Integral expressions of solutions to higher order $\lambda$-weighted Dirac equations valued in the parameter dependent Clifford algebra

1.   Introduction

In 1878, Clifford algebra was defined in ^[1]. In 1982, Brackx et al. ^[2] generalized some results of the complex analysis to Clifford analysis. Malonek and Ren ^[3] studied the Almansi-type decomposition theorems for the $k$-order monogenic functions and $k$-order $\lambda$-weighted monogenic functions in 2002. In the unweighted case, the star-like condition of the domain is needed. This fact accounts for the greater generality of the decomposition in the weighted case, which indeed holds in any domain. When $k = 1$, the origin of the notion of $\lambda$-weighted monogenic functions is given. In 2017, García et al. ^[4] studied an integral representation for the solution to the sandwich Dirac equation in Clifford analysis. Yang et al. ^[5] obtained the Cauchy theorem for the solution to the $k$-order Dirac equation with $\alpha$-weight in 2018, where $k > 0$ is an integer and $\alpha$ is a nonzero real number. In 2020, Blaya et al. ^[6] gave the integral representation for the solution of the bilateral higher-order Dirac equation and proved some properties for Cauchy and Teodorescu transforms. In 2022, Peláez et al. ^[7] took the sum of the left Dirac operator multiplied by $\alpha$ and the right Dirac operator multiplied by $\beta$ as a new operator, and studied the integral representation of solutions to higher-order new operators, where $\alpha, \beta$ are real numbers. In 2023, Dinh ^[8] introduced $(\alpha, \beta)$-monogenic functions and isotonic functions, where $\alpha, \beta$ are real numbers and $\alpha\neq\beta$; they gave the integral representation formulae of these functions respectively by using the new proof method and proved the series representation of polynomial Dirac equations. In 2024, Gao et al. ^[9] got an integral representation for the solution of the bilateral higher order Dirac equation with $\alpha$-weight, where $\alpha$ is a nonzero real number. Liu et al. ^[10] investigated some Riemann-Hilbert boundary value problems for perturbed Dirac operators in the Clifford algebra $Cl(V_{3, 3})$. D. A. Santiesteban et al.^[11] examined well-posed boundary value problems for second-order elliptic systems of partial differential equations in bounded regular domains of Euclidean spaces.

In 2008, Clifford algebras depending on parameters emerged as an extension of the classical Clifford algebra. Its applications in partial differential equations were introduced by Tutschke and Vanegas ^[12]. In 2012, Di Teodoro et al. ^[13] studied solutions for the first order homogeneous meta-Dirac equation and then gave a solution of the inhomogeneous equation by using Fubini$^{\prime}$s theorem. In 2013, the integral representations for the meta-Dirac operator of $n$-order and its conjugate operators of $n$-order are derived by Balderrama et al. ^[14]. In 2014, some achievements of hypercomplex analysis were expounded and some of its development trends were presented in reference ^[15]. Ariza et al. ^[16] gave the integral formulae to solutions for second order elliptic Dirac equation in 2015. In 2017, Ariza García et al. ^[17] obtained the correlation between first-order differential operators and $q$-Dirac operators, with the aim of studying initial value problems, where $q$ is a $n$-dimensional vector. In 2021, Cuong et al. ^[18] studied the integral expression of monogenic functions in the Clifford algebra depending on three parameters and solved two boundary value problems related to this function.

Based on the above work, we have conducted certain work with the aim of extending the results from the classical Clifford algebra to the framework of parameter dependent Clifford algebra. In Section 2, we investigate some important properties of functions valued in this Clifford algebra. In Section 3, integral representations for $\mathfrak{p}$-order $\lambda$-weighted monogenic functions and right $\mathfrak{q}$-order $\lambda$-weighted monogenic functions are derived. Furthermore, in Section 4, we present an integral representation for $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted monogenic functions. Finally, Section 5 contains the conclusion and discussion of this paper. This paper mainly generalizes some results of references ^[5,9].

2.   Preliminaries

In this section, we present some basic results on the parameter dependent Clifford analysis, meanwhile, we prove some important properties of some functions valued in the parameter dependent Clifford algebra.

2.1.   Some basic results on the parameter dependent Clifford analysis

Suppose that $\alpha_{j}$, $\gamma_{ij} = \gamma_{ji}$ are nonnegative real numbers for $i, j = 1, 2, \ldots, n, i\neq j$, the set of base element is $\{e_{0} = 1, e_{1}, \ldots, e_{n}\}$, and the base element satisfies the following multiplication rule

From this, we obtain a parameter dependent Clifford algebra $\mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})$ which is generated by the structural relationship (2.1). Every element of the algebra is of the form $c = \sum\limits_{A_{1}}c_{A_{1}}e_{A_{1}}$, $c_{A_{1}}\in\mathbf{R}$, where $A_{1} : = \{j_{1}, \ldots, j_{k}\}\subseteq\{1, \ldots, n\}$, $j_{1} < j_{2} < \cdots < j_{k}$, $e_{A_{1}} = e_{j_{1}}\cdots e_{j_{k}}$, and $e_{0} = e_{\varnothing} = 1$. As indices we use the elements $A_{1}$ of the set containing the ordered subsets of $\{1, 2, \ldots, n\}$, with the empty subset corresponding to the index $0$. The set $A_{1}$ runs over all the possible ordered sets $A_{1} = \{1 \leq j_{1} < \ldots < j_{k} \leq n\}$, or $A_{1} = \varnothing$. The dimension of this algebra is $2^{n}$.

Let $\mathbf{N}^{\ast}$ be the set of positive integers. If $1\leq j\leq n$ and $j\in \mathbf{N}^{\ast}$, the base element satisfies the involution $\overline{e_{j}} = -e_{j}$. If $e_{A} = e_{h_{1}\cdots h_{r}} = e_{h_{1}}\cdots e_{h_{r}}$, then $\overline{e_{A}} = \overline{e_{h_{r}}}\cdots \overline{e_{h_{1}}} = (-1)^{r}e_{h_{r}}\cdots e_{h_{1}}$. For any $\xi = \sum\limits_{A_{1}}\xi_{A_{1}} e_{A_{1}}\in \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})$, we define $\overline{\xi} = \sum\limits_{A_{1}}\xi_{A_{1}}\overline{e_{A_{1}}}$, $|\xi|^{2} = \sum\limits_{A_{1}}\xi_{A_{1}}^{2}$, where $\xi_{A_{1}}\in\mathbf{R}$.

The Euclidean Clifford algebra $\mathcal{B}_{n}(2, 1, 0)$ is one of the special cases of $\mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})$.

The function $f: \Omega\rightarrow \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})$ is denoted by $f(x) = \sum\limits\limits_A f_{A_{1}}(x)e_{A_{1}}$, where $f_{A_{1}}(x)$ is a real-valued function and $\Omega$ is an open connected bounded domain in $\mathbf{R}^{n}$. $f$ is a $r$-times continuously differentiable function, which means $f_{A_{1}}$ is a $r$-times continuously differentiable function, where $r\in \mathbf{N}^{\ast}$. The set consisting of the $r$-times continuously differentiable function is denoted by $\mathbf{F}^{r}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$.

When $f\in \mathbf{F}^{1}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, Dirac operators and its conjugate operators acting on function $f$ are defined respectively as follows:

After a direct calculation, we have

the corresponding quadratic form is

which has a coefficient matrix

Denote

See references ^[19,20]. By using the Sylvester$^{\prime}$s criterion, (2.2) is a positive definite quadratic form if and only if the determinant of each $B_{j}$ is a positive number for all $j = 1, 2, \ldots, n$, i.e.,

In this situation, $D_{x}\overline{D_{x}} = \overline{D_{x}}D_{x}$ becomes an elliptic Dirac operator, so we denote $D_{x}\overline{D_{x}} = \overline{D_{x}}D_{x}$ by $\widetilde{\Delta}_{n}$.

Suppose that (2.4) holds in this paper, then the inverse matrix of matrix $B$ exists and can be represented by

where $a_{ij} = a_{ji}$, $i, j = 1, 2, \ldots, n$.

See reference ^[12]. For two points $x = (x_{1}, \cdots, x_{n})$ and $\zeta = (\zeta_{1}, \cdots, \zeta_{n})$ in $\mathbf{R}^{n}$, $x\neq\zeta$, the representation of the non-Euclidean distance $\rho$ as follows:

the representation of the Euclidean distance is $\iota = |x-\zeta|$.

See reference ^[14]. Suppose that for some $Y\in \mathbf{R}^{n}$ and $Y$ satisfying $|Y| = 1$, we denote $x-\xi = \iota Y$, then the infimum of $\rho(Y, 0)$ for all $Y$ is positive, i.e., $\rho^{2}(Y, 0)\geq c_{0} > 0$, where $c_{0}$ is a constant, so $\rho^{2}(x, \xi)\geq c_{0}\iota^{2}$.

For $f\in\mathbf{ F}^{1}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, the first-order $\lambda$-weighted Dirac operators acting on the function $f$ are defined as follows:

where $\rho_{x} = \Big(\sum_{i, j = 1}^{n}a_{ij}x_{i}x_{j}\Big)^{\frac{1}{2}}$, $H(x) = \sum_{i, j = 1}^{n}\overline{e_{i}}a_{ij}x_{j}$, and $\lambda$ is a fixed nonzero real number.

Definition 2.1. ^[12] Suppose $f\in \mathbf{F}^{1}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, then a solution $f$ of the Dirac equation $D_{x}f(x) = 0$ ($f(x)D_{x} = 0$) is called a left (right) monogenic function.

See reference ^[12]. We know that $\rho_{x}^{-n}H(x)$ is not only a left monogenic function but also a right monogenic function.

Definition 2.2. Suppose $f\in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, $\mathfrak{p}, \mathfrak{q}$ are positive integers.

(i) A solution $f$ of the $\mathfrak{p}$-order $\lambda$-weighted Dirac equation $(D_{x}^{\lambda})^{\mathfrak{p}}f(x) = 0$ is called a left $\mathfrak{p}$-order $\lambda$-weighted monogenic function, where $(D_{x}^{\lambda})^{\mathfrak{p}} = D_{x}^{\lambda}\circ \cdots \circ D_{x}^{\lambda}$ ($\mathfrak{p}$-times), $\circ$ is a composite operation of operators.

(ii) A solution $f$ of the right $\mathfrak{q}$-order $\lambda$-weighted equation $f(x)(D_{x}^{\lambda})^{\mathfrak{q}} = 0$ is called a right $\mathfrak{q}$-order $\lambda$-weighted monogenic function, where $(D_{x}^{\lambda})^{\mathfrak{q}} = D_{x}^{\lambda}\circ \cdots \circ D_{x}^{\lambda}$ ($\mathfrak{q}$-times).

Remark 2.1. (i) When $\mathfrak{p} = 1 (\mathfrak{q} = 1)$ in Definition 2.2, a solution $f$ of the $\lambda$-weighted Dirac equation $D_{x}^{\lambda}f(x) = 0\, (f(x)D_{x}^{\lambda} = 0)$ is called a left (right) $\lambda$-weighted monogenic function.

(ii) A left $\mathfrak{p}$-order $\lambda$-weighted monogenic function can be called a $\mathfrak{p}$-order $\lambda$-weighted monogenic function for short. A left $\lambda$-weighted monogenic function can be called a $\lambda$-weighted monogenic function for short.

Definition 2.3. Suppose $f\in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, $\mathfrak{p}$ and $\mathfrak{q}$ are positive integers, then the solution $f$ of equation $\Big((D_{x}^{\lambda})^{\mathfrak{p}}f(x)\Big)(D_{x}^{\lambda})^{\mathfrak{q}} = 0$ is called a $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted monogenic function.

If $f$ is a left $\mathfrak{p}$-order $\lambda$-weighted monogenic function, then $f$ is a $(\mathfrak{p}+\mathfrak{q})$-order monogenic $\lambda$-weighted function.

Remark 2.2. $\rho_{x}^{-n}H(x)$, $\rho_{x}^{-n+(\mathfrak{p}-1)\lambda}H(x)$, and $\rho_{x}^{-n+(\mathfrak{p}+\mathfrak{q}-1)\lambda}H(x)$ are $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted monogenic functions, where $\lambda$ is a fixed nonzero real number.

For $x\in\partial\Omega$, its outer unit normal vector is $N(x) = (N(x_{1}), \ldots, N(x_{n})) = (N_{1}, \ldots, N_{n})$, $d\sigma_{x} = \sum\limits_{i = 1}^{n}N_{i}e_{i}d\mu$ is the Clifford-algebra-valued measure element of $\partial\Omega$, $d\mu$ represents the scalar measure element of $\partial\Omega$, and $\partial\Omega$ is a sufficiently smooth boundary.

Lemma 2.1. ^[12] Suppose $f, g\in \mathbf{F}^{1}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})),$ then

where $dx = dx_{1}\wedge dx_{2}\wedge \cdots \wedge dx_{n}.$

Similar to the proof of the theorem in reference ^[21], we can prove that Lemma 2.2 holds.

Lemma 2.2. Suppose $f, \, g\in \mathbf{F}^{1}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, then

Proof. We suppose that $f(x) = \sum\limits_{A_{1}}f_{A_{1}}(x)e_{A_{1}}$, $g(x) = \sum\limits_{A_{2}}g_{A_{2}}(x)e_{A_{2}}$, where $f_{A_{1}}(x)$ and $g_{A_{2}}(x)$ are real-valued functions, then

Similarly, we can prove the other equality. □

2.2.   Some important properties of some functions valued in $\mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})$

Suppose that $M_{s}^{\lambda}(x) = E_{s}(x)\rho_{x}^{-\lambda}H(x)$, where $E_{s}(x) = \frac{C_{s}}{\rho_{x}^{n-s\lambda}}$, $C_{s} = \frac{1}{\omega_{n}\lambda^{s-1}(s-1)!}$, $s\in\mathbf{N^{\ast}}$, $\omega_{n}$ represents the Euclidean surface measure of the unit sphere.

Proposition 2.1. When $s > 1$, we have

Proof. Since $AB = E$ and $E$ is the identity matrix, we obtain that for $m, k = 1, 2, \ldots, n$,

then

therefore,

Also,

Consequently,

By $\overline{H(x)} = -H(x)$, we can conclude that $H(x)\overline{H(x)} = -H^{2}(x) = \rho_{x}^{2}$.

By $AB = E$, we have

Similarly, we can prove that $H(x)D = n$.

By equalities $\overline{H(x)}H(x) = \rho_{x}^{2}$ and $D_{x}H(x) = n$, we can conclude that

Similarly, we have $M_{s}^{\lambda}(x)D_{x} = E_{s-1}(x)$. □

Proposition 2.2. Let $f \in \mathbf{F}^{k}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, $k\in\mathbf{N^{\ast}}$, $s = 1, 2, \ldots, k$.

$(1)$ Suppose that $f$ is a solution of the Dirac equation $D_{x}f = 0$, then

$(2)$ Suppose that $f$ is a solution of the Dirac equation $fD_{x} = 0$, then

Proof. (1) When $s = 1$, by using the equality $H(x)\overline{H(x)} = \rho_{x}^{2}$, it is easy to deduce that

and

We suppose that $(D_{x}^{\lambda})^{s-1}\Big(\rho_{x}^{k\lambda}f(x)\Big) = \frac{k!}{(k-s+1)!}\lambda^{s-1}\rho_{x}^{(k-s+1)\lambda}f(x)$ holds, then

According to the mathematical induction, we get the conclusion.

Similarly, we can prove that $(2)$ holds □

Proposition 2.3. Suppose $1\leq s\leq \mathfrak{p}$, $1\leq t\leq \mathfrak{q}$, and $1\leq k\leq \mathfrak{p}+\mathfrak{q}$, where $s, t, k, \mathfrak{p}, \mathfrak{q } \in\mathbf{N^{\ast}}$.

$(1)$ Let $f\in \mathbf{F}^{\mathfrak{p}}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$ be a solution of the equation $D_{x}f = 0$. Then, $\rho_{x}^{(\mathfrak{p}-s)\lambda}f(x)$ is a solution of the $\mathfrak{p}$-order $\lambda$-weighted Dirac equation $(D_{x}^{\lambda})^{\mathfrak{p}}f(x) = 0$.

$(2)$ Let $f\in \mathbf{F}^{\mathfrak{q}}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$ be a solution of the right equation $fD_{x} = 0$. Then, $\rho_{x}^{(\mathfrak{q}-t)\lambda}f(x)$ is a solution of the right $\mathfrak{q}$-order $\lambda$-weighted Dirac equation $f(x)(D_{x}^{\lambda})^{\mathfrak{q}} = 0$.

$(3)$ Let $f\in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}(\Omega, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$ be a solution of the equation system $D_{x}f = 0$ and $fD_{x} = 0$. Then, $\rho_{x}^{(\mathfrak{p}+\mathfrak{q}-k)\lambda}f(x)$ is a solution of the $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted Dirac equation $\Big((D_{x}^{\lambda})^{\mathfrak{p}}f(x)\Big)(D_{x}^{\lambda})^{\mathfrak{q}} = 0$.

Proof. By Proposition 2.2, $(1)$ and $(2)$ hold.

(3) (i) When $\mathfrak{q}\leq k\leq \mathfrak{p}+\mathfrak{q}$, i.e., $1\leq k-\mathfrak{q}\leq \mathfrak{p}$, by (1) in Proposition 2.3, we conclude that $\rho_{x}^{(\mathfrak{p}-(k-\mathfrak{q}))\lambda}f(x)$ satisfies equation $(D_{x}^{\lambda})^{\mathfrak{p}}\Big(\rho_{x}^{(\mathfrak{p}-(k-\mathfrak{q}))\lambda}f(x)\Big) = 0$. Therefore, (3) is clearly valid.

(ii) When $1 < k\leq \mathfrak{q}$, as $f$ satisfies condition $D_{x}f = 0$ and based on (1) in Proposition 2.2, we can deduce that

As $f$ satisfies condition $f D_{x} = 0$ and based on (2) in Proposition 2.3, we obtain

therefore, (3) is established. □

Theorem 2.1. Let $1\leq s\leq \mathfrak{p}$, $1\leq t\leq \mathfrak{q}$, $1\leq k\leq \mathfrak{p}+\mathfrak{q}$, where $s, t, k, \mathfrak{p}, \mathfrak{q} \in\mathbf{N^{\ast}}$.

$(1) \; E_{\mathfrak{p}}(x)\rho_{x}^{-s\lambda}H(x)$ is a solution of the $\mathfrak{p}$-order $\lambda$-weighted Dirac equation $(D_{x}^{\lambda})^{\mathfrak{p}}f(x) = 0$.

$(2) \; E_{\mathfrak{q}}(x)\rho_{x}^{-t\lambda}H(x)$ is a solution of the right $\mathfrak{q}$-order $\lambda$-weighted Dirac equation $f(x)(D_{x}^{\lambda})^{\mathfrak{q}} = 0$.

$(3) \; E_{\mathfrak{p}+\mathfrak{q}}(x)\rho_{x}^{-k\lambda}H(x)$ is a solution of the $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted Dirac equation $\Big((D_{x}^{\lambda})^{\mathfrak{p}}f(x)\Big)(D_{x}^{\lambda})^{\mathfrak{q}}$$=$$0$.

Proof. By Proposition 2.3, $(1)$ and $(2)$ hold.

(3) It is obvious that

By Proposition 2.3 and the equality $D_{x}\big(\rho_{x}^{-n}H(x)\big) = \big(\rho_{x}^{-n}H(x)\big)D_{x} = 0$, we conclude that (3) in Theorem 2.1 is established. □

3.   Integral representations for the solutions of the $\mathfrak{p}$-order $\lambda$-weighted Dirac equation and right $\mathfrak{q}$-order $\lambda$-weighted Dirac equation

In this section, we prove two Cauchy-Pompeiu integral formulae for functions valued in $\mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})$, and obtain the Cauchy integral formulae for the null solution to higher order $\lambda$-weighted Dirac operators as their corollary, respectively.

In this paper, we denote $\{x|y_{0} = x + x_{0} \in \Omega\}$ as $\Omega_{x_{0}}^{\ast}$, for any $x_{0} \in \Omega$.

Theorem 3.1. Let $\mathfrak{p}, \mathfrak{q}\in\mathbf{N^{\ast}}$, $s = 0, 1, \ldots, \mathfrak{p}$; $r = 0, 1, \ldots, \mathfrak{q}$.

$(1)$ If $f \in \mathbf{F}^{\mathfrak{p}}(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, then for any $x_{0} \in \overline{\Omega}$, when $0 < \lambda < \frac{1}{\mathfrak{p}}$, $(D_{x}^{\lambda})^{s}f(y_{0})$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$.

$(2)$ If $f \in \mathbf{F}^{\mathfrak{q}}(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, then for any $x_{0} \in \overline{\Omega}$, when $0 < \lambda < \frac{1}{\mathfrak{q}}$, $f(y_{0})(D_{x}^{\lambda})^{r}$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$.

Proof. (1) When $s = 0$, as $f \in \mathbf{F}^{\mathfrak{p}}(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, $(D_{x}^{\lambda})^{0}f(y_{0})$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$.

When $s = 1, 2, \ldots, \mathfrak{p}$, we denote $H(x)f_{s}(x)$ by $g_{s}(x)$, and let

As $f \in \mathbf{F}^{\mathfrak{p}}(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, $f_{1}, f_{2}, ..., f_{\mathfrak{p}}$ are bounded functions in $\overline{\Omega_{x_{0}}^{\ast}}$.

When $s = 1$, we have $D_{x}^{\lambda}f(y_{0}) = \rho_{x}^{-\lambda}H(x)\Big(D_{x}f(y_{0})\Big) = \rho_{x}^{-\lambda}g_{1}(x).$

We suppose that $t < \mathfrak{p}$, $t\in\mathbf{N^{\ast}}$, and $(D_{x}^{\lambda})^{t}f(y_{0}) = \rho_{x}^{-t\lambda}g_{t}(x),$ then

According to the mathematical induction, we get

So for any $x_{0}\in\overline{\Omega}$, if $0 < \lambda < \frac{1}{s}$, then we conclude that $(D_{x}^{\lambda})^{s}f(y_{0})$ is bounded in $\overline{\Omega_{x_{0}}^{\ast}}$.

Hence, for any $x_{0}\in\overline{\Omega}$, when $\lambda\in\bigcap_{s = 1}^{\mathfrak{p}}\bigg(0, \frac{1}{s}\bigg) = \bigg(0, \frac{1}{\mathfrak{p}}\bigg)$, we conclude that $(D_{x}^{\lambda})^{s}f(y_{0})$ is bounded in $\overline{\Omega_{x_{0}}^{\ast}}$, where $s = 1, 2, \ldots, \mathfrak{p}$.

Similarly, we can prove that $(2)$ holds. □

Theorem 3.2. Suppose that $f\in \mathbf{F}^{\mathfrak{p}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big),$ $\mathfrak{p}\leq n, \; 0 < \lambda < \frac{1}{\mathfrak{p}}, \; \mathfrak{p}\in \mathbf{N^{\ast}}$, for arbitrary $x_{0}\in \Omega,$ then we have

where $c_{1}(\alpha_{j}, \gamma_{ij})$ is a Clifford constant. If $c_{1}(\alpha_{j}, \gamma_{ij})$ has a single inverse element, this formula is called the Cauchy-Pompeiu integral formula of the function $f\in \mathbf{F}^{\mathfrak{p}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big)$.

Proof. For any $x_{0}\in\Omega$, we have $0+x_{0}\in\Omega$, so $0\in \Omega_{x_{0}}^{\ast}$.

We can choose an arbitrarily small positive number $\delta$, and make a small ball $B_{\delta} = \{x:|x| < \delta\}$ such that $\overline{B_{\delta}}$ is a subset of $\Omega_{x_{0}}^{\ast}$.

For any $s = 2, 3, \ldots, \mathfrak{p}$, by Lemma 2.1, Proposition 2.1, and Theorem 3.1, we have

From Theorem 3.1, it can be derived that $(D_{x}^{\lambda})^{s-1}f(y_{0})$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$, then $|(D_{x}^{\lambda})^{s-1}f(y_{0})|\leq M_{1}$.

For $x\in\partial B_{\delta}$, suppose that $x = \delta X$, where $X\in\partial B_{1} = \{X:|X| = 1\}$, $d\mu = \delta^{n-1}d\mu_{1}$, $d\mu_{1}$ is the surface element of the unit sphere $\partial B_{1}$, and since $\rho_{x}^{2}\geq c_{0}\delta^{2}$, we can obtain

where $M_{i} > 0$ are constants, $i = 1, 2, 3$, then we can conclude that

Hence,

where $s = 2, 3, \ldots, \mathfrak{p}$.

For $s = 1$, by Lemma 2.1 and the equality $\Big(\rho_{x}^{-n}H(x)\Big)D = 0$, we have

We can calculate that

we can conclude that $c_{1}(\alpha_{j}, \gamma_{ij})$ is a Clifford constant, and $c_{1}(\alpha_{j}, \gamma_{ij})$ does not depend on $\delta$ but only on the values of the parameters $\alpha_{j}$ and $\gamma_{ij}$; see Remark 2.6 in reference ^[14].

Hence,

therefore,

By Equalities (3.2) and (3.4), we have

Consequently, we prove that the conclusion holds. □

Remark 3.1. When $c_{1}(\alpha_{j}, \gamma_{ij})$ is not required to be invertible, the value of $f(x_{0})$ is not uniquely determined by the integral transform.

Corollary 3.1. Suppose that $f\in \mathbf{F}^{\mathfrak{p}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big)$ is a solution of the equation $(D_{x}^{\lambda})^{\mathfrak{p}}f(y_{0}) = 0$ in $\overline{\Omega_{x_{0}}^{\ast}}$, $\mathfrak{p}\leq n, \; 0 < \lambda < \frac{1}{\mathfrak{p}}$, $\mathfrak{p}\in \mathbf{N^{\ast}}$, for arbitrary $x_{0}\in \Omega$, then we have

where $c_{1}(\alpha_{j}, \gamma_{ij})$ is a Clifford constant. If $c_{1}(\alpha_{j}, \gamma_{ij})$ has a single inverse element, this formula is called the Cauchy integral formula of the $\mathfrak{p}$-order $\lambda$-weighted monogenic function.

Theorem 3.3. Suppose that $f\in \mathbf{F}^{\mathfrak{q}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big),$ $\mathfrak{q}\leq n, \; 0 < \lambda < \frac{1}{\mathfrak{q}}, \; \mathfrak{q}\in \mathbf{N^{\ast}}$, for arbitrary $x_{0}\in \Omega$, then we have

where $c_{1}(\alpha_{j}, \gamma_{ij})$ is a Clifford constant. If $c_{1}(\alpha_{j}, \gamma_{ij})$ has a single inverse element, this formula is called the Cauchy-Pompeiu integral formula of the function $f\in \mathbf{F}^{\mathfrak{q}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big)$.

Proof. Similar to the proof of Theorem 3.2, we can prove Theorem 3.3. □

Corollary 3.2. Suppose that $f\in \mathbf{F}^{\mathfrak{q}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big)$ is a solution of right $\mathfrak{q}$-order $\lambda$-weighted Dirac equation $f(y_{0})(D_{x}^{\lambda})^{\mathfrak{q}} = 0$ in $\overline{\Omega_{x_{0}}^{\ast}}$, $\mathfrak{q} \leq n, \; 0 < \lambda < \frac{1}{\mathfrak{q}}$, $\mathfrak{q}\in \mathbf{N^{\ast}}$, for arbitrary $x_{0}\in \Omega$, then we have

where $c_{1}(\alpha_{j}, \gamma_{ij})$ is a Clifford constant. If $c_{1}(\alpha_{j}, \gamma_{ij})$ has a single inverse element, this formula is called the Cauchy integral formula of the right $\mathfrak{q}$-order $\lambda$-weighted monogenic function.

4.   The integral representation for the $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted monogenic function

In this section, we obtain the integral representation for the $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted monogenic function.

Theorem 4.1. Let $\mathfrak{p}, \mathfrak{q}\in \mathbf{N}^{\ast}$, $s = 0, 1, \ldots, \mathfrak{p}$; $r = 0, 1, \ldots, \mathfrak{q}$.

Suppose that $f \in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, when $0 < \lambda < \frac{1}{\mathfrak{p}+\mathfrak{q}}$, then for arbitrary $x_{0} \in \overline{\Omega}$, $\Big((D_{x}^{\lambda})^{s}f(y_{0})\Big)(D_{x}^{\lambda})^{r}$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$.

Proof. (i) For arbitrary $x_{0}\in\overline{\Omega}$, when $s = 0$ and $r = 0, 1, \ldots, \mathfrak{q}$, from Theorem 3.1 and the inequality $0 < \lambda < \frac{1}{\mathfrak{p}+\mathfrak{q}} < \frac{1}{\mathfrak{q}}$, it follows that $f(y_{0})(D_{x}^{\lambda})^{r}$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$.

(ii) When $s = 1, 2, \ldots, \mathfrak{p}$, and $r = 1, \ldots, \mathfrak{q}$, we denote $H(x) f_{s, 0}(x)$ by $g_{s, 0}(x)$, we denote $f_{s, r}(x)H(x)$ by $g_{s, r}^{\prime}(x)$, and let

As $f \in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}))$, $f_{1, 0}, \ldots, f_{s, 0}, f_{s, 1}, \ldots, f_{s, r}$ are bounded functions in $\overline{\Omega_{x_{0}}^{\ast}}$.

($a$) When $s = 1$ and $r = 0$, by directly calculating, we can obtain

When $s = 2, \ldots, \mathfrak{p}$ and $r = 0$, we suppose that $s = t$, where $t < \mathfrak{p}, \; t\in \mathbf{N^{\ast}}$, and

then

According to the mathematical induction, we have

For any $x_{0}\in\overline{\Omega}$, when $0 < \lambda < \frac{1}{\mathfrak{p}+\mathfrak{q}}$, we conclude that $0 < \lambda\leq\frac{1}{s}$, so $(D^{\lambda})^{s}f(y_{0})$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$.

($b$) When $s = 1, \ldots, \mathfrak{p}$ and $r = 1$, by Equality (4.1), we have

When $s = 1, \ldots, \mathfrak{p}$ and $r = 2, \ldots, \mathfrak{q}$, we suppose that $r = l$, where $l < \mathfrak{q}, \; l\in \mathbf{N^{\ast}}$, and

then

According to the mathematical induction, we have

For arbitrary $x_{0}\in\overline{\Omega}$, when $0 < \lambda < \frac{1}{\mathfrak{p}+\mathfrak{q}}$, we conclude that $0 < \lambda < \frac{1}{s+r}$, so $\Big((D_{x}^{\lambda})^{s}f(y_{0})\Big)(D_{x}^{\lambda})^{r}$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$, where $s = 1, 2, \ldots, \mathfrak{p}$; $r = 1, \ldots, \mathfrak{q}.$

From the above, for any $x_{0} \in \overline{\Omega}$, when $0 < \lambda < \frac{1}{\mathfrak{p}+\mathfrak{q}}$, we conclude that $0 < \lambda < \frac{1}{s+r}$, it can be concluded that $\Big((D_{x}^{\lambda})^{s}f(y_{0})\Big)(D_{x}^{\lambda})^{r}$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$, where $s = 0, 1, \ldots, \mathfrak{p}$; $r = 0, 1, \ldots, \mathfrak{q}$. □

Theorem 4.2. Suppose that $f\in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big), \; \mathfrak{p}+\mathfrak{q} \leq n, \; 0 < \lambda < \frac{1}{\mathfrak{p}+\mathfrak{q}}, \; \mathfrak{p}, \mathfrak{q}\in \mathbf{N^{\ast}}$, for arbitrary $x_{0}\in \Omega$, then we have

where $c_{1}(\alpha_{j}, \gamma_{ij})$ is a Clifford constant. If $c_{1}(\alpha_{j}, \gamma_{ij})$ has a single inverse element, this formula is called the Cauchy-Pompeiu integral formula of the function $f\in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big)$.

Proof. We can conclude that Theorem 4.2 holds by applying Theorem 3.2, once we prove that the following equality holds, that is,

For any $x_{0}\in\Omega$, we know that $0+x_{0}\in\Omega$, so $0\in\Omega_{x_{0}}^{^{\ast}}$. We can choose an arbitrarily small positive number $\delta$ and make a small ball $B_{\delta} = \{x:|x| < \delta\}$ such that $\overline{B_{\delta}}$ is a subset of $\Omega_{x_{0}}^{\ast}$.

When $r = 1, 2, 3, \ldots, \mathfrak{q}$, by Lemma 2.1 and Proposition 2.1, we have

By Theorem 4.1, it follows that $\Big((D_{x}^{\lambda})^{\mathfrak{p}}f(y_{0})\Big)(D_{x}^{\lambda})^{r-1}$ is a bounded function in $\overline{\Omega_{x_{0}}^{\ast}}$, then $\big|\Big((D_{x}^{\lambda})^{\mathfrak{p}}f(y_{0})\Big)(D_{x}^{\lambda})^{r-1}\big|\leq M_{4}$, hence,

where $M_{i} > 0$ are positive constants, $i = 4, 5, 6$, and we can conclude that

Hence,

By Equality (4.3), we can deduce that

We complete the proof. □

Corollary 4.1. Suppose that $f\in \mathbf{F}^{\mathfrak{p}+\mathfrak{q}}\big(\overline{\Omega}, \mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})\big)$ is a solution of the equation $\Big((D^{\lambda})^{\mathfrak{p}}f(y_{0})\Big) (D^{\lambda})^{\mathfrak{q}}$$=$$0$ in $\overline{\Omega_{x_{0}}^{\ast}}$, $\mathfrak{p}+\mathfrak{q} \leq n, \; 0 < \lambda < \frac{1}{\mathfrak{p}+\mathfrak{q}}, \; \mathfrak{p}, \mathfrak{q}\in \mathbf{N^{\ast}}$, for arbitrary $x_{0}\in \Omega$, then we have

where $c_{1}(\alpha_{j}, \gamma_{ij})$ is a Clifford constant. If $c_{1}(\alpha_{j}, \gamma_{ij})$ has a single inverse element, this formula is called the Cauchy integral formula of the $(\mathfrak{p}+\mathfrak{q})$-order $\lambda$-weighted monogenic function.

Remark 4.1. Theorem 4.1 is used to prove Theorem 4.2. As $\mathfrak{p}\in \mathbf{N}^{\ast}$, where $\mathbf{N}^{\ast}$ is a set of positive integers, there is no direct relationship between Theorems 3.3 and 4.2. However, when $\mathfrak{p} = 0$ in Theorem 4.2, if $\sum\limits^{\mathfrak{p}}_{s = 1}(-1)^{s}\int_{\partial\Omega_{x_{0}}^{\ast}} M_{s}^{\lambda}(x)d\sigma_{x} \Big((D_{x}^{\lambda})^{s-1}f(y_{0})\Big) = 0$ in Equality (4.2), then the right end of the equality in Theorem 4.2 is reduced to the right end of the equality in Theorem 3.3. When $\mathfrak{q} = 0$ in Theorem 4.2, if $\sum\limits^{\mathfrak{q}}_{r = 1}(-1)^{\mathfrak{p}+r}\int_{\partial\Omega_{x_{0}}^{\ast}} \Big[\Big((D_{x}^{\lambda})^{\mathfrak{p}}f(y_{0})\Big)(D_{x}^{\lambda})^{r-1}\Big] d\sigma_{x}M_{\mathfrak{p}+r}^{\lambda}(x) = 0$ in Equality (4.2), then the equality in Theorem 4.2 is reduced to the equality in Theorem 3.2.

5.   Conclusions

In recent years, the integral representations for the solution to the higher order Dirac equation in $\mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij})$ have been studied, which generalize the integral representation in the classical Clifford algebra. In this paper, we not only prove three Cauchy-Pompeiu integral formulae for functions valued in the dependent parameter Clifford algebra, but also obtain integral representations for three different higher order $\lambda$-weighted monogenic functions.

If $\mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}) = \mathcal{B}_{n}(2, 1, 0)$, then Corollary 3.1 in this paper is reduced to one result of Theorem 3.7 in reference ^[5], that is,

Theorem 5.1. ^[5] Suppose that $\Omega\subseteq \mathbf{R}^{n}$ is a domain, $\Omega^{\ast}: = \{x|y_{0} = x + x_{0} \in \Omega\}$, $H_{j}(x) = \frac{A_{j}}{|x|^{n-j\alpha}}$, $A_{j} = \frac{(-1)^{j-1}}{\omega_{n}\alpha^{j-1}(j-1)!}$, $0 < \alpha < \frac{1}{k}$. If $f(x+ x_{0})$ is a $k$-monogenic function with $\alpha$-weight in $\Omega^{\ast}$, for arbitrary $x_{0}\in \Omega$, then we have

If $\mathcal{B}_{n}(2, \alpha_{j}, \gamma_{ij}) = \mathcal{B}_{n}(2, 1, 0)$, Corollary 4.1 in this paper is reduced to Corollary 3.5 in ^[9], that is,

Theorem 5.2. ^[9] Suppose $f\in C^{r}(\Omega, Cl_{0, n}(\mathbf{R}))$, where $r\geq p+q$, $n\geq p+q$, $\Omega\subseteq \mathbf{R}^{n}$ is a domain, $\Omega^{\ast}: = \{x|y_{0} = x + x_{0} \in \Omega\}$, $H_{p+j}(x) = \frac{A_{p+j}}{|x|^{n-(p+j)\alpha}}$, $A_{p+j} = \frac{(-1)^{p+j-1}}{\omega_{n}\alpha^{p+j-1}(p+j-1)!}$, $0 < \alpha < \frac{1}{p+q}$. If $f(x+ x_{0})$ is a $(p, q)$-monogenic function with $\alpha$-weight in $\Omega^{\ast}$, then for any $x_{0}\in \Omega$, we have

With the method of the Clifford analytic approach and Newton embedding method, reference ^[10] proved the existence and uniqueness of solutions of the nonlinear Riemann-Hilbert problems. For a $k$-vector field $F_{k}$, reference ^[11] obtained the solution of boundary value problems for the associated with the equations $(D_{x})^{2s-1}(F_{k})D_{x} = f_{k}$, where $f_{k}\in \mathbf{F}\big(\Omega, \mathcal{B}_{m}^{(k)}(2, 1, 0)\big)$, $\mathcal{B}_{m}^{(k)}(2, 1, 0)$ is the space of pseudo-scalars in the classical Clifford algebra $\mathcal{B}_{m}(2, 1, 0)$. We hope to solve the boundary value problem related to the equation $(D_{x})^{2s-1}(F_{k})D_{x} = f_{k}$ in the dependent parameter Clifford algebra in our future work.

Author contributions

Xiaojing Du: Conceptualization, Writing-original draft, Writing-review and editing; Xiaotong Liang: Validation and Writing-review; Yonghong Xie: Supervision, Validation and Funding acquisition. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The work was supported by the Natural Science Foundation of Hebei Province (Nos. A2023205006, A2023205045, A2022208007 and A2024208005), the Key Development Foundation of Hebei Normal University (No. L2024ZD08), the National Natural Science Foundation of China (No. 12431005), and the Funding Project of Central Guidance for Local Scientific and Technological Development (No. 246Z7608G).

Conflict of interest

The authors state that there is no conflicts of interest in this paper.