International Journal of Analysis and Applications

Volume 16, Number 3 (2018), 445-453

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-445

QK-TYPE SPACES OF QUATERNION-VALUED FUNCTIONS

M.A. BAKHIT∗

Department of Mathematics, Faculty of Science, Jazan university, Jazan, Saudi Arabia.

∗Corresponding author: mabakhit@jazu.edu.sa

Abstract. In this paper we develop the necessary tools to generalize the QK -type function classes to the

case of the monogenic functions defined in the unit ball of R3, some important basic properties of these

functions are also considered. Further, we show some relations between QK (p,q) and α-Bloch spaces of

quaternion-valued functions.

1. Introduction

1.1. Analytic function spaces. The so called QK-type spaces of analytic functions on D = {z ∈ C : |z| <

1} the unit open complex disk, were introduced by Wulan and Zhou in [12]. For K : [0,∞) → [0,∞) is a

non-decreasing and non-negative function, and 0 < p < ∞,−2 < q < ∞, an analytic function f in D belongs

to the QK(p,q) if

sup
a∈D

∫
D
|f′(z)|p(1 −|z|2)qK(1 −|ϕa(z)|2)dxdy < ∞.

Moreover, if

lim
|a|→1

∫
D
|f′(z)|p(1 −|z|2)qK(1 −|ϕa(z)|2)dxdy = 0,

then f ∈ QK,0(p,q), where ϕa(z) = (a− z)/(1 − āz) is the automorphism of the unit disk D that changes

0 and a. The QK(p,q) class is Banach under the norm ‖f‖ = ‖f‖QK(p,q) + |f(0)|, when p ≥ 1. If K(t) =

ts, 0 ≤ s < ∞, then QK(p,q) = F(p,q,s) see [3]. For more results of QK(p,q) classes see [3] and [7].

Received 2017-10-27; accepted 2018-01-08; published 2018-05-02.

2010 Mathematics Subject Classification. 46E15, 30G35.

Key words and phrases. Clifford analysis; Bloch-type classes of quaternion-valued functions; QK -type spaces.
c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

445

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-445


Int. J. Anal. Appl. 16 (3) (2018) 446

1.2. Quaternion function spaces. We will work throughout this paper in the field H (the skew field of

quaternion-valued functions), i.e. each element a ∈ H with basis 1,e1,e2,e3, can be given in the form

a := a0 + a1e1 + a2e2 + a3e3, ak ∈ R,k = 0, 1, 2, 3

The multiplication rules of these elements are given by

e21 = e
2
2 = e

2
3 = −1,

e1e2 = −e2e1 = e3,

e2e3 = −e3e2 = e1,

e3e1 = −e1e3 = e2.

The conjugation element ā of an element a is ā = a0 −a1e1 −a2e2 −a3e3, with the property

aā = āa = |a|2 = a20 + a
2
1 + a

2
2 + a

2
3.

If a ∈ H \{0}, then a−1 := ā/|a|2 and |ab| = |a||b| for each a,b ∈ H. Let x = (x0,x1,x2) ∈ R3 of the form

x = x0 + x1e1 + x2e2 be a quaternion point.

Given Ω ⊂ R3 a domain and let f : Ω −→ H the quaternion-valued functions defined in Ω. For p ∈ N∪{0},

thus the notation Cp(Ω; H) has the usual componentwise meaning. We consider D and D the generalization

of a Cauchy-Riemann operator and it’s conjugate, respectively, and they are defined on C1(Ω; H) by

Df =
∂f

∂x0
+ e1

∂f

∂x1
+ e2

∂f

∂x2
,

Df =
∂f

∂x0
−e1

∂f

∂x1
−e2

∂f

∂x2
.

The equation Df = 0 has the solutions for all x ∈ Ω, are called left-hyperholomorphic functions and they

are generalized of the analytic function classes from the functions in one complex variable theory. For more

details about monogenic function classes and general Clifford analysis, we refer to [2, 6, 11] and others.

Let B be the unit ball in ⊂ R3, with boundary S = ∂B. The class M(B) consists of all monogenic functions

on B. For r > 0 and a ∈ R3, let B(a,r) denote by the ball with center a and radius r. Also, for a ∈ B and

0 < R < 1, an Euclidean ball U(a,R) = {x : |ϕa(x)| < R}, with center and radius, respectively,
(1−R2)a
1−R2|a|2 and

(1−|a|2)R
1−R2|a|2 , is called the pseudo-hyperbolic ball. Where ϕa(x) : B → B is defined by ϕa(x) = (a−x)/(1− āx),

for a ∈ B.

Let α > 0, the quaternion α-Bloch space Bα (see [4, 9]) defined by :

Bα = {f ∈M(B) : ‖f‖Bα = sup
x∈B
|Df(x)|(1 −|x|2)α < ∞}.



Int. J. Anal. Appl. 16 (3) (2018) 447

If α = 3
2
, we have the standard quaternion Bloch space B. The space Bα0 is called the quaternion little

α-Bloch, which consists of all f ∈Bα such that

lim
|x|→1−

|Df(x)|(1 −|x|2)α = 0.

For f ∈M(B), the weighted quaternion Dirichlet space Dp,q, (0 < p < ∞,−2 < q < ∞), is given by:

Dp,q =
{
f ∈M(B) : ‖f‖Dp,q =

∫
B

|Df(x)|p(1 −|x|2)qdx < ∞
}
.

If q = 0, we have the space D2,0 (the quaternion Dirichlet space D).

Through this work, we let K(t), 0 < t < ∞, be a non-decreasing and non-negative (righ-continuous)

function, which is not equal to 0 identically. For 0 < p < ∞,−2 < q < ∞ and f ∈ M(B), define

JK,p,qf : B → [0,∞) by

J
p,q
K f(a) =

∫
B

|Df(x)|p(1 −|x|2)qK(1 −|ϕa(x)|2)dx, a ∈ B.

The set QK(p,q) given by

QK(p,q) :=
{
f ∈M(B) : ‖f‖QK(p,q) = sup

a∈B
J
p,q
K f(a) < ∞

}
,

and the little quaternion QK,0(p,q) is defined by

QK,0(p,q) :=
{
f ∈M(B) : ‖f‖QK,0(p,q) = lim|a|→1−

J
p,q
K f(a) = 0

}
.

Remark 1.1. If we put s < 3 and K(t) = ts, then QK(p,q) = F(p,q,s) (see [8]). If p = 2,q = 0, then

QK(2, 0) = QK (see [1]). Also, if K(t) = 1, then QK(p,q) = Dp,q, the quaternion Dirichlit space.

For 0 < p < ∞,−1 < q < ∞, define the DK(p,q) quaternion Dirichlet-type space as the set of f ∈M(B)

satisfying

J
p,q
K f(0) < ∞.

From the definition of QK(p,q) spaces the following lemma become immediate with a = 0.

Lemma 1.1. Let 0 ≤ p < ∞,−1 < q < ∞, then QK(p,q) ⊂DK(p,q).

From now, we assume that
1∫

0

(1 −ρ2)qK
(
1 −ρ2

)
ρ2dρ < ∞. (1.1)

Otherwise, QK(p,q) contains only constant functions.

Fact 1

Let 0 ≤ p < ∞,−1 < q < ∞, and let f ∈ M(B) be a non-constant function. If ( 1.1) does not hold, then



Int. J. Anal. Appl. 16 (3) (2018) 448

f /∈QK(p,q).

Proof. Let f ∈ QK(p,q) be a non constant function. Then, there is x0 ∈ B and 0 < R < 1 such that

|Df(x)| > 0 for each x ∈ B(x0,R). Thus by Lemma 1.1 and subharmonicity of |Df|p where A(x0,R) =

B\B(0, |x0|−R), we obtain

∞ > Jp,qK f(0)

≥
∫

A(x0,R)

|Df(x)|p(1 −|x|2)qK
(
1 −|x|2

)
dx

≥
1∫

|x0|

(1 −ρ2)qK
(
1 −ρ2

)
ρ2
∫
S

|Df(ρζ)|pdσ(ζ)dρ

≥
∫
S

|Df(|x0|ζ)|pdσ(ζ)
1∫

|x0|

(1 −ρ2)qK
(
1 −ρ2

)
ρ2dρ = ∞,

where dσ denotes the normalized surface element in S. This is a contradiction; therefor f is constant and

the fact is proved.

In this work, we introduce a classes of H-valued functions on R3. These classes are so called QK(p,q)

spaces of monogenic function. We will study these classes and their relations to the quaternion α-Bloch

space. We shall prove some basic properties concerning QK(p,q) and QK,0(p,q) spaces in hyperholomorphic

functions. Our results in this work are extensions of our results in [1] and the results due to Essén and

Wulan (see [3]) in hyperholomorphic functions case. For simplicity we restricted us toR3 the lowest non-

commutative case and quaternion-valued functions. Next, the hyperholomorphic function spaces were the

aim of many works as [1, 4, 8] and [9].

In particular, we will need the following results for quaternion sense in the sequel:

Lemma 1.2. [5] Let 1 ≤ p < ∞,f ∈M(B) and let 0 < R < 1. Then, we have

|Df(0)|p ≤
3

4πR2

∫
U(a,R)

|Df(x)|pdx, for all a ∈ B. (1.2)

Lemma 1.3. [9] Let 1 < p < ∞,f ∈M(B) and let 0 < R < 1. Then, for every a ∈ B, we have

|Df(a)|p ≤
C4p

R3(1 −R2)2p(1 −|a|2)3

∫
U(a,R)

|Df(x)|pdx, for all a ∈ B, (1.3)

where C = 48
π
.



Int. J. Anal. Appl. 16 (3) (2018) 449

Remark 1.2. The problem in quaternion sense is that, Df(x) is monogenic, but Df(ϕa(w)) is not mono-

genic. From [10] we know that 1−w̄a|1−āw|3 Df(ϕa(w)) is hyperholomorphic. So, by the Jacobian determinant(
1−|a|2
|1−āw|2

)3
, which has no singularities we can solve this problem.

Lemma 1.4. [8] Let f ∈M(B),fa = f ◦ϕa and let Ψfa : B → H given by

Ψfa(x) =
1 −xa
|1 −ax|3

Df(ϕa(x)). (1.4)

Then Ψfa ∈M(B) and |Ψfa| is a subharmonic function.

2. Characterizations of QK(p,q) classes

In this part, we prove some essential properties of quaternion QK(p,q) spaces as basic scale properties.

Proposition 2.1. Let K satisfy (1.1) and let f ∈M(B), 1 ≤ p < ∞, and −2 < q < ∞. Then, we have

(1 −|a|2)q+3|Df(a)|p ≤
4q−p+3

C(R)
J
p,q
K f(a), for 0 < R < 1.

Proof. Since 0 < R < 1, by Lemma 1.4 after the change of variable x = ϕa(w) then we deduce that

J
p,q
K f(a) ≥

∫
U(a,R)

|Df(x)|p(1 −|x|2)qK
(
1 −|ϕa(w)|2

)
dx

≥
(1 −|a|2)q+3

4q−p+3

∫
B(a,R|1−a|)

|Ψfa(w)|
p(1 −|w|2)qK

(
1 −|w|2

)
dw

≥
1

4q−p+3
(1 −|a|2)q+3|Df(a)|p

∫ R
0

(1 −ρ2)qK
(
1 −ρ2

)
ρ2dρ

≥
C(R)

4q−p+3
(1 −|a|2)q+3|Df(a)|p,

with Ψfa(w) =
1−w̄a
|1−āw|3 Df(ϕa(w)). Which implies that

(1 −|a|2)q+3|Df(a)|p ≤
4q−p+3

C(R)
J
p,q
K f(a).

Theorem 2.1. Suppose that K satisfy (1.1), 1 ≤ p < ∞,−2 < q < ∞ and f ∈ M(B). When fn → f,

assume that Dfn → Df uniformly on compact set M ⊂ B, as n → ∞. Then, the space QK(p,q) under the

norm ‖f‖K = |Df(0)| + ‖f‖QK(p,q) is a Banach space.

Proof. Since 1 ≤ p < ∞, t is easy to prove that ‖.‖K is a norm. To show the completeness of (‖.‖K,QK(p,q)),

fix 0 < R < 1. Applying Proposition 2.1, we obtain

(1 −|a|2)q+3|Df(a)|p ≤
4q−p+3

C(R)
J
p,q
K f(a),

which gives

‖f‖
B
q+3
p
≤

4q−p+3

C(R)
‖f‖QK(p,q).



Int. J. Anal. Appl. 16 (3) (2018) 450

By the fact that (1 −|x|2)3 ≈ |U(a,R)|, from Lemma 1.2 and Lemma 1.3, we get

|Df(a) −Df(0)| ≤
(

4q−p+3

C(R)
−
( 3

4πR3
)1
p

)(∫
U(a,R)

|Df(x)|pdx
)1
p

≤ C(R,p,q)‖f‖
B
q+3
p

(∫
U(a,R)

1

(1 −|x|2)q+3
dx

)1
p

≤ C(R,p,q)‖f‖
B
q+3
p

1

|U(a,R)|
q+3
3p

(∫
U(a,R)

dx

)1
p

≤ C1(R,p,q)‖f‖
B
q+3
p
,

where a positive constant C1(R,p,q) is depending on R,p and q, which implies that

|Df(a)| ≤ |Df(0)| + C1(R,p,q)‖f‖
B
q+3
p
.

So, fore each compact set M ⊂ B, there is a constant C ∈ M such that

|Df(a)| ≤ |Df(0)| + C1(R,p,q)‖f‖QK(p,q) ≤ C‖f‖K, (2.1)

where the constant C = max

{
1,C(M,C1(R,p,q),

q+3
p

)

}
.

Now, we let {fn} be a Cauchy sequence in QK(p,q) spaces. From (2.1) we deduce that {fn} is also a Cauchy

sequence in the topology of uniform convergence on compact sets. Thus thereis a function f ∈ M(B) such

that fn → f also, Dfn → Df uniformly on compact subsets of B, as n →∞. To show that ‖fn −f‖K → 0

as n → ∞, we give ε > 0. Since, {fn} is a Cauchy sequence, there is an N > 0 such that ‖fk −fn‖K < ε2
and |Dfn(0) −Df(0)| < ε2 for all n,k ∈≥ N.

For each a ∈ B and n ≥ N, by applying Fatou’s lemma, we obtain∫
B
|Df(x) −Dfn(x)|p(1 −|x|2)qK

(
1 −|ϕa(x)|2

)
dx

=

∫
B

lim
k→∞

|Dfk(x) −Dfn(x)|p(1 −|x|2)qK
(
1 −|ϕa(x)|2

)
dx

≤ lim
k→∞

∫
B
|Dfk(x) −Dfn(x)|p(1 −|x|2)qK

(
1 −|ϕa(x)|2

)
dx

= lim
k→∞

‖fk −fn‖
p
QK(p,q)

<

(
ε

2

)p
.

Thus, for all n ≥ N,

‖fn −f‖K = |Dfn(0) −Df(0)|

+

(
sup
a∈B

∫
B
|Df(x) −Dfn(x)|p(1 −|x|2)qK

(
1 −|ϕa(x)|2

)
dx

)1
p

≤ ε,

which implies that fn → f in QK(p,q). Hence, the norm ‖.‖K is complete, therefore QK(p,q) spaces is a

Banach space in Clifford setting.



Int. J. Anal. Appl. 16 (3) (2018) 451

3. The Quaternion Bloch and QK(p,q) Spaces

In this part of the paper, we consider the relations between QK(p,q) and α-Bloch spaces in quaternion

sense. We characterize the quaternion α-Bloch spaces by the help of integral norms of quaternion QK(p,q)

spaces. Our results extend the results due to Wulan and Zhou [12] in quaternion sense.

Theorem 3.1. Let f ∈M(B) and let 1 ≤ p < ∞,−2 < q < ∞. Then

(i): QK(p,q) ⊂B
q+3
p ,

(ii): QK(p,q) = B
q+3
p ; if

∫ 1
0

(1 −ρ2)−3K(1 −ρ2)ρ2dρ < ∞. (3.1)

Proof. (i) Let 0 < R < 1 be fixed and a ∈ B. From Proposition 2.1, we acquire

(1 −|a|2)q+3|Df(a)|p ≤
4q−p+3

C(R)
J
p,q
K f(a).

If f ∈QK(p,q), then by estimate above we have f ∈B
q+3
p .

(ii) Let f ∈B
q+3
p be non constant. Then, there is M > 0 constant such that

(1 −|a|2)
q+3
p |Df(a)| ≤ M, for all x ∈ B.

Now we change the variable x = ϕa(w), then we acquire

J
p,q
K f(a) ≤

∫
B
Mp(1 −|x|2)−3K

(
1 −|ϕa(x)|2

)
dx

≤ Mp
∫
B
(1 −|ϕa(w)|2)−3K(1 −|w|2)

(1 −|a|2)3

|1 − āw|6
dw

≤ Mp
∫
B
(1 −|w|2)−3K(1 −|w|2)dw

≤ Mp
∫ 1

0

(1 −ρ2)−3K(1 −ρ2)ρ2dρ < ∞.

Thus, f ∈QK(p,q). This show that B
q+3
p ⊂QK(p,q).

Combining Theorem 3.1, we deduce the following corollary:

Corollary 3.1. Let 1 ≤ p < ∞,−2 < q < ∞ and let f ∈M(B). Then

(i): QK,0(p,q) ⊂B
q+3
p

0 ,

(ii): QK,0(p,q) = B
q+3
p

0 ; if (3.1) holds.



Int. J. Anal. Appl. 16 (3) (2018) 452

Proposition 3.1. Let 1 ≤ p < ∞,−2 < q < ∞ and let f ∈M(B). Then f ∈B
q+3
p if and only if there is an

0 < R < 1, such that

sup
a∈B

∫
U(a,R)

|Df(x)|p(1 −|x|2)qK
(
1 −|ϕa(x)|2

)
dx < ∞. (3.2)

Proof. If f ∈B
q+3
p , and a ∈ B. Then for any 0 < R < 1, we deduce

∫
U(a,R)

|Df(x)|p(1 −|x|2)qK
(
1 −|ϕa(x)|2

)
dx

≤
∫
B(0,R)

|Ψfa(w)|
p(1 −|ϕa(w)|2)qK(1 −|w|2)

(1 −|a|2)3

|1 − āw|6−2p
dw

≤ ‖f‖
B
q+3
p

∫
B(0,R)

(1 −|w|2)−3K
(
1 −|w|2

)
dw

≤ C‖f‖
B
q+3
p
.

Conversely, let (3.2) holds then, we deduce

∫
U(a,R)

|Df(x)|p(1 −|x|2)qK
(
1 −|ϕa(x)|2

)
dx

≥ K
(
1 −R2

)∫
B(0,R)

|Df(x)|p(1 −|x|2)qdx

≥
C(R)K

(
1 −R2

)
4q−p+3

(1 −|a|2)q+3|Df(a)|p,

which shows that f ∈B
q+3
p .

Corollary 3.2. Let K : (0,∞) → [0,∞) and f ∈M(B). Then f ∈B
q+3
p

0 if and only if there is an 0 < R < 1,

such that

lim
|a|→1

∫
U(a,R)

|Df(x)|p(1 −|x|2)qK
(
1 −|ϕa(x)|2

)
dx = 0.

Conclusion. Our results in this work will be of important uses in the study of operator theory at the

interface of monogenic function spaces. This work is a try to synthesize the achievements in the properties

of monogenic QK(p,q) function spaces. The problem in quaternion sense is that, Df(x) is monogenic, but

Df(φ(x)) is not monogenic, where φ : B → B is a monogenic function. The following question is open

problem: What properties of operators act between this classes of monogenic functions, like F(p,q,s) and

QK(p,q) classes?

In quaternion case, several authors have studied function spaces and classes like Qp,QK classes and F(p,q,s)

spaces, see [1, 3, 8] and others.



Int. J. Anal. Appl. 16 (3) (2018) 453

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	1. Introduction
	1.1. Ù’Analytic function spaces
	1.2. Quaternion function spaces

	2. Characterizations of QK(p,q) classes 
	3. The Quaternion Bloch and QK(p,q) Spaces 
	References