1. Introduction

In ^[17], Wulan and Wu introduced the so called spaces. These spaces consist of analytic functions on the unit open complex disk such that

where is a non-decreasing and righ-continuous function.

Green's function in the unit disk with logarithmic singularity at is given by

where .

Moreover, if

For more results of spaces see ^{[5, 6, 11]} and ^[16]. It is known that the spaces are Banach spaces under the norm

for every and . Moreover, it is known that the Green's function can be replaced by the weight function .

There are a number of ways we can further generalize the spaces; see ^[4] and ^[14] for example.

Remark 1.1

If then see ^[5]. In particular, if then is the Dirichlet space . Moreover, if then coincides with BMOA, the space of analytic functions of bounded mean oscillation.

Two magnitudes and are similar, denoted by if there exist two non-negative real constants and such that, .

In this paper we will work in , the skew field of quaternions, that is, each element can be written in the form

where and are the basis elements of . For these elements we have the multiplication rules

The product is extended by linearity. The quaternionic conjugation is given by and we have the property

Therefore, if the quaternion

Also, the norm satisfies for each .

We identify each point with a quaternion of the form .

Let be the unit ball in the real three-dimensional space, with boundary . For and , we denote by the ball with center and radius .

Let be a domain in , then we will consider -valued functions defined in (depending on ):

The notation , has the usual component-wise meaning. On we define ageneralized Cauchy-Riemann operator by

and it's conjugate operator by

The solutions of are called (left) hyperholomorphic (or monogenic) functions and generalize the class of holomorphic functions from the one-dimensional complex function theory. For more details about quaternionic analysis and general Clifford analysis, we refer to ^[1], ^[8] and ^[15] and others.

We denote by the class of hyperholomorphic (or monogenic) functions on . For the Möbius transform is defined by

Furthermore, let

be a multiple scalar of the fundamental solution of the Laplacian in composed with the Möbius transform , i.e. is the modified Green's function in quaternion sense.

For and the pseudo-hyperbolic ball is defined by

This is an Euclidean ball, with center and radius given respectively by:

Let the -Bloch space of quaternion valued functions given by (see ^{[2, 9]}):

The space is called the quaternion Bloch space . The little quaternion -Bloch space is a subspace of consisting of all such that

The quaternion Dirichlet space is given by:

Let be a non-decreasing function. Define as

The spaces of quaternion valued functions given by

Moreover, the little quaternion space consists of those for which

Remark 1.2

Obviously, the quaternion spaces are not Banach spaces, also are not linear spaces. Nevertheless, if we consider a small neighborhood of the origin , with an arbitrary but fixed , then we can add the -norm of the function over to the seminorms, so spaces will become Banach spaces.

Remark 1.3

It should be remarked that if we put then (see ^[7]). Also, if , then , the quaternion Dirichlit space.

Let be a non-decreasing function, consider the following problems:

1. What conditions must have in order that to be non-trivial?

2. Which properties of and imply that ?

3. For which a necessary and sufficient conditions on so that ?

The main aim of this paper is to study these spaces and their relations to the above mentioned quaternionic Bloch space. We shall develop a general theory for quaternionic spaces which answers these questions and gives most basic properties of and spaces. Our results are extensions of the results due to Essén and Wulan (see ^[5]) in quaternion sense.

The concept may be generalized in the context of Clifford analysis to arbitrary real dimensions. We will restrict us for simplicity to and quaternion-valued functions as (the lowest non-commutative case) a model case. For more studies on quaternion function spaces, we refer to ^{[2, 3, 7, 10]} and others.

We will need the following lemma in the sequel (see ^[12], Lemma 2.2, if ):

Lemma 1.1

Let and let . Then for every , we have

(1)

where

Remark 1.4

If we change the variables (the Jacobian determinant has no singularities). In quaternion sense, the problem is that, is hyperholomorphic, but after the change of variables is not hyperholomorphic.

But we know from ^[13] that is again hyperholomorphic. So, we can solve this problem by the following lemma (see ^[10], Lemma 2.2):

Lemma 1.2

Let and let and let given by

(2)

Then and is a subharmonic function.

We also refer to ^[15] who studied this problem for the four-dimensional case already in 1979.

2. Q_K –spaces in Clifford Analysis

In this section, relations between and Bloch spaces, which have been attracted considerable attention are given in quaternion sense. Our results are extensions of the results due to Essen and Wulan (see ^[5]) in quaternion sense. We consider some essential properties of spaces of quaternion-valued functions as basic scale properties.

For a non-decreasing function we say that the space is trivial if contains only constant functions. Whether the space is trivial or not depends on the integral

(3)

Proposition 2.1

(i) If the integral (3) is divergent, then the space is trivial.

(ii) If the integral (3) is convergent, then .

Proof:

(i) For and . Let given by (2). Then is a hyperholomorphic function and is a subharmonic function. By Lemma 2.1, after a change of variables we have . Assume that there exists such that for some .

By subharmonicity of , we have

(4)

Thus the integral (3) must be convergent and we have proved (i).

(ii) Conversely, if the integral (3) is convergent and it follows from the inequality (4) that i.e., we have . This completes the proof.

The convergence of (3) is related to the growth order of . The log-order of the real-valued function is defined as

If the log-type of the quaternion-valued function is defined as

We always assume that the non-decreasing function is differentiable and satisfies if and if . We assume also that the integral (3) is convergent, otherwise, contains constant functions only.

The following result was proved in ^[3]:

Proposition 2.2

If the log-order and the log-type of a non-decreasing function satisfy one of the following conditions:

(1) ,

(2) and .

Then the space is trivial.

Remark 2.1

In the critical case and , may be trivial or nontrivial.

From now on and through the remainder of Sections 2 and 3 we assume that the function is non-decreasing and that the integral (3) is convergent.

Theorem 2.1

Assume that and set

Then .

Proof:

Since is non-decreasing and , it is clear that . It remains to prove that . We note that

Thus in . It suffices to deal with integrals over .

Now we let then for , we have

By condition (3), the last integral above is convergent. This shows that and Theorem 3.1 is proved.

The significance of Theorem 3.1 is that the space only depends on the behavior of for close to 0. In particular, when studying spaces, we can always assume that for . However, we do not make this assumption in our main theorems.

Proposition 2.3

Let Then, a monogenic function belongs to the Bloch space if and only if there exists an such that and

(5)

Proof:

If , by the argument in the proof of Theorem 3.1, the supremum in (5) is finite for any

Conversely, if the supremum in (5) is finite, then

The following result gives a characterization of the quaternion Bloch space by quaternion spaces.

Theorem 2.2

Let , then if and only if

(6)

Proof:

Let us first assume that (6) holds. For , we have

Then, for , we deduce that

Here, we used that the Jacobian determinant is

Now, using the equality

we obtain that,

Then, we have .

To prove that , we assume that . For a fixed let

Then, we have

By Lemma 1.1, we obtain

which implies that,

This completes the proof.

The importance of Theorem 2.2 is to give us a characterization for the quaternionic Bloch space by the help of integral norms of spaces of quaternion valued functions.

Also, with the same arguments used to prove the previous theorem, we can prove the following theorem for characterization of little hyperholomorphic Bloch space.

Theorem 2.3

Let , then if and only if (6) holds.

Now we give a characterization for the quaternion spaces in terms of some different weighted functions in the unit ball of .

Define as

Theorem 2.4

For , let . Then,

(7)

Proof:

We consider the equivalence

By the change of variable and Lemma 1.2, we have

with , while

where the Jacobian determinant.

Then, we only need to show

This is obvious because of the assumptions for , and the following obvious facts

• if

• if .

The proof of Theorem 3.4 is completed.

3. Conclusion

Our aim in this paper lies at the interface of hyperholomorphic function spaces and operator theory. This paper is an attempt to synthesizethe achievements in the theory of hyperholomorphic function spaces. Many interesting and seemingly basic problems remain open. One of those open problems is the following question: What kind of operators act between the weighted hyperholomorphic function spaces like Bloch and spaces? In analytic case several authors have studied boundedness and compactness of composition and Toeplitz operators between some weighted classes of function spaces like BMOA (the space of analytic functions of bounded mean oscillation), and spaces (see ^{[4, 9, 14]} and others).

In quaternion sense the problem is that, is hyperholomorphic, but is not hyperholomorphic, where is a hyperholomorphic self-map of the unit ball B.

Acknowledgments

The author would like to thank the referees for their valuable remarks and comments.

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