\varphi (w))\varphi \prime \prime (w) + f \prime \prime j (\varphi (w))(\varphi
\prime (w))2| 2\nu 2(w)d\lambda z(w)+
+ \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| f \prime j(\varphi (w))\varphi \prime \prime (w) + f \prime \prime j (\varphi (w))(\varphi
\prime (w))2| 2\nu 2(w)d\lambda z(w). (2.19)
By taking f(z) = z in \scrF and using the fact that C\varphi : \scrF \rightarrow \scrZ \nu is bounded, we have
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| \varphi \prime \prime (w)| 2\nu 2(w)d\lambda z(w) <\infty . (2.20)
Again, by taking f(z) = z2/2 in \scrF and using the fact that C\varphi : \scrF \rightarrow \scrZ \nu is bounded, we obtain
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| \varphi (w)\varphi \prime \prime (w) + (\varphi \prime (w))2| 2\nu 2(w)d\lambda z(w) <\infty . (2.21)
Thus using an elementary inequality, (2.20), (2.21) and the fact that | \varphi (w)| < 1, we get
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| \varphi \prime (w)| 4\nu 2(w)d\lambda z(w) \leq \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| \varphi (w)\varphi \prime \prime (w) + (\varphi \prime (w))2| 2\nu 2(w)d\lambda z(w)+
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NORM OF A COMPOSITION OPERATOR FROM THE SPACE OF CAUCHY TRANSFORMS . . . 1707
+\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| \varphi \prime \prime (w)| 2\nu 2(w)d\lambda z(w) <\infty . (2.22)
Using (2.17), (2.18), (2.19), Fubini’s theorem and the fact that | fj(\varphi (0))| 2 < \varepsilon , | f \prime j(\psi (0))| 2 < \varepsilon ,
\mathrm{s}\mathrm{u}\mathrm{p}| w| \leq r | f \prime \prime j (w)| 2 < \varepsilon and \mathrm{s}\mathrm{u}\mathrm{p}| w| \leq r | f \prime \prime j (w)| 2 < \varepsilon , for sufficiently large j, say j \geq j0, we have
\| C\varphi fj\| 2\scrZ \nu
\lesssim | fj(\varphi (0))| 2 + f \prime j(\psi (0))| 2 + \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (z)| \leq r
| f \prime j(\varphi (w))| 2 \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| \varphi \prime \prime (w)| 2\nu 2(w)d\lambda z(w)+
+ \mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (z)| \leq r
| f \prime \prime j (\varphi (w))| 2 \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| \leq r
| \varphi \prime (w)| 4\nu 2(w)d\lambda z(w)+
+
\int
\BbbT
\int
| \varphi (z)| >r
\bigm| \bigm| \bigm| \bigm| x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2d\lambda z(w)d| \mu j | (x) <
< C
\biggl(
1 +
\int
\BbbT
d| \mu j | (x)
\biggr)
\varepsilon < C\varepsilon .
Since \varepsilon > 0 is arbitrary, so by Lemma 3, it follows that C\varphi : \scrF \rightarrow \scrZ \nu is compact.
Conversely, suppose that C\varphi : \scrF \rightarrow \scrZ \nu is compact. For each \epsilon > 0, using (2.20) and (2.22), we
can choose r \in (0, 1) such that
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| \varphi \prime \prime (w)| 2\nu 2(w)d\lambda z(w) < \varepsilon (2.23)
and
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| \varphi \prime (w)| 4\nu 2(w)d\lambda z(w) < \varepsilon . (2.24)
Let f \in B\scrF and ft(w) = f(tw), 0 < t < 1. Then for each t \in (0, 1), ft \in \scrF and \mathrm{s}\mathrm{u}\mathrm{p}0<t<1 \| ft\| \scrF \leq
\leq \| f\| \scrF . Moreover, ft \rightarrow f uniformly on compact subsets of \BbbD as t \rightarrow 1. By the compactness of
C\varphi : \scrF \rightarrow \scrZ \nu , we obtain
\mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 1
\| C\varphi ft - C\varphi f\| \scrZ \nu = 0.
Hence, for every \varepsilon > 0, there is a t \in (0, 1) such that
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| (f \prime t(\varphi (w))\varphi \prime \prime (w) + f \prime \prime t (\varphi (w))(\varphi
\prime (w))2) -
- (f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2)| 2\nu 2(w)d\lambda z(w) < \varepsilon . (2.25)
From the inequalities (2.23), (2.24), and (2.25), we have
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1708 AJAY K. SHARMA
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2| 2\nu 2(w)d\lambda z(w) \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| (f \prime t(\varphi (w))\varphi \prime \prime (w) + f \prime \prime t (\varphi (w))(\varphi
\prime (w))2) -
- (f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2)| 2\nu 2(w)d\lambda z(w)+
+ \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| f \prime t(\varphi (w))\varphi \prime \prime (w) + f \prime \prime t (\varphi (w))(\varphi
\prime (w))2| 2\nu 2(w)d\lambda z(w) \leq
\leq \varepsilon C(1 + \| f \prime t\| 2\infty ) for some constant C > 0.
Hence, for every f \in B\scrF , there is a \delta 0 \in (0, 1), \delta 0 = \delta 0(f, \varepsilon ), such that for r \in (\delta 0, 1)
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2| 2\nu 2(w)d\lambda z(w) < \varepsilon . (2.26)
From the compactness of C\varphi : \scrF \rightarrow \scrZ \nu , we have that for every \varepsilon > 0 there is a finite collection of
functions f1, f2, . . . , fm \in B\scrF such that for each f \in B\scrF , there is a k \in \{ 1, 2, . . . ,m\} such that
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
| (f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2) -
- (f \prime k(\varphi (w))\varphi
\prime \prime (w) + f \prime \prime k (\varphi (w))(\varphi
\prime (w))2)| 2\nu 2(w)d\lambda z(w) < \varepsilon . (2.27)
On the other hand, from (2.26) it follows that if \delta := \mathrm{m}\mathrm{a}\mathrm{x}1\leq j\leq m \delta j(fk, \varepsilon ), then, for r \in (\delta , 1) and
all k \in \{ 1, 2, . . . ,m\} , we have
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| f \prime k(\varphi (w))\varphi \prime \prime (w) + f \prime \prime k (\varphi (w))(\varphi
\prime (w))2| 2\nu 2(w)d\lambda z(w) < \varepsilon . (2.28)
From (2.27) and (2.28), we have that for r \in (\delta , 1) and every f \in B\scrF , there is a constant C > 0
such that
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
| f \prime (\varphi (w))\varphi \prime \prime (w) + f \prime \prime (\varphi (w))(\varphi \prime (w))2| 2\nu 2(w)d\lambda z(w) < \varepsilon C. (2.29)
Applying (2.29) to the family of functions fx(w) = 1/(1 - xw), x \in \BbbT , we obtain
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (z)| >r
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2\nu 2(w)d\lambda z(w) < \varepsilon C.
Since \varepsilon > 0 is arbitrary, (2.16) follows.
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NORM OF A COMPOSITION OPERATOR FROM THE SPACE OF CAUCHY TRANSFORMS . . . 1709
Corollary 2. Let \nu be a normal weight, \varphi be a holomorphic self-map of \BbbD and C\varphi : \scrF \rightarrow \scrZ \nu
are bounded. If
\mathrm{l}\mathrm{i}\mathrm{m}
r\rightarrow 1
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (w)| >r
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| = 0, (2.30)
then C\varphi : \scrF \rightarrow \scrZ \nu is compact.
Proof. Suppose that (2.30) holds. Then
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (w)| >r
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2\nu 2(w)d\lambda z(w) \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (w)| >r
\nu 2(w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2 \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
| \varphi (w)| >r
d\lambda z(w) \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (w)| >r
\nu 2(w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| 2 \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbD
\int
\BbbD
(1 - | z| 2)2
| 1 - \=zw| 4
dA(w) \leq
\leq
\biggl(
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\mathrm{s}\mathrm{u}\mathrm{p}
| \varphi (w)| >r
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| \biggr) 2
.
Taking limit as r \rightarrow 1, by Theorem 2, we see that C\varphi : \scrF \rightarrow \scrZ \nu is compact.
3. Boundedness and compactness of \bfitC \bfitvarphi : \bfscrF \rightarrow \bfscrZ \bfitnu ,0 . In this section, we characterize the
boundedness and compactness of C\varphi : \scrF \rightarrow \scrZ \nu ,0.
Theorem 3. Let \nu be a normal weight and \varphi be a holomorphic self-map of \BbbD . Then C\varphi :
\scrF \rightarrow \scrZ \nu ,0 is bounded if and only if C\varphi : \scrF \rightarrow \scrZ \nu is bounded and
\mathrm{l}\mathrm{i}\mathrm{m}
| w| \rightarrow 1
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| = 0 (3.1)
for every x \in \BbbT .
Proof. First suppose that C\varphi : \scrF \rightarrow \scrZ \nu ,0 is bounded. Then it is obvious that C\varphi : \scrF \rightarrow \scrZ \nu is
bounded. Once again consider the family of test functions in (2.3). Then \| fx\| \scrF = 1 and
fkx (\varphi (w))\varphi
\prime \prime (w) + f \prime \prime x (\varphi (w))(\varphi
\prime (w))2 =
x\varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2(x)2(\varphi \prime (w))2
(1 - x\varphi (w))3
.
Thus, by the boundedness of C\varphi : \scrF \rightarrow \scrZ \nu ,0, we have C\varphi fx \in \scrZ \nu ,0 for every x \in \BbbT and so
\mathrm{l}\mathrm{i}\mathrm{m}
| z| \rightarrow 1
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| = 0
for every x \in \BbbT .
Conversely, suppose that C\varphi : \scrF \rightarrow \scrZ \nu is bounded and (3.1) hold. By (3.1), the integrand in
(2.9) tends to zero for every x \in \BbbT , as | z| \rightarrow 1, and is dominated by M, where M is as in (b)
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1710 AJAY K. SHARMA
of Theorem 1. Thus by the Lebesgue convergence theorem, the integral in (2.9) tends to zero as
| w| \rightarrow 1, implying
\mathrm{l}\mathrm{i}\mathrm{m}
| w| \rightarrow 1
\nu (w)| (C\varphi f)\prime \prime (z)| = 0.
Hence, for every f \in \scrF , we have that C\varphi f \in \scrZ \nu ,0, from which the boundedness of C\varphi : \scrF \rightarrow \scrZ \nu ,0
follows.
Theorem 4. Let \nu be a normal weight and \varphi be a holomorphic self-map of \BbbD . Then C\varphi :
\scrF \rightarrow \scrZ \nu ,0 is compact if and only if
\mathrm{l}\mathrm{i}\mathrm{m}
| w| \rightarrow 1
\mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| = 0. (3.2)
Proof. By Lemma 2, a closed set E in \scrZ \nu ,0 is compact if and only if it is bounded and satisfies
\mathrm{l}\mathrm{i}\mathrm{m}
| w| \rightarrow 1
\mathrm{s}\mathrm{u}\mathrm{p}
f\in E
\nu (w)| f \prime \prime (w)| = 0.
Thus, the set \{ C\varphi f : f \in \scrF , \| f\| \scrF \leq 1\} has compact closure in \scrZ \nu ,0 if and only if
\mathrm{l}\mathrm{i}\mathrm{m}
| w| \rightarrow 1
\mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
\nu (w)| (C\varphi f)\prime \prime (w)| : f \in \scrF , \| f\| \scrF \leq 1
\bigr\}
= 0. (3.3)
Let f \in B\scrF , then there is a \mu \in \frakM such that \| \mu \| = \| f\| \scrF and
f(z) =
\int
\BbbT
d\mu (x)
1 - xz
.
Thus, we easily get that for each f \in B\scrF
\nu (w)| (C\varphi f)\prime \prime (z)| \leq
\int
\BbbT
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| d| \mu | (x) \leq
\leq \| \mu \| \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| \leq
\leq \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbT
\nu (w)
\bigm| \bigm| \bigm| \bigm| \varphi \prime \prime (w)
(1 - x\varphi (w))2
+
2x(\varphi \prime (w))2
(1 - x\varphi (w))3
\bigm| \bigm| \bigm| \bigm| . (3.4)
Using (3.2) in (3.4), we get (3.3). Hence C\varphi : \scrF \rightarrow \scrZ \nu ,0 is compact. Conversely, suppose that
C\varphi : \scrF \rightarrow \scrZ \nu ,0 is compact. Taking the test functions in (2.3), we can easily obtain that (3.2) follows
from (3.3).
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NORM OF A COMPOSITION OPERATOR FROM THE SPACE OF CAUCHY TRANSFORMS . . . 1711
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Received 28.11.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 12
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| id | umjimathkievua-article-367 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:02:34Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/34/31a500d997f533354ed10dea2df6f234.pdf |
| spelling | umjimathkievua-article-3672020-01-09T12:22:37Z Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces Норма оператора композиции из пространства преобразований Коши в пространства типа Зигмунда Норма оператора композицiї з простору перетворень Кошi у простори типу Зигмунда Sharma, Ajay K. Sharma, Ajay K. Sharma, Ajay K. composition operator, space of Cauchy transforms, Zygmund-type space, little Zygmund-type space The problem of evaluation of the norm of a composition operator acting between the spaces of holomorphic functions is quite difficult. In the Hardy and weighted Bergman spaces, the norm of the composition operator is unknown even for the choice of a fairly simple symbol. We compute the operator norm of composition operator acting between the space of Cauchy transforms and Zygmund-type spaces. We also characterize bounded and compact composition operators acting between these spaces. Проблема визначення норми оператора композиції, що діє між просторами голоморфних функцій, є дуже складною. В просторах Гарді та зважених просторах Бергмана норма оператора композиції є невідомою навіть у випадку, коли вибрано досить простий символ.Ми знаходимо операторну норму оператора композиції, що діє між просторами перетворень Коші та просторами типу Зигмунда.Крім того, наведено характеристику обмежених та компактних операторів композиції, що діють між цими просторами. Institute of Mathematics, NAS of Ukraine 2019-12-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/367 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 12 (2019); 1699-1711 Український математичний журнал; Том 71 № 12 (2019); 1699-1711 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/367/1529 https://umj.imath.kiev.ua/index.php/umj/article/view/367/1530 |
| spellingShingle | Sharma, Ajay K. Sharma, Ajay K. Sharma, Ajay K. Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces |
| title | Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces |
| title_alt | Норма оператора композиции из пространства преобразований Коши в пространства типа Зигмунда Норма оператора композицiї з простору перетворень Кошi у простори типу Зигмунда |
| title_full | Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces |
| title_fullStr | Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces |
| title_full_unstemmed | Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces |
| title_short | Norm of a composition operator from the space of Cauchy transforms into Zygmund-type spaces |
| title_sort | norm of a composition operator from the space of cauchy transforms into zygmund-type spaces |
| topic_facet | composition operator space of Cauchy transforms Zygmund-type space little Zygmund-type space |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/367 |
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