Int. J. Anal. Appl. (2023), 21:21

The Möbius Invariant QTH Spaces

Munirah Aljuaid∗

Department of Mathematics, Northern Border University, Arar 73222, Saudi Arabia

∗Corresponding author: moneera.mutlak@nbu.edu.sa

Abstract. In this article, we introduce a new space of harmonic mappings that is an extension of the

well known space QT in the unit disk D in term of non decreasing function. Several characterizations
of the space QTH are investigated. We also define the little subspace of Q

T
H. Finally, the boundedness

of the composition operators Cϕ mapping into the space QTH and Q
T
H,0 are considered.

1. Introduction

A harmonic mapping on a simply connected domain ψ is a complex-valued function k such that the

Laplace’s equation satisfied

∆k := 4kηη ≡ 0, on ψ,

where kηη represents the mixed complex derivative of k.

The harmonic mapping k admits a representaion of the form f + g, where f and g are analytic

functions. This representaion is unique up to an additive constant. In this work, we consider all the

functions defined on the open unit disk D := {η ∈ C : |η| < 1} so, the representaion of k is given by
k = f + g and g(0) = 0.

Let H(D) denotes the collection of all analytic functions on D and H(D) be the collection of
harmonic mappings on D.

The operator theory of spaces of analytic functions on a various settings on the unit disk has been

completely analyzed and a enormous amount of research papers on this matter have appeared in the

literature, but the study of a similarly coverage in the harmonic setting is still limited.

Received: Jan. 27, 2023.

2020 Mathematics Subject Classification. 47B33.

Key words and phrases. QT space; harmonic mapping; composition operators.

https://doi.org/10.28924/2291-8639-21-2023-21
ISSN: 2291-8639

© 2023 the author(s).

https://orcid.org/0000-0002-5748-9738
https://doi.org/10.28924/2291-8639-21-2023-21


2 Int. J. Anal. Appl. (2023), 21:21

In recent years, some papers have concentrated on the study of harmonic mappings. Besides [2],

for characterization of Bloch type spaces of harmonic mapping, see [6], for harmonic zygmund spaces.

In [18], the authors investigate the compactness and boundedness of Cϕ mapping into weighted Banach

spaces of harmonic mappings. We also encourage the reader to see the additional references related to

the harmonic mappings such as [ [21] [5], [16], [14], [15], [17], [13], [7], [8], [10], [11], [12], [17], [9]].

The results carried out in [19] bring the interesting question for whether we can extend the space

QT to the harmonic setting and study the operator theoretic properties of Cϕ.

2. preliminaries and background

We start this section with several preliminaries facts on the spaces that will be used in this work.

Harmonic Bloch space BH can be seen as the collection of k ∈ H(D) and the a semi-norm bk
satisfies the following condition

bk := sup
η∈D

(1 −|η|2)(|f ′(η)| + |g′(η)|) < ∞. (2.1)

BH is a Banach space when it is equipped with the harmonic Bloch norm defined as

‖k‖BH := |k(0)| + bk.

BH space extends the well known Bloch space B. An analytic function f ∈B if and only if

bf = sup
η∈D

(1 −|η|2)|f ′(η)| < ∞, (2.2)

with norm

‖f‖B = |f (0)| + bf .

In [3], the author obtains that the Bloch constant of k can be written as follows

bk := sup
η∈D

(1 −|η|2)(|kη(η)| + |kη̄(η)|) < ∞. (2.3)

and

max{bf ,bg}≤ bk ≤ bf + bg. (2.4)

Consequently, a harmonic mapping k belongs to the harmonic Bloch space if and only if the functions

f ,g ∈ H(D) such that k = f + ḡ with g(0) = 0 are in the classical Bloch space. For more details,
see [2].

The little harmonic Bloch space BH,0 is the subspace of BH such that

BH,0 := {k ∈BH : lim
|η|→1

(1 −|η|2)
(
|kη(η)| + |kη̄(η)|

)
= 0}.



Int. J. Anal. Appl. (2023), 21:21 3

and the little Bloch spaces B0 defined as

B0 := {f ∈B : lim
|η|→1

(1 −|η|2)|f ′(η)| = 0}.

Consider nondecreasing function T : [0, +∞) → [0, +∞). The logarithmic order of T (r) is given
by

λ = lim
r→∞

log∗ log∗T (r)

log r
,

where log∗γ = max{0, log γ}
If λ > 0, the logarithmic type of the function T (r) is given by

Γ = lim
r→∞

log∗T (r)

rλ
,

The space QT is the collection of analytic functions f defined on D and

qT (f ) = sup
ν∈D

(∫
D

(|f ′(η)|2T (g(η,ν))dA(η)
)1

2

< ∞,

where dA(η) represents the area measure on the unit disk and g(η,ν) = − log |σν(η)| is the Green

function of D with pole at ν ∈ D and σν(η) =
(ν −η)

(1 − ν̄η)
be a Möbius transformation of D.

3. The Möbius invariant QTH spaces

We now introduce the harmonic QTH space of harmonic mapping by a nondecreasing function T (r)
on r ∈ [0,∞).

Definition 3.1. For nondecreasing function T : [0, +∞) → [0, +∞). A harmonic mapping k ∈H(D)
is said to be in the class QTH if

[qT (k)]2 = sup
ν∈D

∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η) < ∞,

and the norm of QTH is defined as:

‖k‖QT
H

:= |k(0)| + qT (k). (3.1)

The little harmonic QTH,0 is the subspace of Q
T
H such that

QTH,0 :=
{
k ∈H(D) : lim

|η|→1

∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η) = 0
}
.

Remark 3.1. As a special case when k ∈ H(D), the functions f ,g in the canonical decomosition of k
are given by k = f and g ≡ 0. Moreover, the collections of analytic function on the unit disk in the
QTH is just the space Q

T .



4 Int. J. Anal. Appl. (2023), 21:21

Corollary 3.1. For T : [0, +∞) → [0, +∞) be non-decreasing function. Let f ∈ H(D), if k ∈H(D)
be the real part of f or imaginary part of f then

qT (k) = qT (f )

Proof. Assume f = Re(k). Then we have,

k =
1

2
(f + f̄ ).

Therefore,

qT (k) =

(
sup
ν∈D

∫
D

(
1

2
|f ′(η)| +

1

2
|f ′(η)|)2T (g(η,ν))dA(η)

)1
2

=

(
sup
ν∈D

∫
D
|f ′(η)|2T (g(η,ν))dA(η)

)1
2

= qT (f )

In a similar way, assume f = Im(k), then we have

k =
1

2i
f −

1

2i
f̄ .

Thus,

qT (k) = (sup
ν∈D

∫
D

(
1

2
|f ′(η)| +

1

2
|f ′(η)|)2T (g(η,ν))dA(η))

1
2

= (sup
ν∈D

∫
D
|f ′(η)|2T (g(η,ν))dA(η))

1
2

= qT (f )

Theorem 3.1. For T : [0, +∞) → [0, +∞) be non-decreasing function. Let k = f +ḡ ∈H(D) where
f ,g ∈ H(D).Then f ,g ∈QT if and only if k ∈QTH. Moreover, if g(0) = 0, then

1

2
(‖f‖QT + ‖g‖QT ) ≤‖k‖QT

H
≤ 2((‖f‖QT + ‖g‖QT )).

Proof. Consider f ,g ∈QT and let k = f + ḡ. Then

f ′ = kη and g
′ = kη̄.

Therefore,

(|kη(η)| + |hη̄(η)|)2 < 22(|kη(η)|2 + |kη̄(η)|2)

The above inequality follows from the fact that for c1,c2 ≥ 0,(
c1 + c2

2

)2
≤ [max{c1,c2}]2 = max{c21 ,c

2
2}≤ c

2
1 + c

2
2 ,



Int. J. Anal. Appl. (2023), 21:21 5

we have

qT (k)2 = sup
ν∈D

∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η)

≤ 22
[

sup
ν∈D

∫
D

(|kη(η)|)2T (g(η,ν))dA(η) + sup
ν∈D

∫
D

(|kη̄(η)|)2T (g(η,ν))dA(η)
]
< ∞.

Therefore k ∈QTH and,

qT (f + ḡ)2 ≤ 4(qT (f )2 + qT (g)2). (3.2)

Taking the square root, we get

qT (k) ≤ 2
√(

qT (f )2 + qT (g)2
)
< 2
(
qT (f ) + qT (g)

)
.

Moreover, using |k(0)| ≤ |f (0)| + |g(0)|, the upper estimate holds

Conversely, let k ∈QTH and note that

|f ′(η)|2 + |g′(η)|2 ≤ (|f ′(η)| + |g′(η)|)2,

Thus

sup
ν∈D

∫
D

(|kη(η)|)2T (g(η,ν))dA(η) + sup
ν∈D

∫
D

(|kη̄(η)|)2T (g(η,ν))dA(η))

≤ sup
ν∈D

∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η) < ∞.

Therefore, both f and g are in the space QT and

qT (f )2 + qT (g)2 ≤ qT (k)2.

Hence, by 3.2

1

2
[qT (f ) + qT (g)] ≤

√
qT (f )2 + qT (g)2.

Then, we combine these two inequalities to get

1

2
[qT (f ) + qT (g)] ≤ qT (k).

By the assumption g(0) = 0, we have

1

2
|f (0)| ≤ |f (0)| = |k(0)|.

Therefore,

1

2
[‖f‖QT + ‖g‖QT ] ≤‖k‖QT

H
,

We deduce the lower estimate.



6 Int. J. Anal. Appl. (2023), 21:21

Lemma 3.1. For T : [0, +∞) → [0, +∞) be non-decreasing function. Then k ∈QTH if and only if

sup
ν∈D

(∫
D

(|kη(η)| + |kη̄(η)|)2T (1 −|σν(η)|2)dA(η)
)1

2
< ∞, (3.3)

Proof. Recall that for s ∈ (0, 1], we have

−2 log s ≥ 1 − s2

and for s ∈ ( 1
4
, 1) we have

− log s ≤ 4(1 − s2)

Assume k ∈QTH then we have,

qT (k) = sup
ν∈D

(∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η)
)1

2

(3.4)

≤ sup
ν∈D

(∫
D

(|kη(η)| + |kη̄(η)|)2T (1 −|σν(η)|2)dA(η)
)1

2

(3.5)

Since
∫
D

(|kη(η)| + |kη̄(η)|)2|dη| is increasing function on δ ∈ (0, 1), we have

∫
D

(|kη(η)| + |kη̄(η)|)2|dη| ≤
∫

D/D(0, 1
4

)

(|kη(η)| + |kη̄(η)|)2T (1 −|σν(η)|2)dA(η) ≤ (qT (k))2.

This inequality with 3.4, prove the theorem. �

We now study the relationship between k ∈QTH and the associated real and imaginary parts.

Proposition 3.1. For T : [0, +∞) → [0, +∞) be non-decreasing function. Let k ∈H(D) and assume
that τ be the real part of k and θ is the imaginary part of k such that

τ = Re(k) and θ = Im(k).

Then k ∈QTH, if and only if τ,θ ∈Q
T
H. Moreover

1

4

(
‖τ‖QT

H
+ ‖θ‖QT

H

)
≤‖k‖QT

H
≤‖τ‖QT

H
+ ‖θ‖QT

H
.

Proof. Assume τ,θ ∈ QTH . Due to linearity, k ∈ Q
T
H and the upper estimate hold directly by the

property of the norm (triangle inequality ) .

Let k ∈QTH and recall that

J(τ,θ) = τxθy −θxτy

We have

2|J(τ,θ)| ≤ ‖∇τ‖2 + ‖∇θ‖2, (3.6)



Int. J. Anal. Appl. (2023), 21:21 7

where ∇τ = (τx,τy ) , and ∇θ = (θx,θy ).
From this, we get

(‖∇τ‖2 + ‖∇θ‖2 + 2J(τ,θ))
1
2 + (‖∇τ‖2 + ‖∇θ‖2 − 2J(τ,θ))

1
2 ≥
√

2(‖∇τ‖2 + ‖∇θ‖2)
1
2 (3.7)

By squaring (3.7), the left-hand side becomes

‖∇τ‖2 + ‖∇θ‖2 + 2J(τ,θ) + ‖∇τ‖2 + ‖∇θ‖2 − 2J(τ,θ) + 2
(
‖∇τ‖2 + ‖∇θ‖2)2 − 4(J(τ,θ)2

)1
2
,

Thus, by neglecting the last term and simple calculation, we obtain

2(‖∇τ‖2 + ‖∇θ‖2).

Now, we may find |kη| + |kη̄| with respect to τ and θ by using the partials with respect to η and η̄,
then calculating the modulus, after that applying (3.7)

|kη| + |kη̄| = |τη + iθη| + |τη̄ + iθη̄|

=
1

2

∣∣τx + θy + i(θx −τy )∣∣ + 1
2

∣∣τx −θy + i(θx + τy )∣∣
=

1

2

√((
τx + θy

)2
+
(
θx −τy

)2)
+

1

2

√((
τx −θy

)2
+
(
θx + τy

)2)
=

1

2

√(
‖∇τ‖2 + ‖∇θ‖2 + 2J(τ,θ)

)
+

1

2

√(
‖∇τ‖2 + ‖∇θ‖2 − 2J(τ,θ)

)
≥

1
√

2

√
‖∇τ‖2 + ‖∇θ‖2

≥
1

2

(
‖∇τ‖ + ‖∇θ‖

)
,

In the last step, we apply the following inequality

‖(η1,η2)‖≥
|η1| + |η2|√

2
f or η1,η2 ∈ C. (3.8)

Therefore,

(qT (k))2 ≥
1

2
sup
η∈D

∫
D

(‖∇τ(η)‖ + ‖∇θη‖)2T (g(η,ν)dA(η)

≥
1

2
max{qTτ ,q

T
θ }

≥
1

4
(qTτ + q

T
θ ) (3.9)

Therefore, by using inequality (3.8) one more time, we obtain

|k(0)| ≥
1
√

2
(|τ(0)| + |θ(0)|) (3.10)

Now, combine (3.9) and (3.10) to get

‖k‖QT
H
≥

1

4
(‖τ‖QT

H
+ ‖θ‖QT

H
)



8 Int. J. Anal. Appl. (2023), 21:21

Thus, τ and θ are in QTH, and that the other estimate is hold.

Theorem 3.2. (QTH,‖ ·‖QT
H

) is a Banach space.

Proof. Obviously, QTH is a normed linear space, we only wish to show completeness.

For each n ∈ N, let {kn} be a Cauchy sequence in QTH . By Theorem 3.1, the analytic functions {fn}
and {gn} such that kn = fn + ḡn with gn(0) = 0 are in QT and {fn} and {gn} are Cauchy sequence in
QT . By proposition 2.2 in [4], QT is complete. Thus, {fn} and {gn} converge to f and g, respectively
in the QT norm.

Define k = f + ḡ. Then, k ∈QTH by the estimates in Theorem 3.1, and

‖kn −k‖QT
H
≤ 2(‖fn − f‖QT + ‖gn −g‖QT ) → 0, as n →∞.

We ends up with kn → k in QTH.

Theorem 3.3. For nondecreasing function T : [0, +∞) → [0, +∞). The space QTH is a subset of
BH. Moreover, for k ∈QTH we have

‖k‖BH ≤ m‖k‖QT
H
,

for some constant m > 0.

Proof. Assume k ∈QTH and let

sup
ν∈D

∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η) = M < ∞,

For δ ∈ (0, 1) define D(�,�) := {η ∈ D : |σν(η)| < δ}. Since T is nondecreasing function and by
the change of variable w = σν(η) we have

M ≥
∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η)

≥
∫

D(�,�)

(|kη(η)| + |kη̄(η)|)2T
(

log
1

σν(η)

)
dA(η)

≥ T
(

log
1

δ

) ∫
D(�,�)

(|kη(η)| + |kη̄(η)|)2dA(η)

= T
(

log
1

δ

) ∫
|w|<δ

(|(k ◦σν)w (w)| + |(k ◦σν)w̄ (w)|)2dA(w)

≥ πδ2T
(

log
1

δ

)
(|(k ◦σν)ν(0)| + |(k ◦σν)ν̄(0)|)2

= πδ2T
(

log
1

δ

)
(|(kν(ν)| + |(kν̄(ν)|)2(1 −|ν|2)2



Int. J. Anal. Appl. (2023), 21:21 9

Fix δ0 ∈ (0, 1). Thus

sup
ν∈D

(1 −|ν|2)[|(kν(ν)| + |(kν̄(ν)|] ≤
√

M

πδ20T
(

log 1
δ0

)
Therefore,

bk ≤
qT (k)

δ0

√
πT
(

log 1
δ0

) (3.11)
We obtained that k ∈BH and QTH ⊂BH.

�

Theorem 3.4. If the logarithmic type Γ and the logarithmic order λ of T (r) satisfying one of the

following cases,

(1) λ > 1,

(2) Γ > 2 and λ = 1,

then the space QTH has only constant functions(trivial space).

Proof. By theorem 3.3, it is sufficient to prove that for each non constant harmonic Bloch function

k can not be in the space QTH. Indeed, if either λ > 1 or Γ > 2 and λ = 1, there is a sequence {rj}
as j →∞, the sequence {rj}→∞ as follows

lim
j→∞

log∗ log∗T (rj)

log rj
= λ > 1, (3.12)

or

lim
j→∞

log∗T (rj)

rj
= Γ > 2, (3.13)

In the case 3.12 or 3.13, we get

lim
j→∞

T (rj)

e2rj
= ∞. (3.14)

Set hj = e−rj , for j ∈ N, then

lim
j→∞

h2j T
(

log
1

hj

)
= ∞. (3.15)

Assume k ∈BH be a non-constant. Then it is clear that the semi-norm bk 6= 0.
However, by 3.11, and 3.15, as j →∞ we obtain

sup
ν∈D

∫
D

(|kη(η)| + |kη̄(η)|)2T (g(η,ν))dA(η) ≥ πb2k h
2
j T (log

1

hj
) →∞.

That implies k /∈QTH which proves the theorem. �

The next theorem shows that the Möbius invariance of QT space extends to the harmonic setting.



10 Int. J. Anal. Appl. (2023), 21:21

Theorem 3.5. For T : [0, +∞) → [0, +∞) be non-decreasing function. QTH is a Möbius invariant
space.

Proof. It is obvious that rotations have no effect on the semi-norm qT (k). We wish to show qT (k ◦
ϕν) = q

T (k), for ν ∈ D and k ∈QTH.

For ν ∈ D, and since ϕν is its own inverse, we have

(1 −|η|2)|ϕ
′
(η)| = 1 −|ϕν(η)|2

and

ϕ
′
ν(ϕν(η)) =

1

ϕ
′
ν(η)

By change of variables ξ = ϕν(η), we get

qT (k ◦ϕν)2 = sup
ν∈D

∫
D
T (1 −|ϕν(η)|2)[|(k ◦ϕν)η(η)| + |(k ◦ϕν)η̄(η)|]2dA(η)

= sup
ν∈D

∫
D
T (1 −|ϕν(η)|2)[|kη(ϕν(η))ϕ

′
ν(η)| + |(kη̄(ϕν(η))ϕ

′
ν(η))|]

2dA(η)

= sup
ν∈D

∫
D
T (1 −|ϕν(η)|2)|ϕ

′
ν(η)|

2[|kη(ϕν(η))| + |kη̄(ϕν(η))|]2dA(η)

= sup
ν∈D

∫
D
T (1 −|ξ|2)|ϕ

′
ν(ϕν(ξ))|

2[|kη(ξ)| + |(kη̄(ξ))|]2|ϕ
′
ν(ξ)|

2dA(ξ)

= sup
ν∈D

∫
D
T (1 −|ξ|2)

1

|ϕ′ν(ξ)|2
[|hη(ξ)| + |hη̄(η)|]2|ϕ

′
ν(ξ)|

2dA(ξ)

= sup
ν∈D

∫
D
T (1 −|ξ|2)[|kη(ξ)| + |kη̄(ξ)|]2dA(ξ)

= qT (k)2

as desired.

Finally, we move our attention to study the boundedness of composition operator Cϕ from the

harmonic Bloch space BH to QTH and Q
T
H,0.

4. Boundedness

Due to the representation of the harmonic mapping, the composition operator Cϕ induced by

analytic or a conjugate analytic self-maps of D is given by

Cϕk = k ◦ϕ,

for all k belonging to a class of harmonic mappings.

The following is a basic property of the harmonic Bloch space was introduced in [20].



Int. J. Anal. Appl. (2023), 21:21 11

Lemma 4.1. For η ∈ D. If k1 , k2 ∈BH we have

(1 −|η|2)−1 ≤ (k1)η(η)| + |(k1)η̄(η)| + |(k2)η(η)| + |(k2)η̄(η)|.

The next result which will be used in the proof of the main theorem of this section is a special case

of Theorem 3.6 in [1]

Lemma 4.2. For k ∈BH and ϕ : D → D,

|k(ϕ(0))| ≤ |k(0)| +
1

2
log

1 + |ϕ(0)|
1 −|ϕ(0)|

bk.

Theorem 4.1. For T : [0, +∞) → [0, +∞) be non-decreasing function. Let ϕ be analytic function
such that ϕ : D → D. Then Cϕ : BH →QTH is bounded operator if and only if

sup
ν∈D

∫
D

|ϕ′(η)|2

(1 −|ϕ(η)|2)2
T (g(η,ν))dA(η) < ∞. (4.1)

Proof. Let us assume 4.1 holds and let ρ21 be the supremum in 4.1. Let η ∈ D and k ∈BH, then∫
D
T (g(η,ν))[|(k ◦ϕ)η(η)| + |(k ◦ϕ)η̄(η)|]2dA(η)

=

∫
D
T (g(η,ν))|ϕ

′
(η)|2[|kη(ϕ(η))| + |kη̄(ϕ(η))|]2dA(η)

≤ b2k
∫
D
T (g(η,ν))

|ϕ
′
z (ξ)|2

(1 −|ϕ(η)|2)2
dA(η)

≤ ρ21b
2
k.

Therefore, qT (k ◦ϕ) ≤ ρ1 bk. Since k ∈BH we have

‖Cϕk‖2QT
H

=
(
|k ◦ϕ(0)| + qT (Cϕk)

)2
≤

(
|k(0)| +

1

2
log

1 + |ϕ(0)|
1 −|ϕ(0)|

bk + ρ1 bk)
)2

≤ ρ2
(
|k(0)| + bk

)2
= ρ2‖k‖2BH.

where ρ = max{1,ρ1 +
1

2
log

1 + |ϕ(0)|
1 −|ϕ(0)|

}.

Therefore, ‖Cϕk‖QT
H
≤ ρ‖k‖BH which implies that Cϕ : BH →Q

T
H is bounded. Conversely, Assume

the boundedness of Cϕ : BH → QTH holds, then there is a positive constant ρ > 0 for all k ∈ BH, we
have

‖Cϕk‖QT
H
≤ ρ‖k‖BH.

On the other hand, by Lemma 4.1 for all η ∈ D, there exist k1 , k2 ∈BH such that

(1 −|η|2)−1 ≤ |(k1)η(η)| + |(k1)η̄(η)| + |(k2)η(η)| + |(k2)η̄(η)|



12 Int. J. Anal. Appl. (2023), 21:21

Therefore,

|ϕ(η)′|2[
1 −|ϕ(η)|2

]2 ≤ 2|(k1 ◦ϕ)η(η)|2 + 2|(k1 ◦ϕ)η̄(η)|2 + 2|(k2 ◦ϕ)η(η)|2 + 2|(k2 ◦ϕ)η̄(η)|2
≤ 2[|(k1 ◦ϕ)η(η)| + |(k1 ◦ϕ)η̄(η)|]2 + 2[|(k2 ◦ϕ)η(η)| + |(k2 ◦ϕ)η̄(η)|]2

where the last inequity follows from the fact that for c1,c2 ≥ 0 and m > 1 we have

cm1 + c
m
2 ≤ (c1 + c2)

m

Moreover,∫
D
T (g(η,ν))

|ϕ(η)′|2(
1 −|ϕ(η)|2

)2 dA(η)
≤ 2

∫
D

[
[|(k1 ◦ϕ)η(η)| + |(k1 ◦ϕ)η̄(η)|]2 + [|(k2 ◦ϕ)η(η)| + |(k2 ◦ϕ)η̄(η)|]2

]
T (g(η,ν))dA(η)

≤ 2ρ2
(
‖k1‖2BH + ‖k2‖

2
BH
)
,

Thus, take the supremum over all η ∈ D, the quantity 4.1 holds since ρ is a constant and k ∈BH. �

Theorem 4.2. For nondecreasing function T : [0, +∞) → [0, +∞). Let ϕ be analytic function such
that ϕ : D → D. Then Cϕ : BH →QTH,0 is bounded operator if and only if

lim
|ν|→1

∫
D

|ϕ′(η)|2

(1 −|ϕ(η)|2)2
T (g(η,ν))dA(η) = 0. (4.2)

Proof. By theorem 4.1, we know that Cϕ : BH →QTH is bounded since the condition 4.2 implies the
following

sup
ν∈D

∫
D

|ϕ′(η)|2

(1 −|ϕ(η)|2)2
T (g(η,ν))dA(η) < ∞.

We only wish to show that Cϕk ∈QTH,0 for each k ∈BH and this comes from the inequality∫
D
T (g(η,ν))[|(k ◦ϕ)η(η)| + |(k ◦ϕ)η̄(η)|]2dA(η)

=

∫
D
T (g(η,ν))|ϕ

′
(η)|2[|kη(ϕ(η))| + |kη̄(ϕz (η))|]2dA(η)

≤ b2k
∫
D
T (g(η,ν))

|ϕ
′
z (η)|2

(1 −|ϕ(η)|2)2
dA(η)

Thus, Cϕk ∈QTH,0.
Conversely, consider Cϕ : BH → QTH,0 is bounded. By Lemma 4.1 there exist k1 , k2 ∈ BH such

that

(1 −|η|2)−1 ≤ |(k1)η(η)| + |(k1)η̄(η)| + |(k2)η(η)| + |(k2)η̄(η)|



Int. J. Anal. Appl. (2023), 21:21 13

Then Cϕk1 , Cϕk2 ∈QTH,0.

Therefore,

lim
|ν|→1

∫
D
T (g(η,ν))

|ϕ(η)′|2[
1 −|ϕ(η)|2

]2 dA(η)
≤ 2 lim

|ν|→1

∫
D
T (g(η,ν))

(
[|(k1 ◦ϕ)η(η)| + |(k1 ◦ϕ)η̄(η)|]2 + [|(k2 ◦ϕ)η(η)| + |(k2 ◦ϕ)η̄(η)|]2

)
dA(η) = 0

Then 4.2 holds and this complete the proof. �

Conflicts of Interest: The author declares that there are no conflicts of interest regarding the publi-

cation of this paper.

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	1. Introduction
	2. preliminaries and background 
	3. The Möbius invariant QTH spaces
	4. Boundedness
	References