1. Introduction
Quaternions were introduced for the first time by William Rowan Hamilton in 1843 [
1]. The generalizations of the theory of holomorphic functions in one complex variable is known as Quaternion analysis [
2,
3,
4,
5]. Quaternions are also recognized as a powerful tool for modeling and solving problems in theoretical as well as applied mathematics [
6]. The emergence of a large of software packages to perform computations in the algebra of the real quaternions [
7], or more generally, Clifford algebra has been enhanced by the increasing interest in using quaternions and their applications in almost all applied sciences [
8,
9].
Definition 1.
Let \(\;0< p< \infty\), \(\;-2< q< \infty\;\) and \(\;0< s< \infty\) and let \(f\) be an analytic function in \(\mathbb D.\) If
$$
\|f\|^{p}_{F(p,q,s)}=\sup_{a\in{\mathbb D}}\int_{{\mathbb
D}}|f'(z)|^{p}(1-|z|^2)^{q} g^{s}(z,a)dA(z)< \infty,
$$
then \(f\in F(p,q,s).\) Moreover, if
$$
\lim_{|a|\rightarrow1}\int_{\mathbb D}|f'(z)|^{p}(1-|z|^2)^{q}
g^{s}(z,a)dA(z)=0,
$$
then \(\;f\in{F_{0}(p,q,s)}.\)
To introduce the meaning of hyperholomorphic functions, let \(\mathbb{H}\) be the skew field of quaternions. The element \(w\in\mathbb{H}\) can be written in the form:
$$
w=w_0+w_1 i+w_2 j+w_3 k,\quad w_0,w_1,w_2,w_3\in\mathbb{R},
$$
where \(1,i,j,k\) are the basis elements of \(\mathbb{H}\). For these elements we have the multiplication rules
$$
i^2=j^2=k^2=-1,ij=-ji=k, kj=-jk=i, ki=-ik=j.
$$
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
$$
w\bar w=\bar w w=\| w\|^2=w_0^2+w_1^2+w_2^2+w_3^2.
$$
Moreover, we can identify each vector \(\vec x=(x_0,x_1,x_2)\in\mathbb{R}^3\) with a quaternion \(x\) of the form
$$
x=x_0+x_1 i+x_2 j.
$$
We will work in the unit ball in the real three-dimensional space, \(\mathbb B_1(0)\subset\mathbb{R}^3\). We will consider functions \(f\) defined on \(\mathbb B_1(0)\)
with values in \(\mathbb{H}\). We define a generalized Cauchy-Riemann operator \(D\) and it’s conjugate \(\bar D\) by
$$
Df=\frac{\partial f}{\partial x_0}+i\frac{\partial f}{\partial x_1}+j\frac{\partial f}{\partial x_2},
$$
and
$$
\bar Df=\frac{\partial f}{\partial x_0}-i\frac{\partial f}{\partial x_1} -j\frac{\partial f}{\partial x_2}.
$$
For these operators, we have
$$D \bar D= \bar DD=\Delta_3,$$
where \(\Delta_3\) is the Laplacian for functions defined over domains in \(\mathbb{R}^3.\) We denote by
\(
\varphi_a(x)=(a-x)(1-\bar a x)^{-1},\;|a|< 1,
\)
the Möbius transform, which maps the unit ball onto itself.
Let
$$
g(x,a)=\frac{1}{4\pi}\left(\frac{1}{|\varphi_a(x)|}-1\right)
$$
be the modified fundamental solution of the Laplacian in \(\mathbb{R}^3.\) Let \(f:\mathbb B\mapsto \mathbb{H}\) be a hyperholomorphic function. Then [
4]:
- \({\cal B}(f)=\sup\limits_{x\in \mathbb B} (1-|x|^2)^{3/2} |\bar Df(x)|\),
- \(Q_p(f)=\sup\limits_{a\in \mathbb B}\int_{\mathbb B}|\bar Df(x)|^2g^p(x,a)d{\mathbb B}_x\).
Definition 2.
Let \(0< \alpha< \infty.\) The hyperholomorphic \(\alpha\)-Bloch space is defined as follows (see [2]):
$$
{\cal B}^\alpha=\{f\in \ker D :\;\;\sup\limits_{x\in \mathbb B}(1-{\vert x\vert}^2)^{\frac{3\alpha}{2}} \vert \bar Df(x)\vert < \infty\}.
$$
The little \(\alpha\)-Bloch type space \({\cal B}^\alpha_0\) is a subspace of \({\cal B}\) consisting of all \(f\in {\cal B}^\alpha\) such that
$$\lim_{\vert x\vert \to 1^-}(1-{\vert x\vert}^2)^{\frac{3\alpha}{2}} \vert \bar Df(x)\vert=0.$$
Definition 3.
[10]
Let \(f\) be quaternion-valued function in \(\mathbb B.\) For \(0< p< \infty,\) \(-2< q< \infty\) and \(0< s< \infty.\) If
$$
\|f\|^{p}_{F(p,q,s)}=\sup_{a\in{\mathbb B}}\int_{{\mathbb
B}}|{\overline D} f(x)|^{p}(1-|x|^2)^{\frac{3q}{2}} \biggl(1-\vert
\varphi_a(x)\vert^2\biggr)^sd{\mathbb B}_x< \infty,
$$
then \(f\in F(p,q,s).\) Moreover, if
$$
\lim_{|a|\rightarrow1}\int_{{\mathbb B}}|{\overline D}
f(x)|^{p}(1-|x|^2)^{\frac{3q}{2}} \biggl(1-\vert
\varphi_a(x)\vert^2\biggr)^sd{\mathbb B}_x=0,
$$
then \(\;f\in{F_{0}(p,q,s)}.\)
The green function in \(\mathbb{R}^3 \) is defined as (see [
11]):
$$G(x,a)=\frac{1-|\varphi_a(x)|^2}{|1-\overline{a}x|}.$$
We introduce following new definition of so called hyperholomorphic \(F^{\alpha}_{G}(p,q,s)\) spaces.
Definition 4.
Let \(1< \alpha,\; p < \infty,\) \(-2< q0.\) Assume that \(f\) be hyperholomorphic function in the unit ball \(\mathbb{B}_{1}(0).\) Then, \(f \in F^{\alpha}_{G}(p,q,s),\) if
\begin{eqnarray*}
F^{\alpha}_{G}(p,q,s)=\bigg\{f\in ker D: \sup\limits_{a\in {\mathbb B_1(0)}} \int_{{\mathbb B_1(0)}}|{\overline D} f(x)|^{p}{(1-|x|^2)^{{3\alpha q}{2}+2s}} \big( G(x,a)\big)^{s}d{\mathbb B}_x< \infty\bigg\}.
\end{eqnarray*}
The space \(F^{\alpha}_{G,0}(p,q,s)\) is subspace of \(F^{\alpha}_{G}(p,q,s)\) consisting of all functions \(f\in F^{\alpha}_{G}(p,q,s),\) such that
\begin{eqnarray*}
\lim_{|a|\rightarrow1^-} \int_{{\mathbb B_1(0)}}|{\overline D} f(x)|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \big( G(x,a)\big)^{s}d{\mathbb B}_x=0.
\end{eqnarray*}
Our objective in this article is twofold. First, we study the generalized quaternion space \(F^{\alpha}_{G}(p,q,s)\) and characterize their relations to the quaternion \(\mathcal B^{\alpha}_{0} \). Second, characterizations \(F^{\alpha}_{G}(p,q,s)\) function space by the coefficients of Hadamard gap expansions.
The following lemma, we will need in the sequel:
Lemma 5.[12]
Let \(0< R< 1\), \(1< q,\) \(a\in \mathbb{B}_1(0)\) and \(f\; : \; \mathbb{B}_1(0)\longrightarrow \mathbb{H} \) be a hyperholomorphic function. Then
\[\vert\bar{D}f(a)\vert^q\leq \displaystyle\frac{3\cdot4^{2+q}}{\pi R^3(1-R^2)^{2q}(1-\vert a\vert^2)^3}\int_{\mathcal{M}(a,R)}\bigl\vert\bar{D}f(x)\bigr\vert^q \, d\mathbb{B}_x\ .\]
2. Power series expansions in \(\mathbb{R}^3\)
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
$$y_1 = x_1-ix_0\; and\; y_2 = x_2 – jx_0,$$
and a multi-index \( \nu= (\nu_1, \nu_2),\; |\nu| = (\nu_1 + \nu_2)\) to define the \(\nu\)-power of \(\underline{y}\)
by a \(|\nu|\)-ary product [
5,
13,
14].
Definition 6.
Let \(\nu_1\) elements of the set \(a_1,…. ,\; a_{|\nu|} \) 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{eqnarray}\label{eq 3.1}
\underline{y}:=\frac{1}{|\nu|!}\sum_{(i_1,…,i_{|\nu|})\in\pi(1,…|\nu|)}a_{i1}a_{i2}…a_{{i}_{|\nu|}},
\end{eqnarray}
(1)
where the sum runs over all permutations of \((1,…., |\nu|).\)
The general form of the Taylor series of left monogenic functions in the neighborhood of the origin is given as [
14]:
\begin{eqnarray}\label{eq 3.2.1}
P(\underline{y}):=\sum_{n=0}^{\infty}\bigg(\sum_{|\nu|=n}\underline{y}^\nu c_\nu\bigg),\;\; \; c_\nu\in\mathbb{H}.
\end{eqnarray}
(2)
Theorem 7.[5, 15]
Let \(g(x)\) be left hyperholomorphic with the Taylor series (2). Then
\begin{eqnarray}\label{eq 3.111}
\bigg|\frac{1}{2}\overline {D} g(x)\bigg|\leq \sum\limits_{n=1}^{\infty}n\bigg(\sum\limits_{|\nu|=n}|c_\nu|\bigg)|x|^{n-1}.
\end{eqnarray}
(3)
We introduce the notation \(\mathbf{H}_n(x):=\sum\limits_{|\nu|=n}\underline{y}^{|\nu|}c_\nu\) and consider monogenic functions composed by
\(\mathbf{H}_n(x)\) in the following form:
$$f(x)=\sum\limits_{n=0}^{\infty}\mathbf{H}_n(x)b_n,\;\;\;\;\; b_n\in\mathbb{H}.$$
Using (3), we have
\begin{eqnarray}\label{eq 3.112}
\bigg|\frac{1}{2}\overline D f(x)\bigg|\leq \sum\limits_{n=1}^{\infty}n\bigg(\sum\limits_{|\nu|=n}|c_\nu|\bigg)|b_n||x|^{n-1}.
\end{eqnarray}
(4)
This is the motivation for another notation,
\begin{eqnarray}\label{eq 3.112.x} a_n:= \bigg(\sum\limits_{|\nu|=n}|c_\nu|\bigg)|b_n|\;\;\; (a_n \geq 0),\end{eqnarray}
(5)
finally, we have
\begin{eqnarray}\label{eq 3.113}
\bigg|\frac{1}{2}\overline D f(x)\bigg|\leq \sum\limits_{n=1}^{\infty}na_n|x|^{n-1}.
\end{eqnarray}
(6)
3. Lacunary series expansions in \(F^{\alpha}_{G}(p,q,s)\) spaces
In this section, we give a sufficient and necessary condition for the hyperholomorphic function \(f\) on \({\mathbb B_1(0)} \) of \(\mathbb{R}^3\) with Hadamard gaps to belong to the weighted hyperholomorphic \(F^{\alpha}_{G}(p,q,s)\) spaces.
The function
\begin{eqnarray}\label{eq4.1}
f(r)=\sum\limits_{k}^{\infty}a_k r^{n_{k}}\;\;\;\;\; (\; n_k\in \mathbb{N};\; \forall\; k\in \mathbb{N})
\end{eqnarray}
(7)
belong to the Hadamard gap class (Lacunary series) if there exists a constant \(\lambda>1\) such that \(\frac{n_{k+1}}{n_k}\geq \lambda,\; \forall\; k\in \mathbb{N}.\)
Characterizations in higher dimensions using several complex variables and quaternion sense [
16,
17,
18].
Theorem 8.
Let
\(f(r)=\sum_{n=1}^{\infty}a_{n}r^{n},\)
with \(a_{n} \geq 0 .\) If \( \alpha > 0,\;p > 0.\)
Then
\begin{eqnarray}
\int_{0}^{1} (1-r)^{\alpha-1}(f(r))^{p}\; dr
\approx
\sum_{n=0}^{\infty} \; 2^{-n \alpha} \; t_n^{p},
\end{eqnarray}
(8)
where \(t_n=\sum_{k\in I_n}a_{k},\; n\in \mathbb{N},\) \(I_n = \{k: 2^{n}
\leq k < 2^{n+1};\;\;k\in \mathbb{N}\}.\)
Proof.
The prove of this theorem can be obtained easily from Theorem 2.5 of [19] with the same steps.
Theorem 9.
Let \(\alpha,\; p\geq 1,\) \(-2< q0,\) and \(I_n=\{k:2^n\leq k< 2^{n+1};k\in\mathbb{N}\}.\)
Suppose that \(f(x)=\sum\limits_{n=0}^{\infty}H_n(x) b_n, b_n\in\mathbb{H},\) where \(H_n(x)\) be homogenous hyperholomorphic polynomials of degree \(n,\) and let \(a_n\) be define as {before in (5). If
\begin{eqnarray}\label{eq4.2}
\sum\limits_{n=0}^{\infty}2^{-n(\frac{3}{2}\alpha q +s-p+1)}\bigl({\sum_{k\in I_n}|a_k|}\bigr)^p< \infty,
\end{eqnarray}
(9)
then
\begin{eqnarray}\label{eq4.3}
\sup\limits_{a\in {\mathbb B_1(0)}}\int_{{\mathbb B_1(0)}}\bigg|\frac{1}{2}{\overline D} f(x)\bigg|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \big( G(x,a)\big)^{s}d{\mathbb B}_x< \infty,
\end{eqnarray}
(10)
and \(f\in F^{\alpha}_{G}(p,q,s).\)
Proof.
Suppose that (9) holds. Using the equality
\begin{eqnarray} \label{eq1.1}
G(x,a)=\frac{1-|\varphi_a(x)|^2}{|1-\overline{a}x|}= \frac{(1-|a|^2)(1-|x|^2)}{|1-\overline{a}x|^3},
\end{eqnarray}
(11)
where
\begin{eqnarray}\label{eq1.2}
1-|x|\leq |1-\overline{a}x|\leq 1+|x|,\;\;\;\;\;1-|a|\leq |1-\overline{a}x|\leq 1+|a|\leq2.
\end{eqnarray}
(12)
Then, we get
\begin{eqnarray}\label{eq4.4}
&&\int_{{\mathbb B_1(0)}}\bigg|\frac{1}{2}{\overline D} f(x)\bigg|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \big( G(x,a)\big)^{s}d{\mathbb B}_x\nonumber\\
&&=\int_{{\mathbb B_1(0)}}\bigg|\frac{1}{2}{\overline D} \bigg(\sum\limits_{n=0}^{\infty}H_n(x) b_n \bigg)\bigg|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \frac{(1-|a|^2)^s(1-|x|^2)^s}{|1-\overline{a}x|^{3s}}d{\mathbb B}_x\nonumber\\
&&\leq\int_{{\mathbb B_1(0)}}\bigg(\sum\limits_{n=0}^{\infty}
n a_n x^{n-1} \bigg)^{p} {(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \frac{(1-|a|^2)^s(1-|x|^2)^s}{(1-|a|)^s(1-|x|)^{2s}}d{\mathbb B}_x\nonumber\\
&&\leq2^{\frac{3\alpha q}{2}+4s}\int_{0}^{1}\bigg(\sum\limits_{n=0}^{\infty}
n a_n r^{n-1} \bigg)^{p}{(1-r)^{{3\alpha q}{2}+s}}r^2d{r}\nonumber\\
&&\leq\lambda\int_{0}^{1}\bigg(\sum\limits_{n=0}^{\infty}
n a_n r^{n-1} \bigg)^{p}{(1-r)^{\frac{3\alpha q}{2}+s}}d{r}.
\end{eqnarray}
(13)
Using Theorem 8 in (13), we deduced that
\begin{eqnarray}\label{eq4.5}
\int_{{\mathbb B_1(0)}}\bigg|\frac{1}{2}{\overline D} f(x)\bigg|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \big( G(x,a)\big)^{s}d{\mathbb B}_x
&\leq&\lambda\int_{0}^{1}\bigg(\sum\limits_{n=0}^{\infty}
n a_n r^{n-1} \bigg)^{p}{(1-r)^{\frac{3\alpha q}{2}+s}}d{r}\nonumber\\
&\leq&\lambda\sum\limits_{n=0}^{\infty}
2^{-n({3\alpha q}{2}+s+1)} t_n^P .
\end{eqnarray}
(14)
Since
$$ t_n=\sum_{k\in I_n} k a_{k}< 2^{n+1}\sum_{k\in I_n} a_{k},$$
we obtain that,
\begin{eqnarray}\label{eq4.6}
\int_{{\mathbb B_1(0)}}\bigg|\frac{1}{2}{\overline D} f(x)\bigg|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \big( G(x,a)\big)^{s}d{\mathbb B}_x
&\leq&\lambda_1\sum\limits_{n=0}^{\infty}
2^{-n(\frac{3\alpha q}{2}+s-p+1)} \bigl({\sum_{k\in I_n}|a_k|}\bigr)^p.\nonumber
\end{eqnarray}
Therefore, we have
\begin{eqnarray}\label{eq4.7}
\|f\|_{F^{\alpha}_{G}(p,q,s)}\leq\lambda_1\sum\limits_{n=0}^{\infty}
2^{-n(\frac{3\alpha q}{2}+s-p+1)} \bigl({\sum_{k\in I_n}|a_k|}\bigr)^
p< \infty,\nonumber
\end{eqnarray}
where \(\lambda_1\) is a constant. Then, the last inequality implies that \(f\in {F^{\alpha}_{G}(p,q,s)}\) and the proof of our theorem is completed.
For the converse of Theorem 9, we consider the following theorem.
Proposition 10. (see [5])
Let \(\alpha = (\alpha_1, \alpha_2), \alpha_ i \in\mathbb{R},\; i = 1, 2 \) be the vector of
real coefficients defining \(\mathrm{H}_{n,\alpha}(x)=(y_1\alpha_1+y_2\alpha_2)^n.\) Suppose that \(|\alpha|^2 = \alpha_1^2+\alpha_2^2\neq0.\) Then,
\begin{eqnarray}\label{eq4.10}
\|\mathrm{H}_{n,\alpha}\|^p_{L_p(\partial\mathbb B_1)}=2 \pi \sqrt{\pi}|\alpha|^{np}\frac{\Gamma(\frac{n}{2}p+1)}{\Gamma(\frac{n}{2}p+\frac{3}{2})},\;\;\; where\;0< p< \infty.
\end{eqnarray}
(15)
Moreover, we have (see [
5])
\begin{eqnarray}\label{eq4.11}
\frac{\|-\frac{1}{2}\overline{D}\mathrm{H}_{n,\alpha}\|^p_{L_p(\partial\mathbb B_1)}}{\|\mathrm{H}_{n,\alpha}\|^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)}\geq \lambda n^p>0,\;\;\;\;\;\;
\end{eqnarray}
(16)
where,
\(\mathbf{B}\bigg(\frac{1}{2},\frac{n-1}{2}p+1\bigg)=\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(\frac{n-1}{2}p+1\right)}{\Gamma\left(\frac{n-1}{2}p+\frac{3}{2}\right)},\)
and
\(
\lim\limits_{n\rightarrow\infty}\frac{\mathbf{B}\left(\frac{1}{2},\frac{n-1}{2}p+1\right)}{\mathbf{B}\left(\frac{1}{2},\frac{n}{2}p+1\right)}=1.\)
Corollary 11.[5]
Assume that \(p\geq2.\) Then,
\begin{eqnarray}\label{eq4.12}
\frac{\|-\frac{1}{2}\overline{D}\mathrm{H}_{n,\alpha}\|^2_{L_2(\partial\mathbb B_1)}}{\|\mathrm{H}_{n,\alpha}\|^2_{L_p(\partial\mathbb B_1)}}\geq \lambda n^{\frac{2+3p}{2p}}.
\end{eqnarray}
(17)
Theorem 12.
Let \(\alpha\geq 1,\) \(2\leq p< \infty,\) \(-2< q0,\) and \(0< |x|=r< 1. \) If
\begin{eqnarray}\label{eq4.13}
f(x)=\bigg(\sum\limits_{n=0}^{\infty}\frac{\mathrm{H}_{n,\alpha}}{(1-|x|^2)^{\frac{8s+p}{4p}}\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}} a_n\bigg) \in F^{\alpha}_{G}(p,q,s).
\end{eqnarray}
(18)
Then,
\begin{eqnarray}\label{eq4.14}
\sum\limits_{n=0}^{\infty}2^{-n(\frac{3}{2}\alpha q +s-p+1)}\bigl({\sum_{k\in I_n}|a_k|}\bigr)^p< \infty.
\end{eqnarray}
(19)
Proof.
Since
\begin{eqnarray}\label{eq4.15}
\|f\|_{F^{\alpha}_{G}(p,q,s)}&=&\sup\limits_{a\in {\mathbb B_1(0)}}\int_{{\mathbb B_1(0)}}|{\overline D} f(x)|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \big( G(x,a)\big)^{s}d{\mathbb B}_x\nonumber\\
&=&\sup\limits_{a\in {\mathbb B_1(0)}}\int_{{\mathbb B_1(0)}}|{\overline D} f(x)|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+2s}} \bigg( \frac{(1-|x|^2)(1-|a|^2)}{|1-\overline{a}x|^3}\bigg)^{s}d{\mathbb B}_x\nonumber\\
&\geq&\sup\limits_{a\in {\mathbb B_1(0)}}\int_{{\mathbb B_1(0)}}|{\overline D} f(x)|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+3s}}d{\mathbb B}_x \;\;\; (where \;a=0).
\end{eqnarray}
(20)
Hence, we have
\begin{eqnarray}\label{eq4.16}
\|f\|_{F^{\alpha}_{G}(p,q,s)}&\geq&\int_{{\mathbb B_1(0)}}|-\frac{1}{2}{\overline D} f(x)|^{p}{(1-|x|^2)^{\frac{3\alpha q}{2}+3s}} d{\mathbb B}_x \;\;\; (where \;a=0).\nonumber\\
&=&\int_{{\mathbb B_1(0)}}\bigg|\sum\limits_{n=0}^{\infty}\bigg[\frac{-\frac{1}{2}{\overline D}\mathrm{H}_{n,\alpha}}{(1-|x|^2)^{\frac{8s+p}{4p}}\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}}\bigg] a_n\bigg|^{{p}}{(1-|x|^2)^{\frac{3\alpha q}{2}+3s}}d{\mathbb B}_x.
\end{eqnarray}
(21)
where \(\bigg[\frac{-\frac{1}{2}{\overline D}\mathrm{H}_{n,\alpha}}{\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}}\bigg]\)
is a homogeneous hyperholomorphic polynomial of
degree \(n\)-\(1\) and it can be written in the form
\begin{eqnarray}\label{eq4.17}
\bigg[\frac{-\frac{1}{2}{\overline D}\mathrm{H}_{n,\alpha}}{\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}}\bigg]=r^{(n-1)}\Phi_n(\phi_1,\phi_2),
\end{eqnarray}
(22)
where
\begin{eqnarray}\label{eq4.18}
\Phi_n(\phi_1,\phi_2):=\bigg(\bigg[\frac{-\frac{1}{2}{\overline D}\mathrm{H}_{n,\alpha}}{\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}}\bigg]\bigg)_{\partial\mathbb B_1}.
\end{eqnarray}
(23)
Now, by the inner product \(\langle f, g\rangle_{\partial\mathbb B_1(0)}=\int_{{\partial\mathbb B_1(0)}} \overline{f(x)} g(x)d\Gamma_x,\) the orthogonality of the spherical monogenic \(\Phi_n(\phi_1,\phi_2)\) (see [
20]) in \( L_2(\partial\mathbb B_1(0)).\) From (22) and (23) to (21), we have
\begin{eqnarray}\label{eq4.19}
&&\int_{{\mathbb B_1(0)}}\bigg|\sum\limits_{n=0}^{\infty}\bigg[\frac{-\frac{1}{2}{\overline D}\mathrm{H}_{n,\alpha}}{(1-|x|^2)^{\frac{8s+p}{4p}}\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}}\bigg] a_n\bigg|^{{p}}{(1-|x|^2)^{\frac{3\alpha q}{2}+3s}} d{\mathbb B}_x\nonumber\\
&=&\int_{{0}}^1\int_{{\partial\mathbb B_1(0)}}\bigg(\bigg|\sum\limits_{n=0}^{\infty}\frac{r^{n-1}}{(1-r^2)^{\frac{8s+p}{4p}}}\Phi_n(\phi_1,\phi_2)a_n\bigg|^2 \bigg)^{\frac{{p}}{2}}r^2{(1-r^2)^{\frac{3\alpha q}{2}+3s}}d\Gamma_x dr\nonumber\\
&=&\int_{{0}}^1\int_{{\partial\mathbb B_1(0)}}\bigg(\sum\limits_{n=0}^{\infty}\sum\limits_{j=0}^{\infty}\overline{a_n}\frac{r^{2n-2}}{(1-r^2)^{\frac{8s+p}{2p}}}\overline{\Phi_n}(\phi_1,\phi_2){\Phi_j}(\phi_1,\phi_2)a_j \bigg)^{\frac{{p}}{2}} r^2{(1-r^2)^{\frac{3\alpha q}{2}+3s}}d\Gamma_x dr = \mathrm{L}.
\end{eqnarray}
(24)
From Hölder’s inequality, we have
\begin{eqnarray}\label{eq4.20}
\int_{{\partial\mathbb B_1(0)}}|f(x)|^pd\Gamma_x\geq (4\pi)^{1-p}\bigg|\int_{{\partial\mathbb B_1(0)}}f(x)d\Gamma_x\bigg|^p, \;\;\;\;(where\;1\leq p< \infty).
\end{eqnarray}
(25)
From (25), for \(2\leq p< \infty,\) we have
\begin{eqnarray}\label{eq4.21}
\mathrm{L}
&\geq&(4\pi)^{1-\frac{p}{2}}\int_{{0}}^1\bigg(\sum\limits_{n=0}^{\infty}|{a_n}|^2\frac{r^{2n-2}}{(1-r^2)^{\frac{8s+p}{2p}}}\|{\Phi_n}(\phi_1,\phi_2)\|^2_{L_2(\partial\mathbb B_1)} \bigg)^{\frac{{p}}{2}} r^2{(1-r^2)^{\frac{3\alpha q}{2}+3s}} dr\nonumber\\
&\geq&(4\pi)^{1-\frac{p}{2}}\int_{{0}}^1\bigg(\sum\limits_{n=0}^{\infty}|{a_n}|^2{r^{2n-2}}\|{\Phi_n}(\phi_1,\phi_2)\|^2_{L_2(\partial\mathbb B_1)} \bigg)^{\frac{{p}}{2}} r^3{(1-r^2)^{\frac{3\alpha q}{2}+s-\frac{p}{4}}} dr
\end{eqnarray}
(26)
From Corollary 11, we have
$$\|{\Phi_n}(\phi_1,\phi_2)\|^2_{L_2(\partial\mathbb B_1)}=\frac{\|-\frac{1}{2}{\overline D}\mathrm{H}_{n,\alpha}\|_{L_2(\partial\mathbb B_1)}}{\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}}\geq\lambda n^{\frac{2+3p}{2p}}\geq\lambda n^{\frac{3}{2}}.$$
Then, from above we have
\begin{eqnarray}\label{eq4.22}
\mathrm{L}&\geq&(4\pi)^{1-\frac{p}{2}}\lambda \int_{{0}}^1\bigg(\sum\limits_{n=0}^{\infty}n^{\frac{3}{2}}|{a_n}|^2{r^{2n-2}} \bigg)^{\frac{{p}}{2}} r^3{(1-r^2)^{\frac{3\alpha q}{2}+s-\frac{p}{4}}} dr\nonumber\\
&=&\lambda_1 \int_{{0}}^1\bigg(\sum\limits_{n=0}^{\infty}n^{\frac{3}{2}}|{a_n}|^2{r^{2n-2}} \bigg)^{\frac{{p}}{2}} r^3{(1-r^2)^{\frac{3\alpha q}{2}+s-\frac{p}{4}}} dr\nonumber\\
&=&\frac{\lambda_1}{2} \int_{{0}}^1\bigg(\sum\limits_{n=0}^{\infty}n^{\frac{3}{2}}|{a_n}|^2{\xi^{n-1}} \bigg)^{\frac{{p}}{2}} \xi{(1-\xi)^{\frac{3\alpha q}{2}+s-\frac{p}{4}}} d\xi\nonumber\\
&\geq&\lambda_3 \int_{{0}}^1\bigg(\sum\limits_{n=0}^{\infty}n^{\frac{3}{2}}|{a_n}|^2{\xi^{n-1}} \bigg)^{\frac{{p}}{2}}{(1-\xi)^{\frac{3\alpha q}{2}+s-\frac{p}{4}}} d\xi,
\end{eqnarray}
(27)
where \(\lambda_j,\; j={1,2,3},\) are constants do not depending on \(n.\)
Now, by applying Theorem 8 in equation (27), we deduced that
\begin{eqnarray}\label{eq4.23}
\|f\|_{F^{\alpha}_{G}(p,q,s)}
\geq \mathrm{L}\geq\frac{\lambda_3}{k}\sum\limits_{n=0}^{\infty}2^{{-n}(\frac{3\alpha q}{2}+s-\frac{p}{4}+1)}\big({\sum_{k\in I_n}k^{\frac{3}{2}}|a_k|^2}\big)^{\frac{p}{2}},
\end{eqnarray}
(28)
where
$${\sum_{k\in I_n}k^{\frac{3}{2}}|a_k|^2}> \bigg(2^n\bigg)^{\frac{3}{2}}\bigg({\sum_{k\in I_n}|a_k|^2}\bigg)^{{\frac{{p}}{2}}}.$$
Then,
$$
\|f\|_{F^{\alpha}_{G}(p,q,s)}\geq \mathrm{L}\geq C\sum\limits_{n=0}^{\infty}2^{-n(\frac{3\alpha q}{2}+s-p+1)}\big(\sum_{k\in I_n}|a_k|^2\big)^{\frac{p}{2}},
$$
(29)
From [
21], we have
$$\sum\limits_{n=0}^{N}a_n^p\leq \bigg(\sum\limits_{n=0}^{N}a_n^p\bigg)^p\leq N^{p-1}\sum\limits_{n=0}^{N}a_n^p.$$
Then, we have
\begin{eqnarray}\label{eq4.24}
\|f\|_{F^{\alpha}_{G}(p,q,s)}
\geq \mathrm{L}\geq C_1\sum\limits_{n=0}^{\infty}2^{{-n}(\frac{3\alpha q}{2}+s-{p}+1)}\big({\sum_{k\in I_n}|a_k|}\big)^{{p}},
\end{eqnarray}
(30)
where \(C_1\) be a constants which do not depend on \(n.\) Then,
\begin{eqnarray}
\sum\limits_{n=0}^{ \infty}2^{{-n}(\frac{3 \alpha q}{2}+s-{p}+1)}\bigg({\sum_{k\in I_n}|a_k|}\bigg)^{p}< \infty.
\end{eqnarray}
(31)
This completes the proof of theorem.
Theorem 13.
Let \(\alpha\geq 1,\) \(2\leq p< \infty,\) \(-2< q0,\) then we have
\begin{eqnarray}\label{eq4.25}
f(x)=\bigg(\sum\limits_{n=0}^{\infty}\frac{\mathrm{H}_{n,\alpha}}{(1-|x|^2)^{\frac{8s+p}{4p}}\|\mathrm{H}_{n,\alpha}\|_{L_p(\partial\mathbb B_1)}} a_n\bigg) \in F^{\alpha}_{G}(p,q,s),
\end{eqnarray}
(32)
if and only if,
\begin{eqnarray}\label{eq4.26}
\sum\limits_{n=0}^{\infty}2^{-n(\frac{3}{2}\alpha q +s-p+1)}\bigl({\sum_{k\in I_n}|a_k|}\bigr)^p< \infty.
\end{eqnarray}
(33)
Proof.
This theorem can be proved directly from Theorem 9 and Theorem 13.
4. Conclusion
We have introduce a new class of hyperholomorphic functions, which is also called \(F^{\alpha}_{G}(p,q,s)\) spaces. For this class,
we give some characterizations of the hyperholomorphic \(F^{\alpha}_{G}(p,q,s)\) functions by the coefficients of certain lacunary series expansions in quaternion analysis.
Acknowledgments
The authors wishes to express his profound gratitude to the reviewers for their useful comments on the manuscript.
Author Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Competing Interests
The author(s) do not have any competing interests in the manuscript.
