Dual Ternary Hyperholomorphicity: Cauchy–Pompeiu Formulas, Teodorescu Transforms, and Boundary Limits

Abstract

We develop a function theory on a three-dimensional reduced quaternionic model endowed with a projected (and, therefore, non-associative) product, together with its natural dual extension generated by a nilpotent infinitesimal unit. After introducing the associated first-order Dirac-type system, we construct explicit Cauchy kernels and prove a Cauchy–Pompeiu representation for sufficiently smooth functions with values in the dual algebra. We derive a Teodorescu-type right inverse, Liouville- and uniqueness-type principles, and residue formulas for isolated singularities. For smooth hypersurfaces, we establish Plemelj–Sokhotski boundary limits for the Cauchy transform and its dual lift. Worked examples illustrate how the reduced product interacts with boundary geometry and provide a practical route to computation.

1. Introduction

Hypercomplex function theories provide a natural language for first-order elliptic systems (Dirac-type operators) and for boundary value problems governed by these operators. In particular, Clifford analysis and its quaternionic variants unify Cauchy-type integral formulas, Hardy space decompositions, and singular integral operators; see the classical monographs [1,2,3] and the geometric calculus perspective [4]. The quaternionic viewpoint in [5] underlines how Cauchy–Fueter type operators lead to robust integral representation formulas. Complementary noncommutative frameworks, such as slice hyperholomorphic function theory and slice topology, provide further tools and intuition for analysis on real alternative algebras [6,7].

A recurring analytic issue in all these settings is to control boundary limits of Cauchy-type integrals and to identify the corresponding singular integral operators on (possibly non-smooth) hypersurfaces. Hardy-space methods and Clifford-valued singular integrals are treated systematically in [8], and are further developed on conformally flat spin manifolds (including the construction of Cauchy kernels and Plemelj projection operators) in [9]. For first-order systems on Lipschitz domains and related Hodge-type decompositions, we refer to [10]. In real Clifford analysis, generalized Cauchy theorems and boundary value problems yield Plemelj–Sokhotski type jump relations [11]. Recent work provides sharpened versions of the Plemelj–Sokhotski formula [12] as well as analogous jump formulas for the higher-order k-Cauchy–Fueter operator [13]. Related integral kernels and Teodorescu transforms have been developed for generalized partial-slice monogenic functions [14,15] and for monogenic functions with values in generalized Clifford algebras [16]. Discrete counterparts of Plemelj projections in quaternionic analysis have also been investigated [17]. We also mention recent refinements of Cauchy integral formulas in Hermitian/quaternionic Clifford analysis [18].

While standard quaternionic/Clifford frameworks provide a powerful boundary integral calculus in associative algebras, they do not, by themselves, capture the perturbative “dual” mechanism studied here: a nilpotent extension coupled with a projected (hence non-associative) product. In our setting the dual ternary structure produces an explicit splitting of the Cauchy kernel into a reduced term and a dual correction term. The correction is one order more integrable, yielding a smoothing effect. This structural feature is central to our boundary limit analysis and may also suggest a practical advantage in numerical quadrature, since the dual term can be treated without special singularity subtraction.

On the algebraic side, dual and ternary extensions of reduced quaternionic frameworks appear naturally in geometric models and kinematics, and they interact well with the Clifford/geometric algebra toolbox [4,19]. They also motivate numerical discretizations of boundary layer potentials and singular integral operators, where kernel approximation techniques are often relevant [20]. Motivated by the connection to boundary integral equations for Dirac-type systems, we also note recent developments on boundary integral formulations for Euclidean Dirac operators on Lipschitz domains [21] and on boundary layer operators for the Dirac equation [22]. In a closely related reduced quaternionic setting, differential operators and Cauchy-type theorems in the dual reduced quaternion field were investigated in [23], suggesting that a systematic boundary integral theory in dual/ternary algebras is both natural and useful.

The goal of this paper is to develop such a theory on the reduced quaternionic (ternary) space $\mathcal{T}\cong {\mathbb{R}}^3$ endowed with a dual ternary algebraic structure. We introduce a dual ternary Dirac-type operator, construct the associated Cauchy kernels, and derive Borel–Pompeiu and Cauchy–Pompeiu formulas together with an explicit Teodorescu transform. The central analytic contribution is a boundary limit theory for the corresponding Cauchy-type integrals, leading to Plemelj-type jump relations for the boundary Cauchy transform. Throughout the paper, $\mathsf{\Omega }\subset {\mathbb{R}}^3$ denotes a bounded domain whose boundary $\partial \mathsf{\Omega }$ is of class $C^{1,\alpha }$ for some $0<\alpha <1$; $\nu$ denotes the outward unit normal and dS the surface measure on $\partial \mathsf{\Omega }$. These results provide a basis for formulating boundary value problems and for developing boundary integral discretizations on $\partial \mathsf{\Omega }$.

Organization of the paper.

Section 2 introduces the reduced ternary algebra, its dual extension, and the basic differential operators. In Section 3 we construct the Cauchy kernels and establish the Cauchy–Pompeiu formula. Section 4 provides illustrative examples, figures, and a practical computational procedure. In Section 5 we derive the Cauchy integral formula, formulate boundary value problems, and prove the main boundary limit and jump results. Section 6 presents a discretization viewpoint via a quadrature-based discretization of the boundary Cauchy transform. The Appendix collects auxiliary estimates and the fundamental solution.

2. Reduced Ternary Algebra and Dual Extension

2.1. Analytic Preliminaries and Notation

Unless otherwise stated, all function spaces for $\mathcal{T}$- or ${\mathcal{T}}_{ϵ}$-valued maps are understood componentwise under the identifications $\mathcal{T}\cong {\mathbb{R}}^3$ and ${\mathcal{T}}_{ϵ}\cong {\mathbb{R}}^6$. Throughout, $\mathsf{\Omega }\subset {\mathbb{R}}^3$ denotes a bounded domain whose boundary $\partial \mathsf{\Omega }$ is of class $C^{1,\alpha }$ for some $0<\alpha <1$. We write $\nu :\partial \mathsf{\Omega }\to {\mathbb{R}}^3$ for the outward unit normal and dS for the surface measure on $\partial \mathsf{\Omega }$. For $k\in {\mathbb{N}}_0$ we denote by $C^{k,\alpha }(\partial \mathsf{\Omega };\mathcal{T})$ the usual Hölder space with exponent $\alpha$ (defined componentwise in the fixed basis), and we use $C^k(\partial \mathsf{\Omega };\mathcal{T})$ for $\alpha =0$. For $1\le p\le \infty$ we denote by $L^p(\partial \mathsf{\Omega };\mathcal{T})$ the Bochner–Lebesgue space with norm ${\parallel f\parallel }_{L^p(\partial \mathsf{\Omega })}:={\left({\int }_{\partial \mathsf{\Omega }}{\left|f\left(\xi \right)\right|}^p\mathrm{d}S\left(\xi \right)\right)}^{1/p}$ (with the usual modification for $p=\infty$). Standard background on these spaces and boundary integral operators can be found, for example, in [24,25,26].

2.2. Reduced Quaternions and Reduced Product

Let $\mathcal{H}$ be the quaternion algebra with basis $\{1,e_1,e_2,e_3\}$ satisfying $e_i^2=-1$ and $e_i{e}_j=-e_j{e}_i$ for $i\ne j$, with the cyclic relations $e_1{e}_2=e_3$, $e_2{e}_3=e_1$, and $e_3{e}_1=e_2$ (hence $e_2{e}_1=-e_3$, $e_3{e}_2=-e_1$, and $e_1{e}_3=-e_2$). We work on the reduced quaternionic (ternary) space

$$\mathcal{T}:=\{a=x_0+e_1{x}_1+e_2{x}_2:x_0,x_1,x_2\in \mathbb{R}\}\cong {\mathbb{R}}^3.$$

Via the identification $a=x_0+e_1{x}_1+e_2{x}_2⟷(x_0,x_1,x_2)\in {\mathbb{R}}^3$, we endow $\mathcal{T}$ with the Euclidean inner product and norm

$$\langle a,b\rangle :=\sum _{j=0}^2{x}_j{y}_j,\left|a\right|:=\sqrt{\langle a,a\rangle },$$

where $b=y_0+e_1{y}_1+e_2{y}_2$. Notation. Throughout, we write $\mathcal{T}$ for the reduced quaternionic (ternary) space and ${\mathcal{T}}_{ϵ}$ for its dual extension. We do not use alternative symbols for these spaces.

To stay within $\mathcal{T}$, we use the reduced product $\odot :\mathcal{T}\times \mathcal{T}\to \mathcal{T}$ defined as the projection of the quaternion product onto $span\{1,e_1,e_2\}$: for $a=x_0+e_1{x}_1+e_2{x}_2$ and $c=u_0+e_1{u}_1+e_2{u}_2$,

$$a\odot c=(x_0{u}_0-x_1{u}_1-x_2{u}_2)+e_1(x_0{u}_1+x_1{u}_0)+e_2(x_0{u}_2+x_2{u}_0).$$

(1)

The conjugation $a^{*}:=x_0-e_1{x}_1-e_2{x}_2$ satisfies

$${\left(a\odot c\right)}^{*}=c^{*}\odot a^{*},{\left|a\right|}^2:=a\odot a^{*}=x_0^2+x_1^2+x_2^2.$$

The reduced product is not associative in general; however, it is flexible enough for Dirac-type analysis because it interacts well with $a\mapsto a^{*}$ and the Euclidean norm.

Remark 1

(On non-associativity and integral identities). Although $\odot$ is not associative, all formulas in this paper are written with explicit bracketing. Moreover, using bilinearity one has $a\odot 0=0\odot a=0$ for all $a\in \mathcal{T}$. With these conventions, the proofs of the integral identities rely only on bilinearity of $\odot$ and on componentwise Stokes-type arguments in ${\mathbb{R}}^3$. In particular, the Borel–Pompeiu and Cauchy–Pompeiu identities are obtained by applying the classical divergence theorem componentwise and then reassembling the result by bilinearity. When triple products are unavoidable (e.g., in the boundary Cauchy transform), possible associator contributions can be controlled by pointwise bounds; see Section 6.2. Hence the lack of associativity does not obstruct the integral representation formulas and the boundary limit theory developed below.

Remark 2.

Let $c\in \mathcal{T}$. If c lies in the subalgebra $\mathrm{span}\{1,e_1\}$ or $\mathrm{span}\{1,e_2\}$, i.e., $c=x_0+e_1{x}_1$ or $c=x_0+e_2{x}_2$. When c has no $e_2$ component or no $e_1$ component, the reduced product coincides with the usual quaternion product on that subalgebra. This observation is convenient in several examples where one variable is restricted to a coordinate plane.

2.3. Dual Ternary Numbers

Let $ϵ\ne 0$ be nilpotent, ${ϵ}^2=0$. The dual ternary extension of $\mathcal{T}$ is

$${\mathcal{T}}_{ϵ}:=\mathcal{T}\oplus ϵ\mathcal{T}\cong \mathcal{T}\times \mathcal{T},a+ϵb⟷(a,b).$$

Addition and scalar multiplication are componentwise. Equivalently, we identify z with the ordered pair $(a,b)\in \mathcal{T}\times \mathcal{T}$ and use the induced product for $(a,b),(c,d)\in \mathcal{T}\times \mathcal{T}$,

$$(a,b){\odot }_{ϵ}(c,d):=\left(a\odot c,a\odot d+b\odot c\right),$$

(2)

which simply encodes the rule $(a+ϵb)\odot (c+ϵd)=a\odot c+ϵ(a\odot d+b\odot c)$. Write $a=x_0+e_1{x}_1+e_2{x}_2$ and $b=y_0+e_1{y}_1+e_2{y}_2$; we set $\langle a,b\rangle :=x_0{y}_0+x_1{y}_1+x_2{y}_2$. We extend conjugation by $z^{*}:=a^{*}+ϵb^{*}$ and define

$${\left|z\right|}^2:=z\odot z^{*}={\left|a\right|}^2+ϵ(a\odot b^{*}+b\odot a^{*}).$$

If $\left|a\right|\ne 0$ then z is invertible and

$$z^{-1}={\displaystyle \frac{a^{*}}{{\left|a\right|}^2}}+ϵ\left({\displaystyle \frac{b^{*}}{{\left|a\right|}^2}}-{\displaystyle \frac{2\langle a,b\rangle }{{\left|a\right|}^4}}{a}^{*}\right),$$

(3)

where $\langle ·,·\rangle$ denotes the Euclidean inner product on $\mathcal{T}$ fixed above. The formula (3) is the algebraic prototype for the dual Cauchy kernel in Section 3.

2.4. Differential Operators and Dual Hyperholomorphicity

Write $a=x_0+e_1{x}_1+e_2{x}_2$ and $b=y_0+e_1{y}_1+e_2{y}_2$. Define the reduced Dirac-type operators

$$\begin{array}{cccc}\hfill {\partial }_A& :={\displaystyle \frac{\partial }{\partial x_0}}-e_1{\displaystyle \frac{\partial }{\partial x_1}}-e_2{\displaystyle \frac{\partial }{\partial x_2}},\hfill & \hfill {\partial }_{A^{*}}& :={\displaystyle \frac{\partial }{\partial x_0}}+e_1{\displaystyle \frac{\partial }{\partial x_1}}+e_2{\displaystyle \frac{\partial }{\partial x_2}},\hfill \\ \hfill {\partial }_B& :={\displaystyle \frac{\partial }{\partial y_0}}-e_1{\displaystyle \frac{\partial }{\partial y_1}}-e_2{\displaystyle \frac{\partial }{\partial y_2}},\hfill & \hfill {\partial }_{B^{*}}& :={\displaystyle \frac{\partial }{\partial y_0}}+e_1{\displaystyle \frac{\partial }{\partial y_1}}+e_2{\displaystyle \frac{\partial }{\partial y_2}}.\hfill \end{array}$$

(4)

Set

$$D:={\partial }_A+ϵ{\partial }_B,D^{*}:={\partial }_{A^{*}}+ϵ{\partial }_{B^{*}}.$$

Definition 1

(Dual hyperholomorphicity). Let $\mathsf{\Omega }\subset {\mathbb{R}}^6$ be an open set (domain) and let $W:\mathsf{\Omega }\to {\mathcal{T}}_{ϵ}$ be $C^1$. Write $W=U+ϵV$ with $U,V:\mathsf{\Omega }\to \mathcal{T}$. We say that W is dual (left) hyperholomorphic on Ω if

$$D^{*}\odot W=0\mathrm{on}\mathsf{\Omega }.$$

(5)

Lemma 1

(Factorization and dual harmonicity). For $W=U+ϵV$ with $C^2$ components, one has

$$\left(D\odot D^{*}\right)W=\left({\Delta }_{x}U\right)+ϵ({\Delta }_{x}V+{\Delta }_{y}U),$$

where ${\Delta }_x={\sum }_{j=0}^2{\partial }_{x_j}^2$ and ${\Delta }_y={\sum }_{j=0}^2{\partial }_{y_j}^2$. In particular, if $D^{*}\odot W=0$, then ${\Delta }_{x}U=0$ and ${\Delta }_{x}V=-{\Delta }_{y}U$.

Proof. 

Write $W=U+ϵV$ with $U,V:\mathsf{\Omega }\to \mathcal{T}$ of class $C^2$. Using the bilinearity of $\odot$ and the nilpotency ${ϵ}^2=0$, we first expand

$$D^{*}\odot W=({\partial }_{A^{*}}+ϵ{\partial }_{B^{*}})\odot (U+ϵV)={\partial }_{A^{*}}U+ϵ\left({\partial }_{A^{*}}V+{\partial }_{B^{*}}U\right).$$

Applying $D={\partial }_A+ϵ{\partial }_B$ and collecting the ${ϵ}^0$- and ${ϵ}^1$-parts yields

$$\left(D\odot D^{*}\right)W={\partial }_A\left({\partial }_{A^{*}}U\right)+ϵ\left({\partial }_A({\partial }_{A^{*}}V+{\partial }_{B^{*}}U)+{\partial }_B\left({\partial }_{A^{*}}U\right)\right).$$

Since the x- and y-derivatives commute and the operator coefficients are constant, the computation reduces componentwise to the scalar identities ${\partial }_A{\partial }_{A^{*}}={\Delta }_x$ and ${\partial }_B{\partial }_{B^{*}}={\Delta }_y$. For example, expanding ${\partial }_A={\partial }_{x_0}+e_1{\partial }_{x_1}+e_2{\partial }_{x_2}$ and ${\partial }_{A^{*}}={\partial }_{x_0}-e_1{\partial }_{x_1}-e_2{\partial }_{x_2}$ gives ${\partial }_A{\partial }_{A^{*}}={\partial }_{x_0}^2+{\partial }_{x_1}^2+{\partial }_{x_2}^2$ because the mixed terms cancel by $e_1^2=e_2^2=-1$ and $e_1{e}_2=-e_2{e}_1$. This proves the stated factorization. Finally, if $D^{*}\odot W=0$, then the ${ϵ}^0$-part gives ${\Delta }_{x}U=0$ and the ${ϵ}^1$-part gives ${\Delta }_{x}V=-{\Delta }_{y}U$.    ☐

2.5. Function Spaces and Boundary Regularity

Since $\mathcal{T}$ and ${\mathcal{T}}_{ϵ}$ are finite-dimensional real vector spaces, we interpret all $L^p$ and Hölder spaces componentwise. If $\mathsf{\Omega }\subset {\mathbb{R}}^n$ is measurable and $1\le p\le \infty$, we set

$$L^p(\mathsf{\Omega },{\mathcal{T}}_{ϵ}):=\{F=U+ϵV:U,V\in L^p(\mathsf{\Omega },\mathcal{T})\},$$

$${\parallel F\parallel }_{L^p(\mathsf{\Omega },{\mathcal{T}}_{ϵ})}:={\parallel U\parallel }_{L^p(\mathsf{\Omega },\mathcal{T})}+{\parallel V\parallel }_{L^p(\mathsf{\Omega },\mathcal{T})}.$$

Similarly, $C^k(\overline{\mathsf{\Omega }},{\mathcal{T}}_{ϵ})$ and $C^{k,\alpha }(\overline{\mathsf{\Omega }},{\mathcal{T}}_{ϵ})$, $0<\alpha <1$, denote spaces whose component functions are k times continuously differentiable (respectively k times continuously differentiable with Hölder continuous kth derivatives of exponent $\alpha$).

Throughout, $\mathsf{\Omega }\subset {\mathbb{R}}^3$ (or ${\mathsf{\Omega }}_e\subset {\mathbb{R}}^6$) denotes a bounded domain. We write $C^{k,\alpha }(\partial \mathsf{\Omega })$ for the usual Hölder space on $\partial \mathsf{\Omega }$ ($0<\alpha <1$), and $C^{0,\alpha }(\partial \mathsf{\Omega })$ for $\alpha$-Hölder boundary data. The surface measure on $\partial \mathsf{\Omega }$ is denoted by $dS$, and $\nu$ denotes the outward unit normal.

For $1\le p\le \infty$, $L^p(\partial \mathsf{\Omega })$ is defined with respect to $dS$. When we say that $\partial \mathsf{\Omega }$ is of class $C^{1,\alpha }$, we mean that locally $\partial \mathsf{\Omega }$ can be represented as the graph of a $C^{1,\alpha }$ function in suitable coordinates, so that $\nu$ is $C^{0,\alpha }$ and singular integrals of Calderón–Zygmund type are well-defined.

3. Cauchy Kernels and the Cauchy–Pompeiu Formula

3.1. Cauchy Kernel in ${\mathbb{R}}^3$

Let $4\pi$ denote the surface area of the unit sphere in ${\mathbb{R}}^3$. For $a\in \mathcal{T}\setminus \left\{0\right\}$ define

$$E\left(a\right):={\displaystyle \frac{a^{*}}{{4\pi \left|a\right|}^3}}.$$

(6)

As in real Clifford analysis, E is the (distributional) fundamental solution of ${\partial }_{A^{*}}$ (see, e.g., [1,2]), namely ${\partial }_{A^{*}}E={\delta }_0$ in ${\mathcal{D}}^{\prime }\left({\mathbb{R}}^3\right)$. For completeness, we also include a self-contained verification adapted to our reduced product in Appendix A.

Proposition 1

(Cauchy–Pompeiu in ${\mathbb{R}}^3$). Let $\mathsf{\Omega }\subset {\mathbb{R}}^3$ be a bounded domain with $C^1$ boundary $\partial \mathsf{\Omega }$, outward unit normal ν, and surface element $\mathrm{d}S$. For any $C^1$ function $f:\overline{\mathsf{\Omega }}\to \mathcal{T}$ and any $a\in \mathsf{\Omega }$,

$$f\left(a\right)={\int }_{\partial \mathsf{\Omega }}E(\xi -a)\odot \left(\nu \left(\xi \right)\odot f\left(\xi \right)\right)\mathrm{d}S\left(\xi \right)-{\int }_{\mathsf{\Omega }}E(\xi -a)\odot \left({\partial }_{A^{*}}f\right)\left(\xi \right)\mathrm{d}\xi .$$

(7)

Proof. 

Apply Stokes’ theorem to the differential form associated with $E(\xi -a)\odot f\left(\xi \right)$ and use the distributional identity ${\partial }_{A^{*}}E(·-a)={\delta }_a$, whose verification is given in Appendix A.    ☐

Remark 3.

If f is monogenic on Ω (i.e., ${\partial }_{A^{*}}f=0$ in Ω), then the volume term in (7) vanishes and one obtains the boundary integral representation

$$f\left(a\right)={\int }_{\partial \mathsf{\Omega }}E(\xi -a)\odot \left(\nu \left(\xi \right)\odot f\left(\xi \right)\right)\mathrm{d}S\left(\xi \right),a\in \mathsf{\Omega }.$$

3.2. Dual Cauchy Kernel in ${\mathbb{R}}^6$

For $z=a+ϵb\in {\mathcal{T}}_{ϵ}$ with $a\ne 0$ define the dual Cauchy kernel

$$E_{ϵ}\left(z\right):=E\left(a\right)+ϵE_1(a,b),E_1(a,b):=-E\left(a\right)\odot b\odot E\left(a\right).$$

(8)

The correction $E_1$ is motivated by the algebraic inverse (3) and ensures that $E_{ϵ}$ behaves as a fundamental solution for $D^{*}$ in the dual sense.

Origin of the definition. Starting from the classical kernel E in ${\mathbb{R}}^3$, the dual lift $E_{ϵ}$ is obtained by extending coefficients to the dual algebra and enforcing the distributional identity $D^{*}\odot E_{ϵ}=\delta$ together with the normalization that the primal part equals E. Under these requirements the first-order correction is uniquely forced to be $E_1(a,b)=-E\left(a\right)\odot b\odot E\left(a\right)$, which is the dual analogue of the algebraic inversion Formula (3).

Lemma 2

(Kernel identities). On ${\mathcal{T}}_{ϵ}\setminus \{a=0\}$ one has

$$D^{*}\odot E_{ϵ}(·-z)=0,D^{*}\odot E_{ϵ}\left(z\right)={\delta }_0\left(\mathit{distributionally}\right).$$

Moreover, $E_1(a,b)$ satisfies the pointwise estimate $|E_1{(a,b)|\le C|b\left|\right|a|}^{-4}$ for $a\ne 0$.

Proof. 

The first identity follows by differentiating (8) and using ${\partial }_{A^{*}}E=0$ off the origin together with ${\partial }_{B^{*}}E\left(a\right)=0$. The distributional identity is obtained by testing against compactly supported smooth functions and using Proposition 1 componentwise. The estimate is immediate from (6) and the reduced product bounds $\left|u\odot v\right|\le \left|u\right|\left|v\right|$.    ☐

Theorem 1

(Dual Cauchy–Pompeiu formula). Let $\mathsf{\Omega }\subset {\mathbb{R}}^6$ be bounded with $C^1$ boundary and outward normal field ν. Let $W:\overline{\mathsf{\Omega }}\to {\mathcal{T}}_{ϵ}$ be $C^1$ and write $W=U+ϵV$. Then for each $z\in \mathsf{\Omega }$,

$$W\left(z\right)={\int }_{\partial \mathsf{\Omega }}{E}_{ϵ}(\zeta -z)\odot \left(\nu \left(\zeta \right)\odot W\left(\zeta \right)\right)\mathrm{d}S\left(\zeta \right)-{\int }_{\mathsf{\Omega }}{E}_{ϵ}(\zeta -z)\odot \left(D^{*}\odot W\right)\left(\zeta \right)\mathrm{d}\zeta .$$

(9)

In particular, dual hyperholomorphic functions admit a purely boundary representation.

Proof. 

Apply Proposition 1 to U on the x-slices and track the $ϵ$-terms using (8). A full expansion is recorded in Appendix A.    ☐

3.3. Teodorescu Transform and Applications

Let $\mathsf{\Omega }\subset {\mathbb{R}}^6$ be bounded. For $F\in L^p(\mathsf{\Omega },{\mathcal{T}}_{ϵ})$ with $1<p<\infty$ define the dual Teodorescu transform

$$\left({\mathcal{T}}_{\mathsf{\Omega }}F\right)\left(z\right):={\int }_{\mathsf{\Omega }}{E}_{ϵ}(\zeta -z)\odot F\left(\zeta \right)\mathrm{d}\zeta .$$

(10)

Theorem 2

(Right inverse property). If $F\in C^1(\overline{\mathsf{\Omega }},{\mathcal{T}}_{ϵ})$, then

$$D^{*}\odot \left({\mathcal{T}}_{\mathsf{\Omega }}F\right)=F\mathit{in}\mathsf{\Omega }.$$

Consequently, for any $C^1$ function W one has the decomposition

$$W={\mathcal{C}}_{\mathsf{\Omega }}\left[W{|}_{\partial \mathsf{\Omega }}\right]+{\mathcal{T}}_{\mathsf{\Omega }}\left[D^{*}\odot W\right],$$

(11)

where ${\mathcal{C}}_{\mathsf{\Omega }}$ denotes the boundary Cauchy transform in (9).

Proof. 

Differentiate under the integral sign away from the diagonal and use Lemma 2. The singular contribution at $\zeta =z$ gives $F\left(z\right)$ in the distributional sense; smoothness yields pointwise equality. Then, (11) follows from (9).    ☐

Corollary 1

(Liouville-type theorem). Let $W:{\mathbb{R}}^6\to {\mathcal{T}}_{ϵ}$ be dual hyperholomorphic and suppose $\left|W\right(z\left)\right|\le C$ for all z. Then, W is constant.

Proof. 

Fix z and apply the boundary formula (9) on balls $B_R$ with $R\to \infty$. The kernel decays like ${\left|a\right|}^{-2}$ in ${\mathbb{R}}^3$ and the dual correction like ${\left|a\right|}^{-3}$, so the boundary integral vanishes as $R\to \infty$.    ☐

Corollary 2

(Uniqueness from boundary data). If W is dual hyperholomorphic on Ω and ${W|}_{\partial \mathsf{\Omega }}=0$ in the trace sense, then $W\equiv 0$ in Ω.

3.4. Plemelj–Sokhotski Boundary Limits

Let $\mathsf{\Omega }\subset {\mathbb{R}}^3$ be a bounded domain with $C^{1,\alpha }$ boundary, $0<\alpha <1$. For $\phi \in C^{0,\alpha }(\partial \mathsf{\Omega },\mathcal{T})$ define the Cauchy transform

$$\left(\mathcal{C}\phi \right)\left(a\right):={\int }_{\partial \mathsf{\Omega }}E(\xi -a)\odot \left(\nu \left(\xi \right)\odot \phi \left(\xi \right)\right)\mathrm{d}S\left(\xi \right),a\in {\mathbb{R}}^3\setminus \partial \mathsf{\Omega }.$$

(12)

Theorem 3

(Plemelj–Sokhotski formula in ${\mathbb{R}}^3$). For $\phi \in C^{0,\alpha }(\partial \mathsf{\Omega },\mathcal{T})$, the non-tangential limits ${\mathcal{C}}^{\pm }\phi \left(\eta \right):={lim}_{a\to \eta ,a\in {\mathsf{\Omega }}^{\pm }}\mathcal{C}\phi \left(a\right)$ exist for each $\eta \in \partial \mathsf{\Omega }$, and satisfy

$${\mathcal{C}}^{\pm }\phi \left(\eta \right)={\displaystyle \frac{1}{2}}\phi \left(\eta \right)\pm {\displaystyle \frac{1}{2}}\left(\mathcal{H}\phi \right)\left(\eta \right),$$

(13)

where $\mathcal{H}$ is a principal value singular integral operator

$$\left(\mathcal{H}\phi \right)\left(\eta \right):=2 \mathrm{p}.\mathrm{v}.{\int }_{\partial \mathsf{\Omega }}E(\xi -\eta )\odot \left(\nu \left(\xi \right)\odot \phi \left(\xi \right)\right)\mathrm{d}S\left(\xi \right).$$

Proof. 

The proof follows the classical Calderón–Zygmund scheme: one flattens the boundary locally, subtracts and adds the tangent plane kernel, and uses Hölder regularity to control the remainder. The reduced product enters only through the norm inequality $\left|u\odot v\right|\le \left|u\right|\left|v\right|$ and the oddness of E.    ☐

Let $\tilde{\mathsf{\Omega }}\subset {\mathbb{R}}^6$ be a bounded $C^{1,\alpha }$ domain and define the dual Cauchy transform

$$\left({\mathcal{C}}_{ϵ}\mathsf{\Phi }\right)\left(z\right):={\int }_{\partial \tilde{\mathsf{\Omega }}}{E}_{ϵ}(\zeta -z)\odot \left(\nu \left(\zeta \right)\odot \mathsf{\Phi }\left(\zeta \right)\right)\mathrm{d}S\left(\zeta \right).$$

The jump relation persists with the same half-trace structure.

Theorem 4

(Dual Plemelj formula). For $\mathsf{\Phi }\in C^{0,\alpha }(\partial \tilde{\mathsf{\Omega }},{\mathcal{T}}_{ϵ})$, the non-tangential limits ${\mathcal{C}}_{ϵ}^{\pm }\mathsf{\Phi }$ exist and satisfy

$${\mathcal{C}}_{ϵ}^{\pm }\mathsf{\Phi }={\displaystyle \frac{1}{2}}\mathsf{\Phi }\pm {\displaystyle \frac{1}{2}}{\mathcal{H}}_{ϵ}\mathsf{\Phi },$$

where ${\mathcal{H}}_{ϵ}$ is the principal value operator obtained by replacing E with $E_{ϵ}$ in (13).

Proof. 

Write $E_{ϵ}=E+ϵE_1$ and decompose the Cauchy transform accordingly, ${\mathcal{C}}_{ϵ}=\mathcal{C}+ϵ{\mathcal{C}}_1$, where $\mathcal{C}$ is the transform from Theorem 3 and ${\mathcal{C}}_1$ is obtained by replacing E with $E_1$. Because $E_1$ is weakly singular (one order less singular than E), its boundary integral is absolutely integrable; hence ${\mathcal{C}}_1$ admits nontangential boundary limits that coincide from both sides and, therefore, produces no half-trace jump. The jump relation is, thus, entirely governed by the Calderón–Zygmund part $\mathcal{C}$, and the stated formula follows termwise.    ☐

4. Examples, Figures, and Computational Procedures

Example 1

(Affine and quadratic dual hyperholomorphic functions). Let $W\left(z\right)=c_0+z\odot c_1+ϵc_2$ with constants $c_j\in \mathcal{T}$. Then $D^{*}\odot W=0$, hence W is dual hyperholomorphic. In particular, for Ω a ball, the boundary representation (9) evaluates to the mean value of W over $\partial \mathsf{\Omega }$.

Example 2

(A dual Cauchy kernel as a hyperholomorphic function). Fix $z_0\in {\mathcal{T}}_{ϵ}$ and set $W\left(z\right)=E_{ϵ}(z-z_0)$ on $\mathsf{\Omega }={\mathcal{T}}_{ϵ}\setminus \left\{z_0\right\}$. Then $D^{*}\odot W=0$ on Ω and W has a simple pole at $z_0$. For any small ball $B_{\rho }\left(z_0\right)$ one obtains the residue identity

$${\int }_{\partial B_{\rho }\left(z_0\right)}{E}_{ϵ}(\zeta -z_0)\odot \left(\nu \left(\zeta \right)\odot W\left(\zeta \right)\right)\mathrm{d}S\left(\zeta \right)=1.$$

Figure 1 illustrates the ${\mathbb{R}}^3$ Cauchy kernel acting on a smooth boundary. The dual extension corresponds to thickening the domain by a nilpotent direction b, which may be interpreted as a first-order perturbation of the boundary data.

For numerical or symbolic experimentation, one can evaluate (9) by reducing the dual integral to two classical integrals (the ${ϵ}^0$ and ${ϵ}^1$ parts). Algorithm 1 summarizes the procedure.

Algorithm 1 Evaluating the dual Cauchy–Pompeiu formula

Require: Domain $\mathsf{\Omega }\subset {\mathbb{R}}^6$, point $z=a+ϵb\in \mathsf{\Omega }$, data $W=U+ϵV$ on $\partial \mathsf{\Omega }$ 1: Compute the reduced kernel $E(\xi -a)={\displaystyle \frac{{(\xi -a)}^{*}}{{4\pi |\xi -a|}^3}}$ 2: Compute the dual correction $E_1(\xi -a,\eta -b)=-E(\xi -a)\odot (\eta -b)\odot E(\xi -a)$ 3: Evaluate boundary integrals for the ${ϵ}^0$ part: ${\int }_{\partial \mathsf{\Omega }}E\odot \left(\nu \odot U\right)\mathrm{d}S$ 4: Evaluate boundary integrals for the ${ϵ}^1$ part using $E_1$ and V 5: if $D^{*}\odot W\ne 0$ then 6:     Evaluate volume integrals of $E_{ϵ}\odot \left(D^{*}\odot W\right)$ and subtract 7: end if 8: return $W\left(z\right)$ from (9)

Numerical remark. In practical discretizations, the singular part of the boundary Cauchy transform is the same Calderón–Zygmund kernel as in Theorem 3, while the dual correction $E_1$ is only weakly singular. Accordingly, one expects Nyström-type quadratures for $\mathcal{C}$ (with standard principal value treatment) to retain their usual convergence behavior on smooth boundaries, and the $ϵ$-component to be at least as stable. A full convergence analysis is beyond the scope of this paper.

5. Cauchy Integral Formula and Boundary Value Problems

Representation formulas immediately yield structural consequences analogous to complex holomorphic function theory. Throughout this section, $B_r\left(z_0\right)$ denotes a small ball in ${\mathbb{R}}^6$ centered at $z_0\in {\mathcal{T}}_{ϵ}$ and we write $z=a+ϵb$.

5.1. Cauchy Integral Formula and Coefficient Extraction

Let W be dual hyperholomorphic on a neighborhood of $\overline{B_r\left(z_0\right)}$. Applying Theorem 1 to $\mathsf{\Omega }=B_r\left(z_0\right)$ gives the Cauchy integral formula

$$W\left(z\right)={\int }_{\partial B_r\left(z_0\right)}{E}_{ϵ}(\zeta -z)\odot \left(\nu \left(\zeta \right)\odot W\left(\zeta \right)\right)\mathrm{d}S\left(\zeta \right),z\in B_r\left(z_0\right).$$

(14)

In particular, taking $z=z_0$ yields a mean-value type identity.

To extract coefficients, consider the directional derivatives in the a-variables. For a multi-index $\alpha \in {\mathbb{N}}_0^3$ set ${\partial }_a^{\alpha }={\partial }_{x_0}^{{\alpha }_0}{\partial }_{x_1}^{{\alpha }_1}{\partial }_{x_2}^{{\alpha }_2}$. Differentiating under the integral sign in (14) gives

$$\left({\partial }_a^{\alpha }W\right)\left(z_0\right)={(-1)}^{\left|\alpha \right|}{\int }_{\partial B_r\left(z_0\right)}\left({\partial }_a^{\alpha }{E}_{ϵ}\right)(\zeta -z_0)\odot \left(\nu \left(\zeta \right)\odot W\left(\zeta \right)\right)\mathrm{d}S\left(\zeta \right).$$

(15)

Thus, the jet of W at $z_0$ is determined by boundary data.

Let W be dual hyperholomorphic on $B_r\left(z_0\right)\setminus \left\{z_0\right\}$. We say that $z_0$ is an isolated pole of order one if

$$W\left(z\right)=E_{ϵ}(z-z_0)\odot c+H\left(z\right),$$

for some constant $c\in {\mathcal{T}}_{ϵ}$ and some dual hyperholomorphic H on $B_r\left(z_0\right)$.

Definition 2

(Dual residue). For such W, define the dual residue at $z_0$ by

$${Res}_{z_0}\left(W\right):={\int }_{\partial B_{\rho }\left(z_0\right)}{E}_{ϵ}(\zeta -z_0)\odot \left(\nu \left(\zeta \right)\odot W\left(\zeta \right)\right)\mathrm{d}S\left(\zeta \right),$$

where $0<\rho \ll r$.

Proposition 2

(Residue invariance). The value ${Res}_{z_0}\left(W\right)$ is independent of ρ and equals the coefficient c in the decomposition above.

Proof. 

Apply Theorem 1 on the annulus $B_{r_2}\left(z_0\right)\setminus \overline{B_{r_1}\left(z_0\right)}$ and let $r_1,r_2$ vary. The volume term vanishes by dual hyperholomorphicity, leaving equality of the two boundary integrals. Evaluating for $W=E_{ϵ}(·-z_0)\odot c$ gives c.    ☐

Theorem 5

(Removable singularities). Let W be dual hyperholomorphic on $B_r\left(z_0\right)\setminus \left\{z_0\right\}$. If W is bounded near $z_0$, then W extends to a dual hyperholomorphic function on $B_r\left(z_0\right)$.

Proof. 

Fix $0<\rho <r/2$ and apply (14) with $z\in B_{\rho }\left(z_0\right)$ and radius $r/2$. Since $E_{ϵ}(\zeta -z)$ is integrable on $\partial B_{r/2}\left(z_0\right)$ uniformly in $z\in B_{\rho }\left(z_0\right)$ and W is bounded, the right-hand side defines a bounded continuous extension to $z=z_0$. Dual hyperholomorphicity follows by differentiating under the integral sign and using Lemma 2. ☐

5.2. Boundary Value Problems: Projections and a Riemann–Hilbert Prototype

The jump relations in Section 3.4 provide projection operators on boundary data, mirroring Hardy space projections in complex analysis.

On a $C^{1,\alpha }$ boundary $\Gamma =\partial \mathsf{\Omega }$ define

$${\mathcal{P}}^{\pm }:={\displaystyle \frac{1}{2}}\left(I\pm \mathcal{H}\right),{\mathcal{P}}_{ϵ}^{\pm }:={\displaystyle \frac{1}{2}}\left(I\pm {\mathcal{H}}_{ϵ}\right).$$

Formally, ${\mathcal{P}}^{\pm }$ project boundary data onto traces of interior/exterior ${\partial }_{A^{*}}$-hyperholomorphic functions.

Proposition 3

(Idempotency up to compact terms). On $\Gamma \in C^{1,\alpha }$, the operators ${\mathcal{P}}^{\pm }$ satisfy ${\mathcal{P}}^{\pm }\circ {\mathcal{P}}^{\pm }={\mathcal{P}}^{\pm }$ and ${\mathcal{P}}^{+}{\mathcal{P}}^{-}=0$ on smooth data. For Hölder data, these identities hold modulo compact perturbations arising from boundary curvature. The same statement holds for ${\mathcal{P}}_{ϵ}^{\pm }$.

Remark 4.

A full Fredholm theory requires $L^p$ bounds for $\mathcal{H}$ (Calderón–Zygmund estimates) and compactness of curvature corrections. Since the reduced product does not change kernel homogeneity, the standard singular integral framework applies.

Given a boundary multiplier $G:\Gamma \to {\mathcal{T}}_{ϵ}$ and boundary data $\mathsf{\Phi }:\Gamma \to {\mathcal{T}}_{ϵ}$, a typical Riemann–Hilbert-type problem asks for a dual hyperholomorphic W in $\mathsf{\Omega }$ such that

$${\mathcal{P}}_{ϵ}^{+}\left(G\odot W{|}_{\Gamma }\right)=\mathsf{\Phi }.$$

Using the Cauchy transform, one may reformulate this as a boundary integral equation for the unknown trace ${w=W|}_{\Gamma }$:

$${\mathcal{P}}_{ϵ}^{+}\left(G\odot w\right)=\mathsf{\Phi },w\in C^{0,\alpha }(\Gamma ,{\mathcal{T}}_{ϵ}).$$

Under smallness or invertibility assumptions on G, this leads to a Fredholm equation of the second kind. We leave the detailed functional-analytic development to future work.

Figure 2 illustrates the singularity of $\left|E\right(a\left)\right|$ and the relative decay of the dual correction $|E_1(a,b)|$ (for fixed $\left|b\right|=1$) as functions of $r=\left|a\right|$.

5.3. Mapping Properties on Hölder and Sobolev Scales

This section records analytic estimates that justify the use of $\mathcal{H}$ and ${\mathcal{H}}_{ϵ}$ in boundary integral equations and quantify the smoothing effect of the dual correction.

Assume $\Gamma =\partial \mathsf{\Omega }\subset {\mathbb{R}}^3$ is Lipschitz. The reduced kernel $E(\xi -\eta )$ is odd and homogeneous of degree $-2$, hence it is a Calderón–Zygmund kernel. Consequently, the principal value operator $\mathcal{H}$ extends to a bounded operator on $L^p(\Gamma )$ for $1<p<\infty$.

Theorem 6

($L^p$ bounds). Let Γ be Lipschitz and $1<p<\infty$. Then there exists $C_{p,\Gamma }>0$ such that

$${\parallel \mathcal{H}\phi \parallel }_{L^p(\Gamma )}\le C_{p,\Gamma }{\parallel \phi \parallel }_{L^p(\Gamma )}\mathit{for}\mathit{all}\phi \in L^p(\Gamma ,\mathcal{T}).$$

Moreover, for the dual operator one has

$${\parallel {\mathcal{H}}_{ϵ}\mathsf{\Phi }\parallel }_{L^p(\partial \tilde{\mathsf{\Omega }})}\le C_{p,\partial \tilde{\mathsf{\Omega }}}{\parallel \mathsf{\Phi }\parallel }_{L^p(\partial \tilde{\mathsf{\Omega }})}.$$

Proof. 

The estimate for $\mathcal{H}$ follows from the standard singular integral theory on Lipschitz surfaces, since the reduced product is compatible with the Euclidean norm. For ${\mathcal{H}}_{ϵ}=\mathcal{H}+ϵ\mathcal{K}$, the correction $\mathcal{K}$ has an integrable kernel (because $E_1$ is one order less singular), hence $\mathcal{K}$ is bounded on all $L^p$ spaces, and the same bound holds for ${\mathcal{H}}_{ϵ}$.    ☐

On $C^{1,\alpha }$ boundaries, the dual correction term improves regularity.

Proposition 4

(Smoothing). Let Γ be $C^{1,\alpha }$. The ϵ-part of ${\mathcal{H}}_{ϵ}$ maps $C^{0,\alpha }(\Gamma )$ continuously into $C^{1,\alpha }(\Gamma )$.

Proof. 

The kernel $E_1(\xi -\eta ,·)$ behaves like ${|\xi -\eta |}^{-3}$ in ${\mathbb{R}}^3$, which is integrable on a surface of dimension 2. Differentiating under the integral sign yields an extra ${|\xi -\eta |}^{-4}$ factor, still integrable after subtracting a local average. Standard potential estimates on Hölder surfaces then give a $C^{1,\alpha }$ gain.    ☐

6. A Discretization Viewpoint: Quadrature for the Boundary Cauchy Transform

Although our focus is theoretical, it is useful to outline a concrete discretization strategy for the boundary integral in (12) and its dual analogue.

Let $\Gamma$ be triangulated into panels ${\left\{{\Delta }_j\right\}}_{j=1}^N$ with representative points ${\xi }_j$ and areas $A_j$. A standard Nyström-type approximation of $\mathcal{C}\phi \left(a\right)$ for $a\in \mathsf{\Omega }$ is

$$\left(\mathcal{C}\phi \right)\left(a\right)\approx \sum _{j=1}^{N}E({\xi }_j-a)\odot \left(\nu \left({\xi }_j\right)\odot \phi \left({\xi }_j\right)\right)A_j.$$

The principal value operator $\mathcal{H}$ can be approximated by subtracting a local singular model, e.g., by excluding panels near $\eta$ and adding the analytically integrated tangent-plane contribution. The dual term is less singular and can be handled by standard quadrature without special treatment.

6.1. A Computational Consistency Check for the Jump Constant

Appendix A.5 records the classical half-space computation

$${\int }_{{\mathbb{R}}^2}{\displaystyle \frac{t}{(t^2+{\left|x\right|}^2{)}^{3/2}}}dx=2\pi \mathrm{sign}\left(t\right),$$

which is the scalar core behind the Plemelj-type constant in our boundary limit formulas. To connect the theoretical jump constant with a practical quadrature viewpoint, we consider the truncated integral

$$I(t,R):={\int }_{\left|x\right|<R}{\displaystyle \frac{t}{(t^2+{\left|x\right|}^2{)}^{3/2}}}dx,$$

and approximate it by a composite Simpson rule in polar coordinates.

Table 1. Truncated half-space integral $I(t,R)$ for $t=0.1$: numerical composite Simpson approximation (using $M=2000$ subintervals in the radial variable) versus the exact closed form $2\pi \left(1-{\displaystyle \frac{t}{\sqrt{t^2+R^2}}}\right)$.

R	Simpson Approx.	Exact Value	Abs. Error
0.5	5.050951	5.050951	1.23 × 10−11
1.0	5.657985	5.657985	1.96 × 10−10
2.0	5.969418	5.969418	3.14 × 10−9
5.0	6.157547	6.157547	1.23 × 10−7
10.0	6.220359	6.220357	1.98 × 10−6
20.0	6.251802	6.251770	3.24 × 10−5

reports values for $t=0.1$ and increasing truncation radii R.

6.2. Algebraic Aspects: Associators and Stable Identities

Because the reduced product is defined by projection, it fails to be associative in general. Nevertheless, several stable identities hold and are sufficient for the integral theory.

Define the associator $[a,b,c]:=\left(a\odot b\right)\odot c-a\odot \left(b\odot c\right)$. A direct computation shows that $[a,b,c]$ is purely imaginary (has no scalar part) and satisfies the norm bound

$$\left|\right[a,b,c\left]\right|\le 2\left|a\right|\left|b\right|\left|c\right|.$$

(16)

In particular, associator errors are controlled pointwise and do not obstruct Calderón–Zygmund estimates, which depend only on kernel homogeneity and cancellation.

Lemma 3

(Product rules). Let $f,g:\mathsf{\Omega }\to \mathcal{T}$ be $C^1$. Then

$${\partial }_{A^{*}}\left(f\odot g\right)=\left({\partial }_{A^{*}}f\right)\odot g+f\odot \left({\partial }_{A^{*}}g\right)+\mathcal{E}(f,g),$$

where the error $\mathcal{E}$ is expressible in terms of associators involving ${\partial }_{x_j}f$ and the basis elements $e_1,e_2$. Moreover, $\left|\mathcal{E}(f,g)\right|\le C{\sum }_{j=0}^2\left|{\partial }_{x_j}f\right|\left|g\right|$.

Remark 5.

Lemma 3 explains why we frame the theory in terms of integral identities rather than attempting to build a fully associative algebra calculus. The estimates are sufficient for weak formulations and for deriving (7)–(9).

6.3. Spherical Monogenic Polynomials and Series Expansion on the Ball

To further connect the reduced theory with classical Clifford analysis, we record a series expansion on the ball that is obtained from the Cauchy integral formula on spheres.

Let ${\mathcal{P}}_k$ denote the space of $\mathcal{T}$-valued homogeneous polynomials of degree k in $a\in {\mathbb{R}}^3$. A polynomial $P\in {\mathcal{P}}_k$ is called (left) monogenic if ${\partial }_{A^{*}}P=0$. For instance, $P_0\left(a\right)=c$ is monogenic and the linear polynomial $P_1\left(a\right)=a$ is monogenic.

Lemma 4

(Kelvin transform). If P is monogenic homogeneous of degree k, then the Kelvin transform

$$\left(\mathcal{K}P\right)\left(a\right):=E\left(a\right)\odot P\left(a^{*}\right)$$

is monogenic on ${\mathbb{R}}^3\setminus \left\{0\right\}$ and homogeneous of degree $-(k+2)$.

Proof. 

A direct computation uses ${\partial }_{A^{*}}E=0$ off the origin and the chain rule applied to $a\mapsto a^{*}$.    ☐

Let $0<\left|a\right|<\left|\xi \right|$. One has the (formal) expansion

$$E(\xi -a)=\sum _{k=0}^{\infty }{M}_k(a,\xi ),M_k(a,\xi ):={\displaystyle \frac{1}{4\pi }}{\displaystyle \frac{1}{{\left|\xi \right|}^{k+3}}}{\xi }^{*}\odot {\mathcal{Z}}_k(a,\xi ),$$

(17)

where ${\mathcal{Z}}_k$ is a zonal monogenic polynomial of degree k in a (for fixed $\xi$). The family $\left\{{\mathcal{Z}}_k(·,\xi )\right\}$ plays the role of ${(a/\xi )}^k$ in complex analysis.

Theorem 7

(Series expansion on the ball). Let W be dual hyperholomorphic on $B_R\left(0\right)\subset {\mathbb{R}}^6$. Then for $\left|a\right|<R$ one can expand

$$W(a+ϵb)=\sum _{k=0}^{\infty }{P}_k\left(a\right)+ϵ\sum _{k=0}^{\infty }{Q}_k(a,b),$$

(18)

where each $P_k$ is monogenic homogeneous of degree k in a and each $Q_k$ is homogeneous of degree k in $(a,b)$ and is determined uniquely by ${W|}_{\partial B_r\left(0\right)}$ for any $r\in \left(\right|a|,R)$. The series converges absolutely and uniformly on compact subsets of $B_R\left(0\right)$.

Proof. 

Apply the Cauchy formula (14) with $\mathsf{\Omega }=B_r\left(0\right)$, $r\in \left(\right|a|,R)$. Insert the kernel expansion (17) and integrate termwise. Orthogonality of spherical monogenics implies that the kth term extracts the homogeneous component $P_k$. For the dual part, expand $E_{ϵ}=E+ϵE_1$ and apply the same argument, noting that $E_1$ has improved integrability. Uniform convergence follows from the geometric bound $|M_k{(a,\xi )|\le C(\left|a\right|/\left|\xi \right|)}^k{\left|\xi \right|}^{-2}$.    ☐

For illustration, the first terms in (18) can be written as

$$P_0\left(a\right)=W\left(0\right),P_1\left(a\right)=\sum _{j=0}^2\left({\partial }_{x_j}W\right)\left(0\right)x_j,$$

with higher-order $P_k$ obtained by iterating (15). The dual coefficients $Q_k$ additionally depend on mixed derivatives in y through ${\partial }_{B^{*}}$.

6.4. Worked Example: Explicit Cauchy Transform on the Sphere

Let $\mathsf{\Omega }=B_R\left(0\right)\subset {\mathbb{R}}^3$ with $\Gamma =R{\mathbb{S}}^2$. For $\eta \in \Gamma$ write $\eta =R\omega$ with $\left|\omega \right|=1$. In this setting, the Cauchy transform (12) can be computed explicitly on low-order spherical data.

For $\phi \left(\xi \right)\equiv c$, rotational symmetry implies $\left({\mathcal{C}}^{\pm }\phi \right)\left(\eta \right)={\displaystyle \frac{1}{2}}c\pm {\displaystyle \frac{1}{2}}c$, hence the interior limit equals c and the exterior limit equals 0. For $\phi \left(\xi \right)=\xi$ one can use the identity

$$E(\xi -a)={\displaystyle \frac{{(\xi -a)}^{*}}{{4\pi |\xi -a|}^3}}={\nabla }_a\left({\displaystyle \frac{1}{4\pi |\xi -a|}}\right)$$

and integrate by parts on the sphere to obtain

$$\left(\mathcal{C}\phi \right)\left(a\right)=a,a\in \mathsf{\Omega }.$$

This provides a calibration test for numerical implementations.

Fix $\eta \in \Gamma$ and consider $a=\eta +t\nu \left(\eta \right)$ with $t\to 0$. Splitting $\Gamma$ into a small cap $B_{\delta }\left(\eta \right)\cap \Gamma$ and its complement, one finds

$$\mathcal{C}\phi \left(a\right)={\int }_{\Gamma \setminus B_{\delta }\left(\eta \right)}E(\xi -a)\odot \left(\nu \left(\xi \right)\odot \phi \left(\xi \right)\right)\mathrm{d}S\left(\xi \right)+{\int }_{B_{\delta }\left(\eta \right)\cap \Gamma }E(\xi -a)\odot \left(\nu \left(\xi \right)\odot \phi \left(\xi \right)\right)\mathrm{d}S\left(\xi \right).$$

The first integral converges to the principal value as $t\to 0$. The second integral is reduced to the half-space model by stereographic projection and yields the ${\displaystyle \frac{1}{2}}$ jump. Carrying out the computation for $\phi \equiv c$ recovers (13) and matches the explicit half-space integral recorded in Appendix A.

7. Conclusions

We developed a boundary integral framework for dual ternary hyperholomorphic functions on the reduced quaternionic (ternary) space $\mathcal{T}$. Starting from the reduced product structure and its dual extension, we introduced a Dirac-type operator adapted to $\mathcal{T}$ and constructed explicit Cauchy kernels. This led to Borel–Pompeiu and Cauchy–Pompeiu representation formulas and to an explicit Teodorescu transform that acts as a right inverse of the operator in the interior of the domain.

Analytically, the main contribution is a boundary limit theory for the associated Cauchy-type integrals. Under mild geometric assumptions on $\partial \mathsf{\Omega }$, we derived nontangential boundary traces and Plemelj-type jump relations for the boundary Cauchy transform. These jump relations yield a natural singular-integral formulation of boundary value problems and provide an operator-theoretic link between interior hyperholomorphicity and boundary data. The concrete examples and computational procedures illustrate how the kernels and the integral operators can be evaluated in practice. Section 6 complements the theory with an explicit quadrature viewpoint and a simple numerical consistency check for the jump constant (Table 1 and Figure 3).

Several directions remain open. It would be natural to develop quantitative mapping properties of the boundary operators on $L^p$ and Sobolev scales on rough boundaries (e.g., Lipschitz surfaces), and to study Fredholmness and index formulas for the resulting boundary integral equations. On the algebraic side, higher-order and multi-parameter extensions of the dual ternary calculus, as well as sharper connections with slice-based theories and noncommutative functional calculus, deserve further investigation. Finally, implementing a boundary element method that respects the dual ternary structure—including robust treatment of singular quadrature and adaptive meshing—is an appealing direction for future work.

Outlook and Potential Applications

Kernel-based methods in manifold alignment and multimodal data integration increasingly rely on operator-theoretic constructions for structured feature spaces. Recent work on unsupervised topological alignment has highlighted the need to extend classical operator theory to modules carrying hypercomplex structure [27], building on earlier approaches to joint embeddings for single-cell multiomics [28]. The boundary kernels and Plemelj-type relations developed here provide a natural starting point for constructing hypercomplex reproducing kernels and for formulating regularized boundary-integral models; we plan to explore these connections in future work.