EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 2, 2024, 931-944
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Volterra-Composition Operators Acting on Sp Spaces
and Weighted Zygmund Spaces

Waleed Al-Rawashdeh

Department of Mathematics, Zarqa University, 2000 Zarqa, 13110 Jordan

Abstract. Let φ be an analytic selfmap of the open unit disk D and g be an analytic function on
D. The Volterra-type composition operators induced by the maps g and φ are defined as

(
Iφg f

)
(z) =

∫ z

0

f ′(φ(ζ))g(ζ)dζ and
(
Tφ
g f

)
(z) =

∫ z

0

f(φ(ζ))g′(ζ)dζ.

For 1 ≤ p < ∞, Sp(D) is the space of all analytic functions on D whose first derivative f ′ lies in
the Hardy space Hp(D), endowed with the norm ∥f∥Sp = |f(0)|+ ∥f ′∥Hp . Let µ : (0, 1] → (0,∞)
be a positive continuous function on D such that for z ∈ D we define µ(z) = µ(|z|). The weighted
Zygmund space Zµ(D) is the space of all analytic functions f on D such that supz∈D µ(z)|f ′′(z)|
is finite. In this paper, we characterize the boundedness and compactness of the Volterra-type
composition operators that act between Sp spaces and weighted Zygmund spaces.

2020 Mathematics Subject Classifications: 47B33, 47B38, 30H10, 30H20, 47B37, 30H05,
32C15

Key Words and Phrases: Weighted Zygmund Spaces, Sp spaces, Volterra operators, composi-
tion operators, bounded operators, compact operators

1. Introduction

Let D be the open unit disk {z ∈ D : |z| < 1} in the complex plane C. Let H(D) be the
space of all analytic functions on the open unit disk D. For 1 ≤ p < ∞, the analytic Hardy
space Hp(D on the unit disk D is the Banach space of all analytic functions f ∈ H(D)
such that

∥f∥pHp = sup
0<r<1

∫
∂D

|f(rζ)|pdσ(ζ) < ∞,

where σ is the normalized Lebesgue measure on the boundary of the unit disk. For f
belongs to Hp(D), it is well known from Fatou’s theorem that the radial limit

f∗(ζ) = lim
r→1−

f(rζ)

DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5113

Email address: walrawashdeh@zu.edu.jo (W. Al-Rawashdeh)

https://www.ejpam.com 931 © 2024 EJPAM All rights reserved.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 932

exists for almost all ζ on ∂D. Moreover,

∥f∥pHp =

∫
∂D

|f∗(ζ)|pdσ(ζ),

for all finite values of p. We often use a standard synonym of notation of writing f instead
of f∗. For 1 ≤ p < ∞, Sp(D) is the space of all analytic functions on the unit disk D
whose first derivative f ′ lies in the Hardy space Hp(D) endowed with the norm

∥f∥Sp = |f(0)|+ ∥f ′∥Hp .

One can easily show that Sp is a Banach space with respect to this norm. It is well known
that Sp is a Banach algebra when the norm of f ∈ Sp is defined by ∥f∥∞ + ∥f ′∥Hp . For
more information about these spaces we refer the readers to the monograph [5], the papers
([1], [2]), and the references therein.

Let µ : (0, 1] → (0,∞) be a positive continuous function on D such that for z ∈ D we
define µ(z) = µ(|z|). The weighted Zygmund space Zµ is the space of all analytic functions
f on the open unit disk D such that

sup
z∈D

µ(z)|f ′′(z)| < ∞.

It is a Banach space with the norm

∥f∥Zµ = |f(0)|+ |f ′(0)|+ sup
z∈D

µ(z)|f ′′(z)|.

When µ(z) = 1 − |z|2, this space is known as the classical Zygmund space Z. From
Zygmund’s theorem ([6], Theorem 5.3),we know that f ∈ Z if and only if f is continuous
on D and

∥f∥ = sup

∣∣f (
ei(θ+h)

)
+ f

(
ei(θ−h)

)
− 2f

(
eiθ

)∣∣
h

< ∞,

where the supremum is taken over all θ ∈ R and h > 0.

Let g be an analytic function on D, the Volterra type operator (see [18]) is defined as

(Tgf) (z) =

∫ z

0
f(ζ)g′(ζ)dζ,

where f ∈ H(D) and z ∈ D. Note that Tg can be viewed as a generalization of the cesâro
operator whose first studied by Aleman and Siskkis [4].

It is natural to define another Volterra type operator Ig as follows

(Igf) (z) =

∫ z

0
f ′(ζ)g(ζ)dζ.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 933

Recently, many researchers considered these operators and characterized their bound-
edness and compactness between various spaces of analytic functions, for example see ([3],
[8], [10], [11], [12], [13], [14], [16], [20]) and the references therein.

Let φ be an analytic function maps D into itself, the composition operator induced by
φ is defined on the space H(D) of all analytic functions on D by

Cφf(z) = f(φ(z)),

for all f ∈ H(D) and z ∈ D. It is well known that the composition operator Cφf = f ◦ φ
defines a linear operator Cφ which acts boundedly on various spaces of analytic or har-
monic functions on D. These operators have been studied on many spaces of analytic
functions. During the past few decades much effort has been devoted to the study of these
operators with the goal of explaining the operator-theoretic properties of Cφ in terms of
the function-theoretic properties of the induced map φ. We refer the reader to the mono-
graphs ([5], [7], [9], [15], [17], [22], [23]) and the references therein.

Let g be a fixed analytic function on D, f an analytic function of D and z ∈ D. The
Volterra type composition operators are defined as

(
Tφ
g f

)
(z) =

∫ z

0
f(φ(ζ))g′(ζ)dζ,

(
Iφg f

)
(z) =

∫ z

0
f ′(φ(ζ))g(ζ)dζ.

The classical Volterra operators are obtained in the case when φ(z) = z. These operators
have been studied by many researchers, for example see ([3], [11], [12], [19], [21], [24], [25])
and the references therein.

In this paper, we are investigating the boundedness and compactness of the Volterra
type composition operators Tφ

g and Iφg acting between Sp(D) spaces and weighted Zyg-
mund spaces Zµ.

2. Preliminaries

In this section we present some well known, but useful, information that are curial
for the main results of this paper. The following lemma is a well known fact that can be
proven by using Cauchy estimates, so we omit the proof.

Lemma 1. If {fn} is a sequence converges to zero on compact subsets of D, then {f ′
n}

also converges to zero on compact subsets of D as n → ∞. In particular if K is a compact
subset of D, then lim

n→∞
sup
w∈K

|f ′(w)| = 0.

The following lemma is a know fact, for the readers who are interested in its proof we
refer them to (Theorem 1, [16]).



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 934

Lemma 2. If 1 ≤ p < ∞, the following are true:

(i) Sp(D) ⊂ S1(D) ⊂ H∞;

(ii) ∥f∥∞ ≤ π∥f∥S1(D) ≤ π∥f∥Sp(D);

(iii) Sp(D) is a Banach algebra;

(iv) polynomials are dense in Sp(D).

3. Boundedness and Compactness of Iφg

In this section, we characterize the boundedness and compactness of the operator Iφg
acting between Sp spaces and weighted Zygmund spaces Zµ. The results will be written
in terms of

K1(z) =
µ(z)|g′(z)|

(1− |φ(z)|2)1/p
,

and

K2(z) =
µ(z)|g(z)||φ′(z)|

(1− |φ(z)|2)(1+p)/p
,

where z ∈ D, g ∈ H(D), and φ is the analytic selfmap of D.

In the following Theorem 1, we characterize the boundedness of Iφg that acts between
Sp spaces and weighted Zygmund spaces.

Theorem 1. Let g be an analytic function on D and φ be an analytic selfmap of D. Then
Iφg : Sp → Zµ is bounded if and only if

M1 = sup
z∈D

K1(z) < ∞ and M2 = sup
z∈D

K2(z) < ∞.

Proof. Suppose that Iφg : Sp → Zµ is bounded.
First, for a fixed w ∈ D we consider the test function

f1,w(z) =

(
1− |φ(w)|2

)(2p−1)/p

φ(w)(1− φ(w)z)
.

By direct calculations, we get

∥f1,w∥Sp = ∥f ′
1,w∥Hp ≤ 2(2p−2)/p, (1)

f ′
1,w(φ(w)) =

1

(1− |φ(w)|2)1/p
, (2)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 935

and

f ′′
1,w(φ(w)) =

2φ(w)

(1− |φ(w)|2)(p+1)/p
. (3)

Therefore, we obtain the following(
Iφg f1,w

)′′
(w) =

(
f ′
1,w(φ(w))g(w)

)′
= f ′′

1,w(φ(w))g(w)φ
′(w) + f ′

1,w(φ(w))g
′(w)

=
2φ(w)φ′(w)g(w)

(1− |φ(w)|2)(p+1)/p
+

g′(w)

(1− |φ(w)|2)1/p
(4)

Moreover, we consider another test function

f2,w(z) =

(
1− |φ(w)|2

)(3p−1)/p

2φ(w)(1− φ(w)z)2
.

By direct calculations, we get

∥f2,w∥Sp = ∥f ′
2,w∥Hp ≤ 2(3p−2)/p,

f ′
2,w(φ(w)) =

1

(1− |φ(w)|2)1/p
,

and

f ′′
2,w(φ(w)) =

3φ(w)

(1− |φ(w)|2)(p+1)/p
.

Similarly, we obtain the following

(
Iφg f2,w

)′′
(w) =

3φ(w)φ′(w)g(w)

(1− |φ(w)|2)(p+1)/p
+

g′(w)

(1− |φ(w)|2)1/p
(5)

Now, using equations (4) and (5), we get

(
Iφg f2,w

)′′
(w)−

(
Iφg f1,w

)′′
(w) =

φ(w)φ′(w)g(w)

(1− |φ(w)|2)(p+1)/p
.

Hence, by the boundedness of Iφg : Sp → Zµ we have

µ(w)
∣∣∣φ(w)φ′(w)g(w)

∣∣∣
(1− |φ(w)|2)(p+1)/p

≤ ∥Iφg ∥∥f2,w∥Sp + ∥Iφg ∥∥f1,w∥Sp

≤ ∥Iφg ∥
(
2(3p−2)/p + 2(2p−2)/p

)
≤ C1 (6)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 936

On the other hand, Lemma 2 tells us that polynomials are dense in Sp spaces. Thus
Pn(z) = zn in Sp and we get the following

(
Iφg P1

)′′
(z) =

(∫ z

0
P ′
1(φ(w))g(w)dw

)′′

=

(∫ z

0
g(w)dw

)′′

= g′(z).

Similarly, we obtain the following

(
Iφg P2

)′′
(z) =

(∫ z

0
P ′
2(φ(w))g(w)dw

)′′

=

(∫ z

0
2φ(w)g(w)dw

)′′

= 2φ(z)g′(z) + 2φ′(z)g(z).

Hence, using the previous equations, we get

2
(
φ′g

)
(z) =

(
Iφg P2

)′′
(z)− 2φ(z)

(
Iφg P1

)′′
(z). (7)

Therefore, using equation (7) we get

sup
w∈D

µ(w)|φ′(w)g(w)| ≤ 1

2
∥Iφg P2∥Zµ + sup

w∈D

(
∥Iφg P1∥Zµ sup

w∈D
|φ(w)|

)
≤ ∥Iφg ∥∥P2∥SP + ∥Iφg ∥∥P1∥SP

≤ C2

Now, for a fixed 0 < r < 1, consider w ∈ D such that 0 ≤ |φ(w)| ≤ r < 1. Then we get

µ(w)|φ(w)φ′(w)g(w)|
(1− |φ(w)|2)(p+1)/p

≤ µ(w)|φ′(w)g(w)|
(1− |φ(w)|2)(p+1)/p

≤ C2

(1− r2)1+1/p
(8)

Moreover, consider w ∈ D such that r < |φ(w)| < 1. Then we get

µ(w)|rφ′(w)g(w)|
(1− |φ(w)|2)(p+1)/p

≤ µ(w)|φ(w)φ′(w)g(w)|
(1− |φ(w)|2)(p+1)/p

≤ ∥Iφg ∥
(
2(3p−2)/p + 2(2p−2)/p

)
.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 937

Hence, using equation (6), we get

µ(w)|φ′(w)g(w)|
(1− |φ(w)|2)(p+1)/p

≤ C1

r
. (9)

Therefore, using inequalities (8) and (9), we get

M2 = sup
z∈D

µ(z)|φ′(z)g(z)|
(1− |φ(z)|2)(p+1)/p

≤ max

{
C1

r
+

C2

(1− r2)1+1/p

}
< ∞.

Second, for a fixed w ∈ D, using equations (2), (3) and (4) we get

µ(w)|g′(w)|
(1− |φ(w)|2)1/p

= µ(w)
∣∣g′(w)f ′

1,w(φ(w)) + f ′′
1,w(φ(w))g(w)φ

′(w)− f ′′
1,w(φ(w))g(w)φ

′(w)
∣∣

= µ(w)

∣∣∣∣∣(Iφg f1,w)′′ (w)− 2φ(w)g(w)φ′(w)

(1− |φ(w)|2)(p+1)/p

∣∣∣∣∣
≤ ∥Iφg f1,w∥Zµ +

2µ(w)|φ(w)g(w)φ′(w)|
(1− |φ(w)|2)(p+1)/p

≤ ∥Iφg ∥∥f1,w∥SP + 2M2.

Taking the supremum over all w ∈ D, we get that M1 < ∞.

Conversely, Suppose that conditions M1 and M2 are finite. Let f ∈ Sp, then it is well
known, see [6] or [23], that for all z ∈ D we have

|f ′(z)| ≤ ∥f ′∥Hp

(1− |z|2)1/p
,

and

|f ′′(z)| ≤ ∥f ′∥Hp

(1− |z|2)1+1/p
.

Therefore, for z ∈ D, we have

µ(z)
∣∣∣(Iφg f)′′ (z)∣∣∣ = µ(z)

∣∣∣∣(∫ z

0
f ′(φ(w))g(w)dw

)′′∣∣∣∣
=

(
f ′(φ(z))g(z)

)′
= µ(z)

∣∣f ′′(φ(z))φ′(z)g(z) + f ′(φ(z))g′(z)
∣∣

≤ µ(z)|φ′(z)g(z)|
(1− |φ(z)|2)(p+1)/p

∥f ′∥Hp +
µ(z)|g′(z)|

(1− |φ(z)|2)1/p
∥f ′∥Hp



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 938

≤ (M1 +M2) (∥f∥Sp − |f(0)|)
≤ (M1 +M2) ∥f∥Sp .

Taking the supremum over all z ∈ D, we get

∥
(
Iφg f

)
(z)∥Zµ ≤ (M1 +M2) ∥f∥Sp .

Hence, Iφg is bounded, as desired.

In the following Theorem 2, we characterize the compactness of Iφg that acts between
Sp spaces and weighted Zygmund spaces.

Theorem 2. Let g be an analytic function on D, φ be an analytic selfmap of D and
Iφg : Sp → Zµ be bounded. Then Iφg is compact if and only if

lim
|φ(z)|→1

K1(z) = 0 and lim
|φ(z)|→1

K2(z) = 0. (10)

Proof. Suppose Iφg is compact. Let {zn}n∈N be a sequence in the open unit disk D
such that |φ(zn)| → 1 as n → ∞. For each n ∈ N, consider the test functions f1,w and
f2,w we used in the proof of Theorem 1 with w = zn.

Then, we get

f1,zn(φ(zn)) = φ(zn)
(
1− |φ(zn)|2

)1−1/p
,

and

f2,zn(φ(zn)) =
φ(zn)

2

(
1− |φ(zn)|2

)1−1/p (
2− |φ(zn)|2

)
.

Hence, the sequences {f1,zn} and {f2,zn} converge to zero uniformly on D. Then, by the
compactness of Iφg , we get

lim
n→∞

∥Iφg f1,zn∥Zµ = 0 and lim
n→∞

∥Iφg f2,zn∥Zµ = 0. (11)

Now, following similar argument as in the proof of Theorem 1, we get

µ(zn)
∣∣∣φ(zn)φ′(zn)g(zn)

∣∣∣
(1− |φ(zn)|2)(p+1)/p

≤ ∥Iφg f1,w∥Zµ + ∥Iφg f2,w∥Zµ . (12)

Hence, using equations (11) and (12), we get

lim
n→∞

µ(zn) |φ′(zn)g(zn)|
(1− |φ(zn)|2)(p+1)/p

= 0. (13)

Moreover, following similar argument as in the proof of Theorem 1, we get

µ(zn) |g′(zn)|
(1− |φ(zn)|2)1/p

≤ ∥Iφg f1,w∥Zµ∥+
2µ(zn) |φ′(zn)g(zn)|
(1− |φ(zn)|2)(p+1)/p

. (14)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 939

Hence, using equations (11) and (14), we get

lim
n→∞

µ(zn) |g′(zn)|
(1− |φ(zn)|2)1/p

= 0. (15)

Therefore, equations (13) and (15) give us the desired conditions

lim
|φ(z)|→1

K1(z) = 0 and lim
|φ(z)|→1

K2(z) = 0.

Conversely, suppose conditions (10) hold. Then for ϵ > 0, there is δ ∈ (0, 1) such that
K1(z) < ϵ and K2(z) < ϵ whenever δ < |φ(z)| < 1.

Let {fn} be a bounded sequence in Sp such that sup
n∈N

∥fn∥Sp < L and {fn} converges

to zero uniformly on compact subsets of D. Let U = {z ∈ D : |φ(z)| ≤ δ}.

Now, it is clear that

sup
z∈D

µ(z)
∣∣∣(Iφg fn)′′ (z)∣∣∣

≤ sup
z∈D\U

µ(z)
∣∣∣(Iφg fn)′′ (z)∣∣∣+ sup

z∈U
µ(z)

∣∣∣(Iφg fn)′′ (z)∣∣∣
First, we consider the case |φ(z)| > δ then we have

µ(z)
∣∣∣(Iφg fn)′′ (z)∣∣∣ = µ(z)

∣∣f ′′
n(φ(z))φ

′(z)g(z) + f ′
n(φ(z))g

′(z)
∣∣

≤ µ(z)|φ′(z)g(z)|
(1− |φ(z)|2)(p+1)/p

∥f ′
n∥Hp +

µ(z)|g′(z)|
(1− |φ(z)|2)1/p

∥f ′
n∥Hp

≤ (K2(z) +K1(z)) ∥fn∥Sp

< 2Lϵ. (16)

Second, we consider the case |φ(z)| ≤ δ. Since Iφg is bounded and polynomials are
dense in Sp(D), by taking f(z) = z we get

sup
z∈D

µ(z)
∣∣∣(Iφg f)′′ (z)∣∣∣ = sup

z∈D
µ(z)|g′(z)| < ∞, (17)

and by taking f(z) = z2 we get

sup
z∈D

µ(z)
∣∣∣(Iφg f)′′ (z)∣∣∣ = 2 sup

z∈D
µ(z)|φ(z)g′(z) + φ′(z)g(z)| < ∞. (18)

Using equations (17) and (18), and the boundedness of φ(z) we get



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 940

C1 = sup
z∈D

µ(z)|g′(z)| < ∞,

and
C2 = sup

z∈D
µ(z)|φ′(z)g(z)| < ∞

Hence, for |φ(z)| ≤ δ, using these facts we get

µ(z)
∣∣∣(Iφg fn)′′ (z)∣∣∣ ≤ C2

∣∣f ′′
n(φ(z))

∣∣+ C1

∣∣f ′
n(φ(z))

∣∣ . (19)

Since {fn} is bounded in Sp and converges to zero on {w ∈ D : |w| ≤ δ}, so do the
sequences {f ′

n} and {f ′′
n} by Cauchy’s estimate.

Thus, there exists N ∈ D such that for all n ≥ N and w ∈ D with |w| ≤ δ we have

|f ′
n(w)| < ϵ and |f ′′

n(w)| < ϵ

Hence, using inequality (19), we get

sup
z∈D

µ(z)
∣∣∣(Iφg fn)′′ (z)∣∣∣ ≤ C2 sup

|w|<δ
|f ′′

n(w)|+ C1 sup
|w|<δ

|f ′
n(w)|.

< (C1 + C2)ϵ. (20)

Now, using inequalities (16) and (20), we get

∥Iφg fn∥Zµ = |f ′
n(φ(0))g(0)|+ sup

z∈D
µ
∣∣∣(Iφg fn)′′ (z)∣∣∣

≤ |f ′
n(φ(0))g(0)|+ 2Lϵ+ (C1 + C2)ϵ

Since {f ′
n} converges to zero uniformly on compact subsets of D, it converges point-

wise. Thus, |f ′
n(φ(0))g(0)| → 0 as n → 0.

Hence, for arbitrary ϵ > 0, we get ∥Iφg fn∥Zµ → 0 as n → 0. Therefore, Iφg is compact,
which completes the proof.

4. Boundedness and Compactness of Tφ
g

In this section, we characterize the boundedness and compactness of the operator Tφ
g

acting between Sp spaces and weighted Zygmund spaces Zµ. In the following Theorem 3,
we characterize the boundedness of Tφ

g that acts between Sp spaces and weighted Zygmund
spaces.

Theorem 3. Let g be an analytic function on D and φ be an analytic selfmap of D. Then
Tφ
g : Sp → Zµ is bounded if and only if g ∈ Zµ and

M3 = sup
z∈D

µ(z)|g′(z)||φ′(z)|
(1− |φ(z)|2)1/p

< ∞.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (2) (2024), 931-944 941

Proof. Suppose Tφ
g : Sp → Zµ is bounded. Since polynomials are dense in Sp,

P1(z) = 1 ∈ Sp. By the boundedness of Tφ
g , we get ∥Tφ

g P1∥Zµ < ∞.

Therefore,

sup
z∈D

µ|g′′(z)| = sup
z∈D

µ
∣∣∣(Tφ

g P1

)′′∣∣∣
≤ ∥Tφ

g P1∥Zµ

< ∞,

which gives us that g ∈ Zµ.

Second, consider the test function f1,w we defined in the proof of Theorem 1. Then,(
Tφ
g f1,w

)′′
(w) =

(
f1,w(φ(w))g

′(w)
)′

= f1,w(φ(w))g
′′(w) + f ′

1,w(φ(w))φ
′(w)g′(w)

= f1,w(φ(w))g
′′(w) +

φ′(w)g′(w)

(1− |φ(w)|2)1/p
.

Therefore, by the boundedness of Tφ
g and equation (1), we get

µ(w)|φ′(w)g′(w)|
(1− |φ(w)|2)1/p

≤ µ(w)
∣∣∣(Tφ

g f1,w
)′′

(w)
∣∣∣+ µ(w)|g′′(w)||f1,w(φ(w))|

≤ ∥Tφ
g f1,w∥Zµ + ∥g∥Zµ∥f1,w∥Sp

≤ ∥Tφ
g ∥∥f1,w∥Sp + ∥g∥Zµ∥f1,w∥Sp

≤
(
∥Tφ

g ∥+ ∥g∥Zµ

)
2(2p−2)/p.

Taking the supremum over all w ∈ D, we get

sup
w∈D

µ(w)|φ′(w)g′(w)|
(1− |φ(w)|2)1/p

< ∞.

Conversely, suppose g ∈ Zµ and condition M3 is finite.

Let f ∈ Sp and z ∈ D. Then, by using Lemma 2, we get

µ(z)
∣∣∣(Tφ

g f
)′′

(z)
∣∣∣ = µ(z)

∣∣∣(f(φ(z))g′(z))′∣∣∣
= µ(z)

∣∣f(φ(z))g′′(z) + f ′(φ(z))φ′(z)g′(z)
∣∣

≤ ∥g∥Zµ∥f∥∞ +
µ(z) |φ′(z)g′(z)|
(1− |φ(z)|2)1/p

∥f ′∥Hp

≤ π∥g∥Zµ∥f∥Sp +M3∥f∥Sp .



REFERENCES 942

Since g ∈ Zµ and f ∈ Sp, by taking supremum over all z ∈ D, we get

sup
z∈D

µ(z)
∣∣∣(Tφ

g f
)′′

(z)
∣∣∣ < C∥f∥Sp , (21)

for some constant C.

Finally, using equation (21), we get

∥Tφ
g f∥Zµ =

∣∣(Tφ
g f

)
(0)

∣∣+ ∣∣∣(Tφ
g f

)′
(0)

∣∣∣+ sup
z∈D

µ(z)
∣∣∣(Tφ

g f
)′′

(z)
∣∣∣

≤
∣∣∣(Tφ

g f
)′
(0)

∣∣∣+ C∥f∥Sp

< (C∗ + C)∥f∥Sp ,

for some constant C∗. Hence, Tφ
g is bounded, as desired.

The following Theorem 4 characterizes the compactness of Tφ
g : Sp → Zµ whose proof

is similar to that of Theorem 2 and Theorem 3. So the details are omitted.

Theorem 4. Let g be an analytic function on D and φ be an analytic selfmap of D. Then
Tφ
g : Sp → Zµ is compact if and only if g ∈ Zµ and

lim
|φ(z)|→1

µ(z)|g′(z)||φ′(z)|
(1− |φ(z)|2)1/p

= 0.

Acknowledgements

This research is partially funded by Zarqa University. The author would like to express
his sincerest thanks to Zarqa University for the financial support. The author would like
to express his sincerest thanks to the referees for their valuable comments and various
useful suggestions.

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