Multidimensional Cauchy-Riemann Operator

• The multidimensional Cauchy-Riemann operator is a general framework that extends classical holomorphicity to higher dimensions using complex, hypercomplex, and alternative algebra settings.
• It factors the Laplacian and underpins integral representation formulas, providing explicit solution operators for boundary-value and evolution PDE systems.
• Its applications span complex geometry, hypercomplex analysis, and nonlinear evolution equations, offering rigorous tools for PDE analysis and cohomological methods.

The multidimensional Cauchy-Riemann operator comprises a central framework for analytic function theory in several complex variables, hypercomplex analysis, and related areas. Across various algebraic and geometric settings, it generates a rich structure of partial differential equations governing generalizations of holomorphicity, slice-regularity, and monogenicity, establishes integral representation formulas, and underpins the analysis of boundary-value and evolution equations on complex and hypercomplex domains.

1. Abstract Definition and Algebraic Frameworks

The classical Cauchy-Riemann operator in one complex variable is generalized to higher-dimensional settings as follows:

• Complex case (ℂⁿ): The operator $\overline{\partial}$ acts on smooth (p,q)-forms by

$$\overline{\partial} = \sum_{j=1}^n d\bar{z}_j \wedge \frac{\partial}{\partial \bar{z}_j}.$$

It satisfies $\overline{\partial}^2=0$, producing the Dolbeault complex relevant in several complex variables and complex geometry (Alexander et al., 4 Dec 2025, Laurent-Thiébaut, 2013).

• Real alternative *-algebras: Given an alternative *-algebra $A$ and a hypercomplex subspace $$M = \mathrm{span}_{\mathbb{R}}\{1, v_1, \dots, v_m\}$$, one defines

$$\overline{\partial}_g = \frac{\partial}{\partial x_0} + \sum_{i=1}^m v_i \frac{\partial}{\partial x_i}, \qquad \partial_g = \frac{\partial}{\partial x_0} - \sum_{i=1}^m v_i \frac{\partial}{\partial x_i}$$

on functions $f: M \to A$ (Perotti, 2022).

• Octonionic and Clifford settings: Higher-dimensional analogues involve Dirac-type operators,

$$\partial_x = \partial_{x_0} + e_1\partial_{x_1} + \cdots + e_7\partial_{x_7}$$

for the octonionic case, with alternativity and non-associativity dictating the structure of the corresponding Cauchy-Riemann system (Kauhanen et al., 2017).

• Bicomplex and fractional settings: The bicomplex algebra $$\mathrm{BC} = \{z_1 + z_2 j: z_1, z_2\in \mathbb{C}\}$$ with two commuting imaginary units $i^2 = j^2 = -1$, and $k := ij$ ($k^2 = 1$), allows for a further generalization involving fractional calculus and weighted operators (González-Cervantes et al., 2023).

2. Analytical and PDE Properties

• Factorization of the Laplacian: In all settings above, pairs of first-order Cauchy-Riemann-type operators factor the scalar Laplacian:

$$\partial_g\,\overline{\partial}_g f = \overline{\partial}_g\,\partial_g f = \Delta_g f,$$

where $\Delta_g$ is the (real) Laplacian on $M$ (Perotti, 2022). Analogously, in the octonionic context, $\partial_x \bar{\partial}_x = \Delta_8$ in $\mathbb{R}^8$ (Kauhanen et al., 2017).

• Dolbeault complex: In $\mathbb{C}^n$, the $\overline{\partial}$-operator generates the sequence

$$0 \to \Lambda^{0,0} \xrightarrow{\overline{\partial}^0} \Lambda^{0,1} \to \cdots \to \Lambda^{0, n} \to 0,$$

with $$\overline{\partial}^{q+1} \overline{\partial}^q = 0$$. Cohomology of this complex yields the Dolbeault cohomology groups $H^{p,q}(X, E)$ (Laurent-Thiébaut, 2013, Alexander et al., 4 Dec 2025).

• Slice-regularity: On alternative *-algebras, functions satisfying $\overline{\partial}_g f = 0$ are "slice-regular". In this context, these functions are also polyharmonic, i.e., $\Delta_g^{(m+1)/2} f = 0$ for $m$ odd (Perotti, 2022).

3. Operator Extensions and Integral Representation Formulas

• Fractional and weighted Cauchy-Riemann operators: In bicomplex analysis, the multidimensional Cauchy-Riemann operator is generalized to a proportional fractional $(\vartheta,\varphi)$-weighted form using Riemann–Liouville fractional calculus with respect to hyperbolic-valued functions $\varphi$. This leads to two key operators:

$$D_{a+;\varphi}^{\alpha, +}F(Z,W) := (1-\vartheta)\, \partial (I_{a+;\varphi}^{1-\vartheta}F)(Z,W) + \varphi(Z)^{-1} \partial (I_{a+;\varphi}^{1-\vartheta}F)(Z,W),$$

with a corresponding left-hand variant (González-Cervantes et al., 2023).

• Borel–Pompeiu formula and Borel–Pompeiu kernels: Such generalizations admit a multidimensional fractional Borel–Pompeiu representation, establishing integral formulas for solutions:

$$F(W) = \int_{\partial\Omega} E_{(\vartheta,\varphi)}(V, Z)\,(I_{a+;\varphi}^{1-\vartheta} F)(V, W)\, d p(V) - \int_{\Omega} E_{(\vartheta,\varphi)}(V, Z)\,(D_{a+;\varphi}^{\alpha, +} F)(V, W)\, dV\, dV^* - R_{(\vartheta,\varphi)}(Z, W)$$

(González-Cervantes et al., 2023).

• Explicit solution operators for $\overline{\partial}$: On Cartesian product domains $$D = D_1 \times \ldots \times D_k \subset \mathbb{C}^n$$, explicit solution operators for the equation $\overline{\partial} u = f$ are constructed using iterated partial Cauchy kernels and solid Cauchy integrals, subject to minimal regularity and geometry assumptions (Chen et al., 2019).

4. Functional Analytic and Harmonic Properties

• Banach and Sobolev Spaces: The multidimensional Cauchy-Riemann and Dolbeault complexes extend naturally to $L^p$, Sobolev, and Bochner-Sobolev spaces. For example, the velocity space for the $\overline{\partial}$-Navier–Stokes system in $\mathbb{C}^n$ is given by

$B_{\mathrm{vel}, q}^{k, 2s, s},$

characterized by time and spatial regularity constraints (Alexander et al., 4 Dec 2025).

• $L^p$ and Andreotti-Grauert theory: The $L^p$-theory encompasses local solvability, support control, vanishing theorems, and duality, crucial for both classical and weak boundary value problems in multidimensional analytic function theory (Laurent-Thiébaut, 2013).
• Boundary-value problems and fractional equations: Solutions to boundary-value problems for fractional Cauchy-Riemann operators are given by explicit Borel-Pompeiu integrals, with uniqueness resulting from the integral and harmonic structure of the operator (González-Cervantes et al., 2023).
• Polyharmonicity and Laplacian orders: Slice-regular maps in the alternative algebra context exhibit higher-order harmonicity: for dimension $m+1$, such a function is $(m+1)/2$-polyharmonic when $m$ is odd (Perotti, 2022).

5. Noncommutative and Nonassociative Generalizations

• Octonionic analysis: The extension to the octonion field $\mathbb{O}$ relies on alternativity, Moufang identities, and quaternionic decompositions. The octonionic Cauchy–Riemann operator yields an $8\times 8$ overdetermined real system for $\mathbb{O}$-valued functions, with monogenic functions being precisely null solutions (Kauhanen et al., 2017).
• Slice analysis over real alternative algebras: The generalization to real alternative *-algebras covers both nonassociative and noncommutative structures, where Cauchy–Riemann operators serve as a unifying theme for analyticity, harmonicity, and integrability properties (Perotti, 2022).

6. Applications and Advanced Developments

• Navier–Stokes analogues via $\overline{\partial}$: In $\mathbb{C}^n$, nonlinear evolution systems structurally analogous to the incompressible Navier-Stokes equations are formulated using the multidimensional Cauchy-Riemann operator $\overline{\partial}$, its formal adjoint $\overline{\partial}^*$, and the Dolbeault compatibility complex, allowing the proof of weak and strong solution existence in Bochner-Sobolev spaces (Alexander et al., 4 Dec 2025).
• Product domain methods and regularity theory: Explicit solution schemes for the $\overline{\partial}$ equation on product domains enable sharp norm estimates and extend existing Henkin-Ramírez integral formulas to minimal geometric settings (Chen et al., 2019).
• Fractional and weighted operators in hypercomplex analysis: The multidimensional proportional fractional Cauchy-Riemann operator in the bicomplex setting offers a continuous interpolation between pure integral and pure differential regimes, with flexibility for "steering" fractional differentiation directions via hyperbolic-valued weights (González-Cervantes et al., 2023).
• Boundary-value and extension phenomena: Extensions of Hartogs-type theorems, duality-based solvability criteria, and solution regularity in non-classical settings are established via $L^p$ methods and Banach-complex duality (Laurent-Thiébaut, 2013).

7. Schematic Overview of Operator Variants

Setting	Operator(s)	Algebraic Structure
ℂⁿ (complex)	$\overline{\partial}$	Commutative
ℍ (quaternionic)	Fueter operator	Noncommutative
$\mathbb{O}$ (octonionic)	$\partial_x$	Nonassociative, Alt.
Real alternative *-algebra	$\overline{\partial}_g, \partial_g$	Alternative
Bicomplex, fractional	$D_{a+;\varphi}^{\alpha}$	Commutative, Weighted

Each generalization respects the algebraic properties—commutative, noncommutative, nonassociative, or alternative—of the underlying scalar field or algebra, which in turn dictates the structure of analytic, monogenic, or slice-regular solutions to the resulting PDE system.

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This overview incorporates the principal analytical and algebraic developments surrounding the multidimensional Cauchy-Riemann operator, covering generalizations to complex, hypercomplex, and noncommutative settings, fractional calculus, explicit solution formulas, sophisticated boundary and cohomological problems, and connections to nonlinear evolution equations (González-Cervantes et al., 2023, Kauhanen et al., 2017, Alexander et al., 4 Dec 2025, Perotti, 2022, Laurent-Thiébaut, 2013, Chen et al., 2019).