General weighted class of quaternion-valued functions with lacunary series expansions

Abstract

The purpose of this article is to define a new general weighted class of hyperholomorphic functions, the so called ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ Spaces. For this class we obtain characterizations by weighted Bloch ${{\mathcal {B}}}^{\alpha }_{\omega }$ spaces. Moreover, we characterize the hyperholomorphic ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ functions by the coefficients of certain lacunary series expansions in Clifford analysis.

1 Introduction

Quaternion analysis is the generalizations of the theory of holomorphic functions in one complex variable to Euclidean space. The concept of the hyperholomorphic functions based on the consideration of functions in the kernel of the generalized Cauchy–Riemann operator. The algebraic structure represents the measure difference between the theory of hyperholomorphic functions and the classical theory of analytic functions in the complex plane ${\mathbb {C}}.$ Analytic functions in ${\mathbb {C}}$ form an algebra while the same is not true in the sense of hyperholomorphic functions. The study of hyperholomorphic function spaces began with the interesting papers (see [5, 8, 11, 14]) and others.

Our objective in this article is twofold. First, we introduce a new generalized quaternion space ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ and study relations to the quaternion ${\mathcal {B}}^{\alpha }_{\omega }$ space. Furthermore, we will consider some essential properties of ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ spaces of quaternion-valued function as basic scale properties. Second, characterizations of the hyperholomorphic ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ functions by the coefficients of certain lacunary series expansions in clifford analysis are obtained.

2 Preliminaries

2.1 Analytic function spaces

Let ${{\mathbb {D}}}=\{z\in {{\mathbb {C}}}:\vert z \vert <1\}$ be the complex unit disk. The well known Bloch space is defined by:

$$\begin{aligned} {{\mathcal {B}}}=\left\{ f:f \text{ analytic } \text{ in } {{\mathbb {D}}} \text{ and } {{\mathcal {B}}}(f)=\sup _{z\in {{\mathbb {D}}}}(1-\vert z \vert ^2)\vert f'(z)\vert <\infty \right\} . \end{aligned}$$

Composing the Möbius transform $\varphi _a(z),$ which maps the unit disk ${{\mathbb {D}}}$ onto itself, and the fundamental solution of the two-dimensional real Laplacian on ${{\mathbb {D}}},$ we have the Green’s function $g(z,a)=\ln \vert \frac{1-\bar{a}z}{a-z}\vert$ with logarithmic singularity at $a\in {{\mathbb {D}}}.$ Here, $\varphi _a$ always stands for the Möbius transformation $\varphi _a(z)=\frac{a-z}{1-\bar{a} z}.$ Stroethoff [26] gave the following definition:

Definition 1

Let f be an analytic function in ${{\mathbb {D}}}$ and let $\;0<q<\infty$. If

$$\begin{aligned} \Vert f\Vert ^{q}_{{\mathbf {B}}^{q}}=\sup _{a\in {{{\mathbb {D}}}}}\int _{{{{\mathbb {D}}}}}\vert f'(z)\vert ^{q}(1-\vert z \vert ^2)^{q-2} \biggl (1-\vert \varphi _a(x)\vert ^2\biggr )^2 dA(z)<\infty , \end{aligned}$$

then $f\in {\mathbf {B}}^{q}.$

Definition 2

(See [4, 16]) Let aright-continuous and nondecreasing function $\omega :(0,1]\rightarrow (0,\infty ),$ the weighted Bloch space ${{\mathcal {B}}}_\omega$ is defined as the set of all analytic functions f on ${\mathbb {D}}$ satisfying

$$\begin{aligned} (1-\vert z \vert )\vert f^{\prime }(z)\vert \le C \omega (1-\vert z \vert ),\quad z \in {\mathbb {D}}, \end{aligned}$$

for some fixed $C=C_f > 0.$ In the special case where $\omega \equiv 1,\;{{\mathcal {B}}}_\omega$ reduces to the classical Bloch space ${{\mathcal {B}}}.$

Definition 3

(See [17, 23]) Let $0< \alpha < \infty$ and $\omega :(0,1] \rightarrow (0, \infty ).$ For an analytic function f in ${\mathbb {D}},$ we define the weighted $\alpha$-Bloch space ${{\mathcal {B}}}_{\omega }^\alpha ,$ as follows:

$$\begin{aligned} {{\mathcal {B}}}_{\omega }^{\alpha }=\bigg \{f: f\ \text{ analytic } \text{ in }\ {\mathbb {D}}\;\; \text{ and }\;\;\Vert f\Vert _{{{\mathcal {B}}}_{\omega }^{\alpha }} =\sup _{z\in {\mathbb {D}}}\frac{(1-\vert z \vert )^{\alpha }\big \vert f^{\prime }(z)\big \vert \,}{\omega (1-\vert z \vert )}<\infty \bigg \}. \end{aligned}$$

Also, the little weighted $\alpha$-Bloch space ${{\mathcal {B}}}_{\omega ,0}^{\alpha }$ is a subspace of ${{\mathcal {B}}}_{\omega }^{\alpha }$ consisting of all $f\in {{\mathcal {B}}}_{\omega }^{\alpha },$ such that

$$\begin{aligned} \lim _{\vert z\vert \rightarrow 1^-}\frac{(1-\vert z \vert )^{\alpha }\big \vert f^{\prime }(z)\big \vert \,}{\omega (1-\vert z \vert )}=0. \end{aligned}$$

2.2 Quaternion function spaces

To introduce the meaning of hyperholomorphic functions, let $\mathrm{I\!H}$ be the skew field of quaternions. This means we can write each element $w\in \mathrm{I\!H}$ in the form

$$\begin{aligned} w=w_0+w_1 i+w_2 j+w_3 k,\quad w_0,w_1,w_2,w_3\in \mathrm{I\!R}, \end{aligned}$$

where 1, i, j, k are the basis elements of $\mathrm{I\!H}$. For these elements we have the multiplication rules

$$\begin{aligned} i^2=j^2=k^2=-1,ij=-ji=k, kj=-jk=i, ki=-ik=j. \end{aligned}$$

The conjugate element $\bar{w}$ is given by $\bar{w}=w_0-w_1 i-w_2 j-w_3 k,$ and we have the property

$$\begin{aligned} w\bar{w}=\bar{w} w=\Vert w\Vert ^2=w_0^2+w_1^2+w_2^2+w_3^2. \end{aligned}$$

Moreover, we can identify each vector $\mathbf {x}=(x_0,x_1,x_2)\in \mathrm{I\!R}^3$ with a quaternion x of the form

$$\begin{aligned} x=x_0+x_1 i+x_2 j. \end{aligned}$$

In what follows we will work in ${{\mathbb {B}}}_1(0)\subset \mathrm{I\!R}^3$, the unit ball in the real three-dimensional space. We will consider functions f defined on ${{\mathbb {B}}}_1(0)$ with values in $\mathrm{I\!H}$. We define a generalized Cauchy–Riemann operator D by

$$\begin{aligned} Df=\frac{\partial f}{\partial x_0}+i\frac{\partial f}{\partial x_1} +j\frac{\partial f}{\partial x_2}, \end{aligned}$$

and it’s conjugate operator by

$$\begin{aligned} \overline{D} f=\frac{\partial f}{\partial x_0}-i\frac{\partial f}{\partial x_1} -j\frac{\partial f}{\partial x_2}. \end{aligned}$$

For these operators, we have that

$$\begin{aligned} D \overline{D}= \overline{D} D=\Delta _3, \end{aligned}$$

where $\Delta _3$ is the Laplacian for functions defined over domains in $\mathrm{I\!R}^3.$ For $\vert a \vert <1,$ we will denote by

$$\begin{aligned} \varphi _a(x)=(a-x)(1-\bar{a} x)^{-1}, \end{aligned}$$

the Möbius transform, which maps the unit ball onto itself. Furthermore, let

$$\begin{aligned} g(x,a)=\frac{1}{4\pi }\left( \frac{1}{\vert \varphi _a(x)\vert }-1\right) , \end{aligned}$$

be the modified fundamental solution of the Laplacian in $\mathrm{I\!R}^3$ composed with the Möbius transform $\varphi _a(x)$. Especially, we denote for all $p\ge 0$

$$\begin{aligned} g^p(x,a)=\frac{1}{4^p\pi ^{p}}\left( \frac{1}{\vert \varphi _a(x)\vert }-1\right) ^p. \end{aligned}$$

Let $f:{{\mathbb {B}}}\mapsto \mathrm{I\!H}$ be a hyperholomorphic function. Then from [11], we have the seminorms

• ${{\mathcal {B}}}(f)=\sup _{x\in {{\mathbb {B}}}} (1-\vert x \vert ^2)^{3/2} \vert \overline{D}f(x)\vert$,

• $Q_p(f)=\sup _{a\in {{\mathbb {B}}}}\int _{{\mathbb {B}}}\vert \overline{D}f(x)\vert ^2g^p(x,a)d{{\mathbb {B}}}_x$,

Definition 4

(See [5]) Let $0<\alpha <\infty .$ Recall that the hyperholomorphic $\alpha$-Bloch space is defined as follows:

$$\begin{aligned} {{\mathcal {B}}}^\alpha =\left\{ f\in \ker D({{\mathbb {B}}}_1(0)) :\;\; \sup _{x\in {{\mathbb {B}}}}(1-{\vert x\vert }^2)^{\frac{3\alpha }{2}} \vert \overline{D}f(x)\vert <\infty \right\} , \end{aligned}$$

the little $\alpha$-Bloch type space ${{\mathcal {B}}}^\alpha _0$ is a subspace of ${{\mathcal {B}}}$ consisting of all $f\in {{\mathcal {B}}}^\alpha$ such that

$$\begin{aligned} \lim _{\vert x\vert \rightarrow 1^-} (1-{\vert x\vert }^2)^{\frac{3\alpha }{2}} \vert \overline{D}f(x)\vert =0. \end{aligned}$$

Quite recently, El-Sayed Ahmed in [5], gave the following definition:

Definition 5

Let f be quaternion-valued function in ${{\mathbb {B}}}.$ For $0<q<\infty ,$ and $1\le \alpha <\infty .$ If

$$\begin{aligned} \Vert f\Vert ^{q}_{{\mathbf {B}}^{q}_{\alpha }}=\sup _{a\in {{\mathbb {B}}}} \int _{{{\mathbb {B}}}}\vert {\overline{D}} f(x)\vert ^{q}(1-\vert x \vert ^2)^{\frac{3\alpha q}{2}-3} \biggl (1-\vert \varphi _a(x)\vert ^2\biggr )^3 d{{\mathbb {B}}}_x<\infty , \end{aligned}$$

then $f\in {\mathbf {B}}^{q}_{\alpha }.$ Moreover, if

$$\begin{aligned} \lim _{\vert a \vert \rightarrow 1}\int _{{{\mathbb {B}}}}\vert {\overline{D}} f(x)\vert ^{q}(1-\vert x \vert ^2)^{\frac{3\alpha q}{2}-3} \biggl (1-\vert \varphi _a(x)\vert ^2\biggr )^3 d{{\mathbb {B}}}_x=0, \end{aligned}$$

then $\;f\in {\mathbf {B}}^{q}_{\alpha ,0}.$

Ahmed and Asiri in [8], gave the following definition:

Definition 6

Let aright-continuous and nondecreasing function $\omega :(0,1]\rightarrow (0,\infty ),$ and $1<\alpha <\infty .$ A quaternion-valued function f on ${{\mathbb {B}}}_1(0)$ is said to belong to the weighted $\alpha$-Bloch space ${{\mathcal {B}}}^\alpha _\omega ,$ if

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {B}}}^{\alpha }_{\omega }}=\sup _{x\in {{{\mathbb {B}}}_1(0)}}\frac{(1-\vert x \vert ^2)^{\frac{3\alpha }{2}}}{\omega (1-\vert x \vert )} \vert \overline{D}f(x)\vert <\infty . \end{aligned}$$

Moreover, a quaternion-valued function f on ${{\mathbb {B}}}_1(0)$ is said to belong to the weighted $\alpha$-Bloch space ${{\mathcal {B}}}^\alpha _{\omega ,0},$ if

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {B}}}^{\alpha }_{\omega ,0}}=\lim _{\vert x \vert \rightarrow 1^{-}}\frac{(1-\vert x \vert ^2)^{\frac{3\alpha }{2}}}{\omega (1-\vert x \vert )} \vert \overline{D}f(x)\vert <\infty . \end{aligned}$$

Now, we use the definition of Green function in $\mathrm{I\!R}^3$ (see [2])

$$\begin{aligned} G(x,a)=\frac{1-\vert \varphi _a(x)\vert ^2}{\vert 1-\overline{a}x \vert }. \end{aligned}$$

Then, we introduce the following new definition of the so called the hyperholomorphic ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ spaces.

Definition 7

Let $1\le \alpha <\infty ,$ $0<q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty ).$ Assume that f be hyperholomorphic function in the unit ball ${\mathbb {B}}_{1}(0).$ Then, $f \in {\mathbf {B}}^{q}_{\alpha ,\omega }(G),$ if

$$\begin{aligned} {\mathbf {B}}^{q}_{\alpha ,\omega }(G)=\bigg \{f\in ker D: \sup _{a\in {{{\mathbb {B}}}_1(0)}} \int _{{{{\mathbb {B}}}_1(0)}}\vert {\overline{D}} f(x)\vert ^{q}\frac{(1-\vert x \vert ^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\vert x \vert )} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x<\infty \bigg \}. \end{aligned}$$

The space ${\mathbf {B}}^{q}_{\alpha ,\omega ,0}(G)$ is subspace of ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ consisting of all functions $f\in {\mathbf {B}}^{q}_{\alpha ,\omega }(G),$ such that

$$\begin{aligned} \lim _{\vert a \vert \rightarrow 1^-} \int _{{{{\mathbb {B}}}_1(0)}}\vert {\overline{D}} f(x)\vert ^{q}\frac{(1-\vert x \vert ^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\vert x \vert )} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x=0. \end{aligned}$$

The following lemma, we will need in the sequel:

Lemma 1

(See [24]) Let $f\; {:} \; {\mathbb {B}}_1(0)\longrightarrow \mathrm{I\!H}$ be a hyperholomorphic function. Let $0<R<1,$ $1<q.$ Then for every $a\in {\mathbb {B}}_1(0)$

$$\begin{aligned} \vert \overline{D}f(a)\vert ^q\le \frac{3\cdot 4^{2+q}}{\pi R^3(1-R^2)^{2q}(1-\vert a\vert ^2)^3}\int _{{\mathcal {M}}(a,R)}\bigl \vert \overline{D}f(x)\bigr \vert ^q \, d{\mathbb {B}}_x. \end{aligned}$$

3 Characterization of ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ spaces in Clifford analysis

The relations between ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ space and ${{\mathcal {B}}}^{\alpha }_{\omega }$ spaces are given in quaternion sense. The results in this section are extensions and generalization of the results (see [13]).

Proposition 1

Let $1\le \alpha<\infty ,\;0\le p <\infty ,$ $0<q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty ).$ Assume that f be hyperholomorphic function in the unit ball ${\mathbb {B}}_{1}(0).$ Then

$$\begin{aligned}&\frac{(1-\vert a \vert ^2)^{^{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\vert a \vert )}\vert {\overline{D}} f(a)\vert ^{q}\\&\quad \le \frac{48(2)^{2q}(1+R)^3}{\pi R^3 (1-R^2)^{2q+3}(1-\vert a \vert ^2)^3} \int _{{{{\mathbb {B}}}_1(0)}}\vert {\overline{D}} f(x)\vert ^{q}\frac{(1-\vert x \vert ^2)^{^{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\vert x \vert )} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

Proof

Let ${\mathcal {M}}(a,R)=\{x\in {{{\mathbb {B}}}_1(0)}:\vert \varphi _a(x)\vert =\frac{\;\vert \;a-x\;\vert \;}{\;\vert \;1-\overline{a}x\;\vert \;}<R \}$ be pseudo-hyperbolic ball with center a and radius R. Then

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\nonumber \\&\quad \ge \int _{{\mathcal {M}}(a,R)}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

Since

$$\begin{aligned} \;\vert \;\varphi _a(x)\;\vert \;<R,\quad \forall x\in {\mathcal {M}}(a,R), \end{aligned}$$

and

$$\begin{aligned} G(x,a)=\frac{1-\;\vert \;\varphi _a(x)\;\vert \;^2}{\;\vert \;1-\overline{a}x\;\vert \;}. \end{aligned}$$

Then, we have

$$\begin{aligned} G(x,a)=\frac{1- R^2}{1+R},\quad where\ 1-R\le \;\vert \;1-\overline{a}x\;\vert \;\le 1+R . \end{aligned}$$

Now, for fixed $R \in (0,1)$ and $a\in {{\mathbb {B}}}_1(0).$

Let ${\mathcal {E}}(a,R) \subset {\mathcal {M}}(a,R),$ such that

$$\begin{aligned} {\mathcal {E}}(a,R)=\{x\in {{{\mathbb {B}}}_1(0)}:\;\vert \;x-a\;\vert \;<R \;\vert \;1-a\;\vert \;\}. \end{aligned}$$

Then, we deduce that

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \ge \bigg (\frac{1- R^2}{1+R}\bigg )^3\int _{{\mathcal {M}}(a,R)}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} d{{\mathbb {B}}}_x\\&\quad \ge \bigg (\frac{1- R^2}{1+R}\bigg )^3\int _{{\mathcal {E}}(a,R)}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} d{{\mathbb {B}}}_x\\&\quad \ge \bigg (\frac{1- R^2}{1+R}\bigg )^3 \frac{(1-\;\vert \;a\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;a\;\vert \;)}\int _{{\mathcal {E}}(a,R)}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q} d{{\mathbb {B}}}_x. \end{aligned}$$

Now, using Lemma 1, we obtain

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \ge \bigg (\frac{1- R^2}{1+R}\bigg )^3 \frac{(1-\;\vert \;a\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;a\;\vert \;)} \frac{\pi R^3(1-R^2)^{2q}(1-\;\vert \;a\;\vert \;^2)^3}{3(4)^{q+2}} \vert {\overline{D}} f(a)\vert ^{q} \\&\quad =\frac{\pi R^3(1- R^2)^{2q+3} (1-\;\vert \;a\;\vert \;^2)^{\frac{3}{2}(\alpha q+6)}}{3(4)^{q+2}(1+R)^3\omega ^q(1-\;\vert \;a\;\vert \;)} \vert {\overline{D}} f(a)\vert ^{q}, \end{aligned}$$

which implies that,

$$\begin{aligned}&\frac{ (1-\;\vert \;a\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;a\;\vert \;)} \vert {\overline{D}} f(a)\vert ^{q}\\&\quad \le \frac{48(2)^{2q}(1+R)^3}{\pi R^3(1- R^2)^{2q+3}(1-\;\vert \;a\;\vert \;^2)^3} \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

This completes the proof. $\square$

Corollary 1

From Proposition 1, we get for $1\le \alpha<\infty ,\;0\le p <\infty ,$ $0<q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty )$ that

$$\begin{aligned} {\mathbf {B}}^{q}_{\alpha ,\omega }(G)\subset {{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }. \end{aligned}$$

Proposition 2

Let $1\le \alpha<\infty ,\;0\le p <\infty ,$ $0<q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty ).$ Let f be a hyperholomorphic function in ${B_{1}(0)},$ $\forall a\in {B_{1}(0)};$ $\vert a\vert <1$ and $f\in {{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }.$ Then,  we have that

$$\begin{aligned} \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x \le k({{\mathcal {B}}}^{^{\gamma }}_{\omega }(f)), \end{aligned}$$

where $\gamma =\frac{3}{2q}(\alpha q+4).$

Proof

$$\begin{aligned} \frac{ (1-\;\vert \;x\;\vert \;^2)^{\frac{3\alpha }{2}}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \vert {\overline{D}} f(a)\vert ^{q}\le {{\mathcal {B}}}^{{\alpha }}_{\omega }(f). \end{aligned}$$

(See [8, 11]). Then,

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \le ({{\mathcal {B}}}^{^{\gamma }}_{\omega }(f)) \int _{{{{\mathbb {B}}}_1(0)}} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

Using the equality

$$\begin{aligned} G(x,a)=\frac{1-\;\vert \;\varphi _a(x)\;\vert \;^2}{\;\vert \;1-\overline{a}x\;\vert \;}= \frac{(1-\;\vert \;a\;\vert \;^2)(1-\;\vert \;x\;\vert \;^2)}{\;\vert \;1-\overline{a}x\;\vert \;^3}, \end{aligned}$$

(1)

where

$$\begin{aligned} 1-\;\vert \;x\;\vert \;\le \;\vert \;1-\overline{a}x\;\vert \;\le 1+\;\vert \;x\;\vert \;,\quad 1-\;\vert \;a\;\vert \;\le \;\vert \;1-\overline{a}x\;\vert \;\le 1+\;\vert \;a\;\vert \;\le 2. \end{aligned}$$

(2)

Then, we get

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \le ({{\mathcal {B}}}^{^{\gamma }}_{\omega }(f)) \int _{{{{\mathbb {B}}}_1(0)}} \frac{(1-\;\vert \;a\;\vert \;^2)^3(1-\;\vert \;x\;\vert \;^2)^3}{\;\vert \;1-\overline{a}x\;\vert \;^{9}}d{{\mathbb {B}}}_x.\\&\quad \le 2^{6}({{\mathcal {B}}}^{^{\gamma }}_{\omega }(f))\int _{{{{\mathbb {B}}}_1(0)}} \frac{(1-\;\vert \;a\;\vert \;)^3(1-\;\vert \;x\;\vert \;)^3}{(1-\;\vert \;x\;\vert \;^3)(1-\;\vert \;a\;\vert \;)^{6}}d{{\mathbb {B}}}_x.\\&\quad \le 2^{6}({{\mathcal {B}}}^{^{\gamma }}_{\omega }(f))\int _{{{{\mathbb {B}}}_1(0)}} \frac{1}{(1-\;\vert \;a\;\vert \;)^{3}}d{{\mathbb {B}}}_x.\\&\quad \le k({{\mathcal {B}}}^{^{\gamma }}_{\omega }(f)). \end{aligned}$$

Therefore, the proof of proposition is complete. $\square$

Corollary 2

From Proposition 2, we get for $1\le \alpha<\infty ,\;0\le p <\infty ,$ $0<q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty ),$ that

$$\begin{aligned} {{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }\subset {\mathbf {B}}^{q}_{\alpha ,\omega }(G). \end{aligned}$$

The results in Corollaries 1 and 2 prove the following theorem, which give to us the characterization for the hyperholomorphic weighted Bloch space by the integral norms of ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ spaces of hyperholomorphic functions.

Theorem 3

Let f be a hyperholomorphic function in ${B_{1}(0)}.$ Then for $\alpha ,\; p\ge 1,$ $0< q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty ),$ we have

$$\begin{aligned} {{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }= {\mathbf {B}}^{q}_{\alpha ,\omega }(G). \end{aligned}$$

For characterization the little hyperholomorphic weighted Bloch space, used the same arguments in the previous theorem to prove the following theorem.

Theorem 4

Let f be a hyperholomorphic function in ${B_{1}(0)}.$Then for $\alpha ,\; p\ge 1,$ $0< q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty ),$ we have

$$\begin{aligned} {{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega ,0}= {\mathbf {B}}^{q}_{\alpha ,\omega ,0}(G). \end{aligned}$$

Theorem 5

Let $0<R< 1$ and $\omega :(0,1]\rightarrow (0,\infty ).$ for the hyperholomorphic function f in ${{{\mathbb {B}}}_1(0)}.$ The following are equivalent

• (a) $f\in {{\mathcal {B}}}^{{\frac{3}{2q}(\alpha q+4)}}_{\omega }$.

• (b) For each $1\le \alpha <\infty ,$ and $0<q<\infty$

  $$\begin{aligned} \sup _{a\in {{\mathbb {B}}}_1(0)} \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x<\infty . \end{aligned}$$

• (c) For each $1\le \alpha <\infty ,$ and $0<q<\infty$

  $$\begin{aligned} \sup _{a\in {{\mathbb {B}}}_1(0)} \int _{{{\mathcal {M}}(a,R)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;x\;\vert \;)}d{{\mathbb {B}}}_x<\infty . \end{aligned}$$

• (d) For each $1\le \alpha <\infty ,$ and $0<q<\infty$

  $$\begin{aligned} \sup _{ a\in {{\mathbb {B}}}_1(0)}\frac{\vert {\mathcal {M}}(a,R)\vert ^{\frac{\alpha q+4}{2}}}{\omega ^q(1-\;\vert \;a\;\vert \;)} \int _{{\mathcal {M}}(a,R)}\;\vert \;{\overline{D}}f(x)\;\vert \;^{q}d{{\mathbb {B}}}_x <\infty . \end{aligned}$$

Proof

To prove (a) implies (b). Using (1) and (2), we have

$$\begin{aligned}&\sup _{a\in {{{\mathbb {B}}}_1(0)}} \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \le \sup _{a\in {{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \int _{{{{\mathbb {B}}}_1(0)}} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

Using the same steps as in Proposition 2, we obtain

$$\begin{aligned}&\sup _{a\in {{{\mathbb {B}}}_1(0)}} \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \le 2^{6}\Vert f \Vert _{{{\mathcal {B}}}^{{\frac{3}{2q}(\alpha q+4)}}_{\omega }}^{q} \int _{{{{\mathbb {B}}}_1(0)}}\frac{1}{(1-\;\vert \;a\;\vert \;)^3}d{{\mathbb {B}}}_x\\&\quad \le K_1\Vert f \Vert _{{{\mathcal {B}}}^{{\frac{3}{2q}(\alpha q+4)}}_{\omega }}^{q}\\&\quad <\infty . \end{aligned}$$

(b) implies (c), using the same steps as in Proposition 1, we deduce that

$$\begin{aligned}&\sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \ge \bigg (\frac{1- R^2}{2}\bigg )^3\sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{\mathcal {M}}(a,R)}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;x\;\vert \;)} d{{\mathbb {B}}}_x. \end{aligned}$$

For (c) implies (d), we use the fact $(1-\;\vert \;x\;\vert \;^2)^3 \approx \;\vert \;{\mathcal {M}}(a,R)\;\vert \;,$ $\forall \; x \in {\mathcal {M}}(a,R)$ (see [15]). Then

$$\begin{aligned}&\sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{\mathcal {M}}(a,R)}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;x\;\vert \;)} d{{\mathbb {B}}}_x\\&\quad \approx \frac{\;\vert \;{\mathcal {M}}(a,R)\;\vert \;^{\frac{\alpha q+4}{2}}}{{\omega ^q(1-\;\vert \;a\;\vert \;)}}\sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{\mathcal {M}}(a,R)}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q} d{{\mathbb {B}}}_x. \end{aligned}$$

For (d) implies (a). From Lemma 1, we have

$$\begin{aligned}&\vert \overline{D}f(a)\vert ^q\frac{(1-\;\vert \;a\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;a\;\vert \;)}\\&\quad \le \frac{3\cdot 4^{2+q}(1-\;\vert \;a\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\pi R^3(1-R^2)^{2q}(1-\vert a\vert ^2)^3\omega ^q(1-\;\vert \;a\;\vert \;)} \int _{{\mathcal {M}}(a,R)}\bigl \vert \overline{D}f(x)\bigr \vert ^q \, d{\mathbb {B}}_x \\&\quad =\frac{3\cdot 4^{2+q}(1-\;\vert \;a\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\pi R^3(1-R^2)^{2q}(1-\vert a\vert ^2)^3\omega ^q(1-\;\vert \;a\;\vert \;)}\\&\qquad \times \frac{{(1-R^2\;\vert \;a\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}R^{{\frac{3}{2}(\alpha q+4)}}}{{(1-R^2\;\vert \;a\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}R^{\frac{3}{2}(\alpha q+4)}}}\int _{{\mathcal {M}}(a,R)}\bigl \vert \overline{D}f(x)\bigr \vert ^q d{\mathbb {B}}_x. \end{aligned}$$

Now, since

$$\begin{aligned} \vert \;{\mathcal {M}}(a,R)\;\vert \;=\frac{(1-\;\vert \;a\;\vert \;^2)^3}{(1-R^2\;\vert \;a\;\vert \;^2)^3}R^3. \end{aligned}$$

Also, we used the following inequalities

$$\begin{aligned} 1-R^2\le 1-R^2\;\vert \;a\;\vert \;^2\le 1+R^2 \quad and\quad 1-\;\vert \;a\;\vert \;^2\le 1-R^2\;\vert \;a\;\vert \;^2\le 1+\;\vert \;a\;\vert \;^2. \end{aligned}$$

Then, we have

$$\begin{aligned}&\vert \overline{D}f(a)\vert ^q\frac{(1-\;\vert \;a\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;a\;\vert \;)}\\&\quad \le \frac{3\cdot 4^{2+q}\;\vert \;{\mathcal {M}}(a,R)\;\vert \;^{\frac{\alpha q+4}{2}}}{\pi R^3(1-R^2)^{2q}(1-\vert a\vert ^2)^3\omega ^q(1-\;\vert \;a\;\vert \;)}\times \frac{{(1-R^2\;\vert \;a\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}}{R^{{\frac{3}{2}(\alpha q+4)}}}\int _{{\mathcal {M}}(a,R)}\bigl \vert \overline{D}f(x)\bigr \vert ^q d{\mathbb {B}}_x\\&\quad \le \frac{3\cdot 4^{2+q}\;\vert \;{\mathcal {M}}(a,R)\;\vert \;^{\frac{(\alpha q+4)}{2}}}{\pi R^3(1-R^2)^{2q}(1-R^2)^3\omega ^q(1-\;\vert \;a\;\vert \;)}\times \frac{{(1+R^2)^{{\frac{3}{2}(\alpha q+4)}}}}{R^{{\frac{3}{2}(\alpha q+4)}}}\int _{{\mathcal {M}}(a,R)}\bigl \vert \overline{D}f(x)\bigr \vert ^q d{\mathbb {B}}_x\\&\quad =\frac{3\cdot 4^{2+q}(1+R^2)^{\frac{3}{2}(\alpha q+4)}}{\pi R^{\frac{3}{2}(\alpha q+6)}(1-R^2)^{2q+3}}\times \frac{{\;\vert \;{\mathcal {M}}(a,R)\;\vert \;^{\frac{(\alpha q+4)}{2}}}}{\omega ^q(1-\;\vert \;a\;\vert \;)}\int _{{\mathcal {M}}(a,R)} \bigl \vert \overline{D}f(x)\bigr \vert ^q d{\mathbb {B}}_x. \end{aligned}$$

Therefore, our theorem is proved. $\square$

From Theorem 5, using the same arguments, we directly obtain the following theorem.

Theorem 6

Let $0<R< 1$ and $\omega :(0,1]\rightarrow (0,\infty ).$ Then for the hyperholomorphic function f in ${{{\mathbb {B}}}_1(0)},$ the following are equivalent

• (a) $f\in {{\mathcal {B}}}^{{\frac{3}{2q}(\alpha q+4)}}_{\omega ,0}$.

• (b) For each $1\le \alpha <\infty ,$ and $0<q<\infty$

  $$\begin{aligned} \lim _{\vert a\vert \rightarrow 1^-} \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x=0. \end{aligned}$$

• (c) For each $1\le \alpha <\infty ,$ and $0<q<\infty$

  $$\begin{aligned} \lim _{\vert a\vert \rightarrow 1^-} \int _{{{\mathcal {M}}(a,R)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{{\frac{3}{2}(\alpha q+4)}}}{\omega ^q(1-\;\vert \;x\;\vert \;)}d{{\mathbb {B}}}_x=0. \end{aligned}$$

• (d) For each $1\le \alpha <\infty ,$ and $0<q<\infty$

  $$\begin{aligned} \lim _{\vert a\vert \rightarrow 1^-}\frac{\vert {\mathcal {M}}(a,R)\vert ^{\frac{\alpha q+4}{2}}}{\omega ^q(1-\;\vert \;a\;\vert \;)} \int _{{\mathcal {M}}(a,R)}\;\vert \;{\overline{D}}f(x)\;\vert \;^{q}d{{\mathbb {B}}}_x =0. \end{aligned}$$

The following theorem gives another relation between the quaternion ${{\mathcal {B}}}^{{\alpha }}_{\omega }$ space and the quaternion valued-functions space ${\mathbf {B}}^{q}_{\alpha ,\omega }(G).$

Theorem 7

Let f be a hyperholomorphic function in ${B_{1}(0)}.$ Then for $1\le<\alpha <\infty ,$ $0<q<\infty ,$ and $\omega :(0,1]\rightarrow (0,\infty ),$ we have

$$\begin{aligned} \Vert f\Vert _{{{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }}^{q}\approx \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)}(1-\;\vert \;\varphi _a(x)\;\vert \;^2)^\beta \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

Proof

From Theorem 5, we have

$$\begin{aligned} \Vert f \Vert _{{{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }}^{q}\approx \int _{{{\mathcal {M}}(a,R)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)}d{{\mathbb {B}}}_x. \end{aligned}$$

Let the constant

$$\begin{aligned} C(R)=(1-R^2)^\beta \bigg (\frac{1-R^2}{1+R}\bigg )^3, \end{aligned}$$

since C(R) depending on R is finite, then

$$\begin{aligned}&\Vert f\Vert _{{{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }}^{q}\nonumber \\&\quad \approx \sup _{a\in {{{\mathbb {B}}}_1(0)}} \int _{{{\mathcal {M}}(a,R)}} \vert \;{\overline{D}} f(x)\vert ^{q}\frac{(1-\vert x \vert ^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\vert x\vert )} (1-R^2)^\beta \bigg (\frac{1-R^2}{1+R}\bigg )^3 d{{\mathbb {B}}}_x. \end{aligned}$$

(3)

Since $x \in {{\mathcal {M}}(a,R)},$ then $\;\vert \;\varphi _a(x)\;\vert \;<R,\;\;\vert \;\varphi _a(x)\;\vert \;^2<R^2,$ and $1-\;\vert \;\varphi _a(x)\;\vert \;^2>1-R^2.$

Then, we have

$$\begin{aligned}&\Vert f\Vert _{{{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }}^{q}\nonumber \\&\quad \approx \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{\mathcal {M}}(a,R)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} (1-R^2)^\beta \bigg (\frac{1-R^2}{1+R}\bigg )^3 d{{\mathbb {B}}}_x\nonumber \\&\quad \le \lambda \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\vert {\overline{D}} f(x)\vert ^{q}\frac{(1-\vert x \vert ^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\vert x\vert )}(1-\vert \varphi _a(x)\vert ^2)^\beta \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

(4)

Conversely, we have

$$\begin{aligned}&\sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)}(1-\;\vert \;\varphi _a(x)\;\vert \;^2)^\beta \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\nonumber \\&\quad \le \Vert f\Vert _{{{\mathcal {B}}}^{\frac{3}{2}(\alpha q+4)}_{\omega }}^{q} \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}(1-\;\vert \;\varphi _a(x)\;\vert \;^2)^\beta \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

(5)

Using (1) and (2), we obtain

$$\begin{aligned}&\sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)}(1-\;\vert \;\varphi _a(x)\;\vert \;^2)^\beta \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\nonumber \\&\quad \le \Vert f\Vert _{{{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }}^{q} \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}} \frac{(1-\;\vert \;x\;\vert \;^2)^\beta (1-\;\vert \;a\;\vert \;^2)^\beta }{(1-\;\vert \;x\;\vert \;)^\beta } \frac{(1-\;\vert \;x\;\vert \;^2)^3(1-\;\vert \;a\;\vert \;^2)^3}{(1-\;\vert \;a\;\vert \;)^{6}(1-\;\vert \;x\;\vert \;)^{3}}d{{\mathbb {B}}}_x.\nonumber \\&\quad \le 2^{2(\beta +3)}\Vert f \Vert _{{{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }}^{q}\sup _{a\in {{{\mathbb {B}}}_1(0)}} \int _{{{{\mathbb {B}}}_1(0)}}{(1-\;\vert \;a\;\vert \;)^{\beta -3}} d{{\mathbb {B}}}_x. \end{aligned}$$

(6)

Then, we obtain

$$\begin{aligned}&\Vert f\Vert _{{{\mathcal {B}}}^{\frac{3}{2q}(\alpha q+4)}_{\omega }}^{q}\nonumber \\&\quad \ge \lambda \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}{\overline{D}} \;\vert \;f(x)\vert ^{q}\frac{(1-\vert x \vert ^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\vert x \vert )}(1-\vert \varphi _a(x)\vert ^2)^\beta \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x. \end{aligned}$$

(7)

From (4) and (7), the proof is complete. $\square$

4 Power series expansions of hyperholomorphic functions in $\mathrm{I\!R}^3$

The major difference to power series in the complex case consists in the absence of regularity of the basic variable $x = x_0 + x_1i + x_2j$ and of all of its natural powers $x^n,\; n = 2, 3,\ldots .$ This means that we should expect other types of terms, which could be designated as generalized powers. We use a pair $\underline{y} = (y_1, y_2)$ of two regular variables given by

$$\begin{aligned} y_1 = x_1-ix_0\quad and\quad y_2 = x_2 - jx_0, \end{aligned}$$

and a multi-index $\nu = (\nu _1, \nu _2),\; \;\vert \;\nu \;\vert \; = (\nu _1 + \nu _2)$ to define the $\nu$-power of $\underline{y}$ by a $\;\vert \;\nu \;\vert \;$-ary product (see [10, 14, 21]).

Definition 8

Let $\nu _1$ elements of the set $a_1,\ldots ,\; a_{\;\vert \;\nu \;\vert \;}$ be equal to $y_1$ and $\nu _2$ elements be equal to $y_2$. Then the $\nu$-power of $\underline{y}$ is defined by

$$\begin{aligned} \underline{y}:=\frac{1}{\;\vert \;\nu \;\vert \;!} \sum _{(i_1,\ldots ,i_{\;\vert \;\nu \;\vert \;})\in \pi (1,\ldots \;\vert \;\nu \;\vert \;)}a_{i1}a_{i2}\ldots a_{{i}_{\;\vert \;\nu \;\vert \;}}, \end{aligned}$$

(8)

where the sum runs over all permutations of $(1,\ldots , \;\vert \;\nu \;\vert \;).$

It was shown in [21], that the general form of the Taylor series of left monogenic functions in the neighborhood of the origin is given by

$$\begin{aligned} P(\underline{y}):=\sum _{n=0}^{\infty }\bigg (\sum _{\;\vert \;\nu \;\vert \;=n}\underline{y}^\nu c_\nu \bigg ),\quad with\ c_\nu \in \mathrm{I\!H}. \end{aligned}$$

(9)

The following results, we will need in the following section:

Theorem 8

(See [12, 14]) Let g(x) be left hyperholomorphic in a neighborhood of the origin with the Taylor series given in the form (9). Then there holds

$$\begin{aligned} \bigg \vert \;\frac{1}{2}\overline{D} g(x)\bigg \vert \;\le \sum _{n=1}^{\infty }n\bigg (\sum _{\;\vert \;\nu \;\vert \;=n} \;\vert \;c_\nu \;\vert \;\bigg )\;\vert \;x\;\vert \;^{n-1}. \end{aligned}$$

(10)

In order to formulate the next theorem we added the abbreviated notation ${\mathbf {H}}_n(x):=\sum _{\;\vert \;\nu \;\vert \;=n}\underline{y}^{\;\vert \;\nu \;\vert \;}c_\nu$ for such a homogeneous monogenic polynomial of degree n and consider monogenic functions composed by ${\mathbf {H}}_n(x)$ in the following form:

$$\begin{aligned} f(x)=\sum _{n=0}^{\infty }{\mathbf {H}}_n(x)b_n,\quad b_n\in \mathrm{I\!H}. \end{aligned}$$

Taking into account formula (10), we see that

$$\begin{aligned} \bigg \vert \;\frac{1}{2}\overline{D} f(x)\bigg \vert \;\le \sum _{n=1}^{\infty }n\bigg (\sum _{\;\vert \;\nu \;\vert \;=n} \;\vert \;c_\nu \;\vert \;\bigg )\;\vert \;b_n\;\vert \;\;\vert \;x\;\vert \;^{n-1}. \end{aligned}$$

(11)

This is the motivation for another shorthand notation, namely,

$$\begin{aligned} a_n:= \bigg (\sum _{\;\vert \;\nu \;\vert \;=n}\;\vert \;c_\nu \;\vert \;\bigg )\;\vert \;b_n\;\vert \quad (a_n \ge 0), \end{aligned}$$

finally, we have

$$\begin{aligned} \bigg \vert \;\frac{1}{2}\overline{D} f(x)\bigg \vert \;\le \sum _{n=1}^{\infty }na_n\;\vert \;x\;\vert \;^{n-1}. \end{aligned}$$

(12)

5 Lacunary series expansions in ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ spaces

In this section, we obtain a sufficient and necessary condition for any hyperholomorphic function f on the unite ball ${{{\mathbb {B}}}_1(0)}$ of $\mathrm{I\!R}^3$ with Hadamard gaps to belong to the weighted hyperholomorphic ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ spaces. The function

$$\begin{aligned} f(r)=\sum _{k}^{\infty }a_k r^{n_{k}}\quad (n_k\in {\mathbb {N}};\; \forall \; k\in {\mathbb {N}}) \end{aligned}$$

(13)

is said to belong to the Hadamard gap class (Lacunary series) if there exists a constant $\lambda >1$ such that $\frac{n_{k+1}}{n_k}\ge \lambda ,\; \forall \; k\in {\mathbb {N}}.$ In the past few decades both Taylor and Fourier series expansions were studied by the help of lacunary series (see [6, 7, 18, 20, 25, 27]). On the other hand there are some characterizations in higher dimensions using several complex variables and quaternion sense (see [1, 9, 19]).

Theorem 9

Let

$$\begin{aligned} f(r)=\sum _{n=1}^{\infty }a_{n}r^{n}, \end{aligned}$$

with $a_{n} \ge 0 .$ If $\alpha> 0,\;p > 0$ and $\omega :(0,1]\rightarrow (0,\infty ),$ then

$$\begin{aligned} \int _{0}^{1} (1-r)^{\alpha -1}(f(r))^{p}\; \frac{1}{\omega ^{p}(\log \frac{1}{r})}\; dr \approx \sum _{n=0}^{\infty } \; 2^{-n \alpha } \; t_n^{p} \; \frac{1}{\omega ^{p}( \frac{1}{2^{n}})}, \end{aligned}$$

(14)

where $t_n=\sum _{k\in I_n}a_{k},\; n\in {\mathbb {N}},$ $I_n = \{k: 2^{n} \le k <2^{n+1};\;\;k\in {\mathbb {N}}\}.$

Proof

The proof of this theorem can be obtained easily from Theorem 2.1 in [7] with the same steps, so it will be omitted.

It should be remarked that using simple computations will allow that Theorem 9 is still satisfying for the function $f (r )=\sum _{n=1}^{\infty }a_{n}r^{n-1}.$ $\square$

Theorem 10

Let $1\le \alpha <\infty ,$ $1<q<\infty ,$ $\omega :(0,1]\rightarrow (0,\infty )$ and $I_n=\{k:2^n\le k<2^{n+1};k\in {\mathbb {N}}\}.$ Suppose that $f(x)=\sum _{n=0}^{\infty }H_n(x) b_n,\; b_n\in \mathrm{I\!H},$ where $H_n(x)$ be homogenous hyperholomorphic polynomials of degree n,  and let $a_n$ be defined as before. If

$$\begin{aligned} \sum _{n=0}^{\infty }2^{-n(\frac{3}{2}\alpha q -q+4)}\biggl (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;}{\omega (\frac{1}{2^n})}\biggr )^q<\infty , \end{aligned}$$

(15)

then

$$\begin{aligned} \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\bigg \vert \;\frac{1}{2}{\overline{D}} f(x)\bigg \vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x<\infty , \end{aligned}$$

(16)

and $f\in {\mathbf {B}}^{q}_{\alpha ,\omega }(G).$

Proof

Suppose that (15) is hold. Using equality (1) and inequality (2), we have

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\bigg \vert \;\frac{1}{2}{\overline{D}} f(x)\bigg \vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\nonumber \\&\quad =\int _{{{{\mathbb {B}}}_1(0)}}\bigg \vert \;\frac{1}{2}{\overline{D}} \bigg (\sum _{n=0}^{\infty }H_n(x) b_n \bigg )\bigg \vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \frac{(1-\;\vert \;a\;\vert \;^2)^3(1-\;\vert \;x\;\vert \;^2)^3}{\;\vert \;1-\overline{a}x\;\vert \;^{9}}d{{\mathbb {B}}}_x\nonumber \\&\quad \le \int _{{{{\mathbb {B}}}_1(0)}}\bigg (\sum _{n=0}^{\infty } n a_n x^{n-1} \bigg )^{q} \frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \frac{(1-\;\vert \;a\;\vert \;^2)^3(1-\;\vert \;x\;\vert \;^2)^3}{(1-\;\vert \;a\;\vert \;)^3(1-\;\vert \;x\;\vert \;)^{6}}d{{\mathbb {B}}}_x\nonumber \\&\quad \le 2^{\frac{3}{2}(\alpha q+8)}\int _{0}^{1}\bigg (\sum _{n=0}^{\infty } n a_n r^{n-1} \bigg )^{q}\frac{(1-r)^{\frac{3}{2}(\alpha q+2)}}{\omega ^q(1-r)}r^2d{r}\nonumber \\&\quad \le \lambda \int _{0}^{1}\bigg (\sum _{n=0}^{\infty } n a_n r^{n-1} \bigg )^{q}\frac{(1-r)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-r)}d{r}. \end{aligned}$$

(17)

Using Theorem 9 in (17), we deduced that

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\bigg \vert \;\frac{1}{2}{\overline{D}} f(x)\bigg \vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\nonumber \\&\quad \le \lambda \int _{0}^{1}\bigg (\sum _{n=0}^{\infty } n a_n r^{n-1} \bigg )^{q}\frac{(1-r)^{\frac{3}{2}(\alpha q+2)}}{\omega ^q(1-r)}d{r}\nonumber \\&\quad \le \lambda \sum _{n=0}^{\infty } 2^{-n(\frac{3\alpha q}{2}+4)} t_n^q \frac{1}{\omega ^q(\frac{1}{2^n})}. \end{aligned}$$

(18)

Since,

$$\begin{aligned} t_n=\sum _{k\in I_n} k a_{k}<2^{n+1}\sum _{k\in I_n} a_{k}, \end{aligned}$$

we obtain that,

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\bigg \vert \;\frac{1}{2}{\overline{D}} f(x)\bigg \vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\\&\quad \le \lambda _1\sum _{n=0}^{\infty } 2^{-n(\frac{3\alpha q}{2}-q+4)} \biggl (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;}{\omega (\frac{1}{2^n})}\biggr )^q. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \Vert f \Vert _{{\mathbf {B}}^{q}_{\alpha ,\omega }(G)}^q\le \lambda _1\sum _{n=0}^{\infty } 2^{-n(\frac{3\alpha q}{2}-q+4)} \biggl (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;}{\omega (\frac{1}{2^n})}\biggr )^ q<\infty , \end{aligned}$$

where $\lambda _1$ is constant. Then, the last inequality implies that $f\in {\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ and the proof of our theorem is completed. $\square$

For the converse direction of Theorem 10, we consider the following theorem. We will restrict ourselves to special monogenic homogeneous polynomials of the form

$$\begin{aligned} {\mathrm {H}}_{n,\alpha }(x)=(y_{1}\alpha _{1}+y_{2}\alpha _2)^n=\sum _{k=0}^{n}\bigg (\begin{array}{c} n \\ k \\ \end{array}\bigg )y_{1}^{n-k}\times y_{2}^k\times \alpha _{1}^{n-k}\alpha _{2}^{k}, \end{aligned}$$

(19)

where $\alpha _i\in \mathrm{I\!R},\;i=1,2.$ The hypercomplex derivative is given by

$$\begin{aligned} \bigg (-\frac{1}{2}\overline{D}\bigg ){\mathrm {H}}_{n,\alpha }(x)=n{\mathrm {H}}_{n-1,\alpha }(x)(\alpha _1i+\alpha _2j). \end{aligned}$$

(20)

Proposition 11

(See [14]) Let $\alpha = (\alpha _1, \alpha _2), \alpha _ i \in \mathrm{I\!R},\; i = 1, 2$ be the vector of real coefficients defining the monogenic homogeneous polynomial ${\mathrm {H}}_{n,\alpha }(x)=(y_1\alpha _1+y_2\alpha _2)^n.$ Suppose that $\;\vert \;\alpha \;\vert \;^2 = \alpha _1^2+\alpha _2^2\ne 0.$ Then, 

$$\begin{aligned} \Vert {\mathrm {H}}_{n,\alpha }\Vert ^p_{L_p(\partial {{\mathbb {B}}}_1)}=2 \pi \sqrt{\pi }\;\vert \;\alpha \;\vert \;^{np}\frac{\Gamma (\frac{n}{2}p+1)}{\Gamma (\frac{n}{2}p+\frac{3}{2})},\quad \hbox {where}\ 0<p<\infty . \end{aligned}$$

(21)

Using formula (20), we have

$$\begin{aligned} \frac{\Vert -\frac{1}{2}\overline{D}{\mathrm {H}}_{n,\alpha }\Vert ^p_{L_p(\partial {{\mathbb {B}}}_1)}}{\Vert \vert \;{\mathrm {H}}_{n,\alpha }\ \Vert ^p_{L_p(\partial {{\mathbb {B}}}_1)}}=n^p \frac{{\mathbf {B}}\bigg (\frac{1}{2},\frac{n-1}{2}p+1\bigg )}{{\mathbf {B}}\bigg (\frac{1}{2},\frac{n}{2}p+1\bigg )}\ge \lambda n^p>0, \end{aligned}$$

(22)

(see [14]), where,

$$\begin{aligned} {\mathbf {B}}\bigg (\frac{1}{2},\frac{n-1}{2}p+1\bigg )=\frac{\Gamma (\frac{1}{2})\Gamma (\frac{n-1}{2}p+1)}{\Gamma (\frac{n-1}{2}p+\frac{3}{2})}, \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{{\mathbf {B}}\bigg (\frac{1}{2},\frac{n-1}{2}p+1\bigg )}{{\mathbf {B}}\bigg (\frac{1}{2},\frac{n}{2}p+1\bigg )}=1. \end{aligned}$$

Corollary 3

(See [14]) Assume that $p\ge 2.$ Then, 

$$\begin{aligned} \frac{\Vert -\frac{1}{2}\overline{D}{\mathrm {H}}_{n,\alpha }\Vert ^2_{L_2 (\partial {{\mathbb {B}}}_1)}}{\Vert {\mathrm {H}}_{n,\alpha }\Vert ^2_{L_p(\partial {{\mathbb {B}}}_1)}}\ge \lambda n^{\frac{2+3p}{2p}}. \end{aligned}$$

(23)

Theorem 12

Let $1<\alpha <\infty ,$ $2\le q<\infty ,$ $\omega :(0,1]\rightarrow (0,\infty ),$ and $0<\;\vert \;x\;\vert \;=r<1.$ If

$$\begin{aligned} f(x)=\bigg (\sum _{n=0}^{\infty }\frac{{\mathrm {H}}_{n,\alpha }}{(1-\;\vert \;x\;\vert \;^2)^{\frac{24+q}{4q}}\Vert {\mathrm {H}}_{n,\alpha } \Vert _{L_q(\partial {{\mathbb {B}}}_1)}} a_n\bigg ) \in {\mathbf {B}}^{q}_{\alpha ,\omega }(G). \end{aligned}$$

(24)

Then,

$$\begin{aligned} \sum _{n=0}^{\infty }2^{-n(\frac{3}{2}\alpha q -q+4)}\biggl (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;}{\omega (\frac{1}{2^n})}\biggr )^q<\infty . \end{aligned}$$

(25)

Proof

$$\begin{aligned} \Vert f \Vert _{{\mathbf {B}}^{q}_{\alpha ,\omega }(G)}= & {} \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \big ( G(x,a)\big )^{3}d{{\mathbb {B}}}_x\nonumber \\= & {} \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+4)}}{\omega ^q(1-\;\vert \;x\;\vert \;)} \bigg ( \frac{(1-\;\vert \;x\;\vert \;^2)(1-\;\vert \;a\;\vert \;^2)}{\;\vert \;1-\overline{a}x\;\vert \;^3}\bigg )^{3}d{{\mathbb {B}}}_x\nonumber \\\ge & {} \sup _{a\in {{{\mathbb {B}}}_1(0)}}\int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+6)}}{\omega ^q(1-\;\vert \;x\;\vert \;)}d{{\mathbb {B}}}_x \quad (where \;a=0). \end{aligned}$$

(26)

Hence, we have

$$\begin{aligned}&\Vert f \Vert _{{\mathbf {B}}^{q}_{\alpha ,\omega }(G)}\nonumber \\&\quad \ge \int _{{{{\mathbb {B}}}_1(0)}}\;\vert \;-\frac{1}{2}{\overline{D}} f(x)\;\vert \;^{q}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+6)}}{\omega ^q(1-\;\vert \;x\;\vert \;)}d{{\mathbb {B}}}_x \quad (where\ a=0)\nonumber \\&\quad =\int _{{{{\mathbb {B}}}_1(0)}}\bigg \vert \sum _{n=0}^{\infty }\bigg [\frac{-\frac{1}{2}{\overline{D}}{\mathrm {H}}_{n,\alpha }}{(1- \vert x \vert ^2)^{\frac{24+q}{4q}}\Vert {\mathrm {H}}_{n,\alpha }\Vert _{L_q(\partial {{\mathbb {B}}}_1)}}\bigg ] a_n\bigg \vert \;^{{q}}\frac{((1- \vert x \vert ^2)^{\frac{3}{2}(\alpha q+6)}}{\omega ^q((1- \vert x \vert ^2)}d{{\mathbb {B}}}_x. \end{aligned}$$

(27)

Where $\bigg [\frac{-\frac{1}{2}{\overline{D}}{\mathrm {H}}_{n,\alpha }}{\Vert {\mathrm {H}}_{n,\alpha }\Vert _{L_q(\partial {{\mathbb {B}}}_1)}}\bigg ]$ is a homogeneous hyperholomorphic polynomial of degree $n - 1$ and it can be written in the form

$$\begin{aligned} \bigg [\frac{-\frac{1}{2}{\overline{D}}{\mathrm {H}}_{n,\alpha }}{\Vert {\mathrm {H}}_{n,\alpha }\Vert _{L_q(\partial {{\mathbb {B}}}_1)}}\bigg ]=r^{(n-1)}\Phi _n(\phi _1,\phi _2), \end{aligned}$$

(28)

where

$$\begin{aligned} \Phi _n(\phi _1,\phi _2):=\bigg (\bigg [\frac{-\frac{1}{2}{\overline{D}}{\mathrm {H}}_{n,\alpha }}{\Vert {\mathrm {H}}_{n,\alpha }\Vert _{L_q(\partial {{\mathbb {B}}}_1)}}\bigg ]\bigg )_{\partial {{\mathbb {B}}}_1}. \end{aligned}$$

(29)

Now, using the quaternion-valued inner product

$$\begin{aligned} \langle f, g\rangle _{\partial {{\mathbb {B}}}_1(0)}=\int _{{\partial {{\mathbb {B}}}_1(0)}} \overline{f(x)} g(x)d\Gamma _x, \end{aligned}$$

the orthogonality of the spherical monogenic $\Phi _n(\phi _1,\phi _2)$ (see [3]) in $L_2(\partial {{\mathbb {B}}}_1(0)).$ Then, substituting from (28) and (29) to (27), we obtain

$$\begin{aligned}&\int _{{{{\mathbb {B}}}_1(0)}}\bigg \vert \;\sum _{n=0}^{\infty }\bigg [\frac{-\frac{1}{2}{\overline{D}}{\mathrm {H}}_{n,\alpha }}{(1-\;\vert \;x\;\vert \;^2)^{\frac{24+q}{4q}}\Vert {\mathrm {H}}_{n,\alpha }\Vert _{L_q(\partial {{\mathbb {B}}}_1)}}\bigg ] a_n\bigg \vert \;^{{q}}\frac{(1-\;\vert \;x\;\vert \;^2)^{\frac{3}{2}(\alpha q+6)}}{\omega ^q(1-\;\vert \;x\;\vert \;)}d{{\mathbb {B}}}_x\nonumber \\&\quad =\int _{{0}}^1\int _{{\partial {{\mathbb {B}}}_1(0)}}\bigg (\bigg \vert \;\sum _{n=0}^{\infty }\frac{r^{n-1}}{(1-r^2)^{\frac{24+p}{4q}}}\Phi _n(\phi _1,\phi _2)a_n\bigg \vert \;^2 \bigg )^{\frac{{q}}{2}}r^2\frac{(1-r^2)^{\frac{3}{2}(\alpha q+6)}}{\omega ^q(1-r)}d\Gamma _x dr\nonumber \\&\quad =\int _{{0}}^1\int _{{\partial {{\mathbb {B}}}_1(0)}}\bigg (\sum _{n=0}^{\infty }\sum _{j=0}^{\infty }\overline{a_n}\frac{r^{2n-2}}{(1-r^2)^{\frac{24+q}{2q}}}\overline{\Phi _n}(\phi _1,\phi _2){\Phi _j}(\phi _1,\phi _2)a_j \bigg )^{\frac{{q}}{2}}\nonumber \\&\qquad \times r^2\frac{(1-r^2)^{\frac{3}{2}(\alpha q+6)}}{\omega ^q(1-r)}d\Gamma _x dr = {\mathrm {L}}. \end{aligned}$$

(30)

From Hölder’s inequality, we have

$$\begin{aligned} \int _{{\partial {{\mathbb {B}}}_1(0)}}\;\vert \;f(x)\;\vert \;^qd\Gamma _x\ge (4\pi )^{1-q}\bigg \vert \;\int _{{\partial {{\mathbb {B}}}_1(0)}}f(x)d\Gamma _x\bigg \vert \;^q, \quad (where\ 1\le q<\infty ). \end{aligned}$$

(31)

From (31), for $2\le q<\infty ,$ we have

$$\begin{aligned} {\mathrm {L}}\ge & {} (4\pi )^{1-\frac{q}{2}}\int _{{0}}^1\bigg (\sum _{n=0}^{\infty }\;\vert \;{a_n}\;\vert \;^2\frac{r^{2n-2}}{(1-r^2)^{\frac{24+q}{2q}}}\ \;\vert \;{\Phi _n}(\phi _1,\phi _2)\ \;\vert \;^2_{L_2(\partial {{\mathbb {B}}}_1)} \bigg )^{\frac{{q}}{2}} r^2\frac{(1-r^2)^{\frac{3}{2}(\alpha q+6)}}{\omega ^q(1-r)} dr\nonumber \\\ge & {} (4\pi )^{1-\frac{q}{2}}\int _{{0}}^1\bigg (\sum _{n=0}^{\infty }\;\vert \;{a_n}\;\vert \;^2{r^{2n-2}}\ \;\vert \;{\Phi _n}(\phi _1,\phi _2)\ \;\vert \;^2_{L_2(\partial {{\mathbb {B}}}_1)} \bigg )^{\frac{{q}}{2}} r^3\frac{(1-r^2)^{\frac{3\alpha q}{2}+3-\frac{p}{4}}}{\omega ^q(1-r)} dr. \end{aligned}$$

(32)

From Corollary 3, we have

$$\begin{aligned} \vert \;{\Phi _n}(\phi _1,\phi _2)\ \;\vert \;^2_{L_2(\partial {{\mathbb {B}}}_1)}=\frac{\Vert -\frac{1}{2}{\overline{D}}{\mathrm {H}}_{n,\alpha }\Vert _{L_2(\partial {{\mathbb {B}}}_1)}}{\Vert {\mathrm {H}}_{n,\alpha }\Vert _{L_q(\partial {{\mathbb {B}}}_1)}}\ge \lambda n^{\frac{2+3q}{2q}}\ge \lambda n^{\frac{3}{2}}. \end{aligned}$$

Then, from above we have

$$\begin{aligned} {\mathrm {L}}\ge & {} (4\pi )^{1-\frac{q}{2}}\lambda \int _{{0}}^1\bigg (\sum _{n=0}^{\infty }n^{\frac{3}{2}}\;\vert \;{a_n}\;\vert \;^2{r^{2n-2}} \bigg )^{\frac{{q}}{2}} r^3\frac{(1-r^2)^{\frac{3\alpha q}{2}+3-\frac{q}{4}}}{\omega ^q(1-r)} dr\nonumber \\= & {} \lambda _1 \int _{{0}}^1\bigg (\sum _{n=0}^{\infty }n^{\frac{3}{2}}\;\vert \;{a_n}\;\vert \;^2{r^{2n-2}} \bigg )^{\frac{{q}}{2}} r^3\frac{(1-r^2)^{\frac{3\alpha q}{2}+3-\frac{q}{4}}}{\omega ^q(1-r)} dr\nonumber \\= & {} \frac{\lambda _1}{2} \int _{{0}}^1\bigg (\sum _{n=0}^{\infty }n^{\frac{3}{2}}\;\vert \;{a_n}\;\vert \;^2{\xi ^{n-1}} \bigg )^{\frac{{q}}{2}} \xi \frac{(1-\xi )^{\frac{3\alpha q}{2}+3-\frac{q}{4}}}{\omega ^q(1-\sqrt{\xi })} d\xi \nonumber \\\ge & {} \lambda _3 \int _{{0}}^1\bigg (\sum _{n=0}^{\infty }n^{\frac{3}{2}}\;\vert \;{a_n}\;\vert \;^2{\xi ^{n-1}} \bigg )^{\frac{{q}}{2}}\frac{(1-\xi )^{\frac{3\alpha q}{2}+3-\frac{q}{4}}}{\omega ^q(1-\xi )} d\xi , \end{aligned}$$

(33)

where $\lambda _j,\; j={1,2,3},$ are constants not depending on n.

Now, we apply Theorem 9 in Eq. (33), we deduced that

$$\begin{aligned} \Vert f \Vert _{{\mathbf {B}}^{q}_{\alpha ,\omega }(G)} \ge {\mathrm {L}}\ge \frac{\lambda _3}{k}\sum _{n=0}^{\infty }2^{{-n}(\frac{3\alpha q}{2}+4-\frac{q}{4})}\bigg (\frac{\sum _{k\in I_n}k^{\frac{3}{2}}\;\vert \;a_k\;\vert \;^2}{{\omega ^{2}(\frac{1}{2^n})}}\bigg )^{{\frac{{q}}{2}}}. \end{aligned}$$

(34)

Where

$$\begin{aligned} {\sum _{k\in I_n}k^{\frac{3}{2}}\;\vert \;a_k\;\vert \;^2}> \bigg (2^n\bigg )^{\frac{3}{2}}\bigg ({\sum _{k\in I_n}\;\vert \;a_k\;\vert \;^2}\bigg )^{{\frac{{q}}{2}}}. \end{aligned}$$

Then,

$$\begin{aligned} \Vert f \Vert _{{\mathbf {B}}^{q}_{\alpha ,\omega }(G)} \ge {\mathrm {L}}\ge C\sum _{n=0}^{\infty }2^{{-n}(\frac{3\alpha q}{2}+4-{q})}\bigg (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;^2}{{\omega ^{2}(\frac{1}{2^n})}}\bigg )^{{\frac{{q}}{2}}}, \end{aligned}$$

(35)

where C be a constant not depending on n.

From [22], we have

$$\begin{aligned} \sum _{n=0}^{N}a_n^q\le \bigg (\sum _{n=0}^{N}a_n^q\bigg )^q\le N^{q-1}\sum _{n=0}^{N}a_n^q. \end{aligned}$$

Then, we have

$$\begin{aligned} \Vert f \Vert _{{\mathbf {B}}^{q}_{\alpha ,\omega }(G)} \ge {\mathrm {L}}\ge C_1\sum _{n=0}^{\infty }2^{{-n}(\frac{3\alpha q}{2}+4-{q})}\bigg (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;}{{\omega (\frac{1}{2^n})}}\bigg )^{{q}}, \end{aligned}$$

(36)

where $C_1$ be a constant not depending on n. Then, we deduced that

$$\begin{aligned} \sum _{n=0}^{\infty }2^{{-n}(\frac{3\alpha q}{2}+4-{q})}\bigg (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;}{{\omega (\frac{1}{2^n})}}\bigg )^{{q}}<\infty . \end{aligned}$$

(37)

This completes the proof of theorem. $\square$

Theorem 13

Let $1<\alpha <\infty ,$ $2\le q<\infty ,$ $\omega :(0,1]\rightarrow (0,\infty ),$ and $0<\;\vert \;x\;\vert \;=r<1,$ we have that

$$\begin{aligned} f(x)=\bigg (\sum _{n=0}^{\infty }\frac{{\mathrm {H}}_{n,\alpha }}{(1-\;\vert \;x\;\vert \;^2)^{\frac{24+q}{4q}}\Vert {\mathrm {H}}_{n,\alpha }\Vert _{L_q(\partial {{\mathbb {B}}}_1)}} a_n\bigg ) \in {{\mathbf {B}}^{q}_{\alpha ,\omega }(G)}, \end{aligned}$$

(38)

if and only if,

$$\begin{aligned} \sum _{n=0}^{\infty }2^{-n(\frac{3}{2}\alpha q +4-q)}\biggl (\frac{\sum _{k\in I_n}\;\vert \;a_k\;\vert \;}{\omega (\frac{1}{2^n})}\biggr )^q<\infty . \end{aligned}$$

(39)

Proof

This theorem can be proved directly from Theorems 10 and 12. $\square$

6 Conclusion

The aim of this article was to introduce a new generalized quaternion space ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ and study relations to the quaternion ${{\mathcal {B}}}^{\alpha }_{\omega }$ space. Furthermore, we considered some essential properties of ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ spaces of quaternion-valued function as basic scale properties. Second, characterizations of the hyperholomorphic ${\mathbf {B}}^{q}_{\alpha ,\omega }(G)$ functions by the coefficients of certain lacunary series expansions in clifford analysis are obtained.