Fueter-Sce Theorem in Clifford Analysis

• Fueter-Sce Theorem is a fundamental result in Clifford analysis that converts intrinsic holomorphic functions into axially monogenic functions via differential operators.
• It utilizes the Fueter mapping, employing Laplacian operators and Fourier multipliers, to establish structural links between analytic, harmonic, and monogenic function spaces.
• Generalizations such as CK extension and Vekua systems broaden its applications in boundary value problems and mathematical physics while ensuring surjectivity on axially symmetric domains.

The Fueter-Sce Theorem is a cornerstone in Clifford analysis and higher-dimensional function theory, giving a systematic procedure for generating axially monogenic functions in $\mathbb{R}^{n+1}$ from intrinsic holomorphic functions of one complex variable. Its generalizations link analytic, harmonic, and monogenic categories and establish deep structural correspondences between function spaces via explicit differential and integral transforms.

1. Algebraic Preliminaries: Clifford Algebras and Dirac Operators

Let $e_1,\ldots,e_n$ denote generators of the real Clifford algebra $\mathrm{Cl}_{0,n}$, satisfying $e_i^2 = -1$ and $e_ie_j + e_je_i = 0$ ($i \neq j$). A paravector is defined as $x = x_0 + \vec{x}$, with $\vec{x} = \sum_{i=1}^n x_i e_i$, and $|x|^2 = x_0^2 + |\vec{x}|^2$.

The generalized Cauchy–Riemann (Dirac) operator in $\mathbb{R}^{n+1}$ is

$$D = \partial_{x_0} + \sum_{i=1}^n e_i\,\partial_{x_i}$$

A function $$f: \Omega \subset \mathbb{R}^{n+1} \to \mathrm{Cl}_{0,n}$$ is (left-)monogenic iff $Df \equiv 0$ on $\Omega$.

Axially symmetric domains $\Omega$ are invariant under rotations of the $\vec{x}$-variables. A function $f$ is axially monogenic if it is monogenic and admits the representation

$$f(x) = A(x_0,r) + \frac{\vec{x}}{r}\,B(x_0,r),\quad r = |\vec{x}|$$

with $A$, $B$ scalar functions.

2. The Fueter Mapping: Definitions and Differential Operators

Given a holomorphic intrinsic function $f_0: O\to \mathbb{C}$ on an intrinsic domain $O\subset \mathbb{C}$, where $f_0(\bar{z}) = \overline{f_0(z)}$, one forms the slice extension

$$\vec{f}_0(x_0 + \vec{x}) = u(x_0, r) + \frac{\vec{x}}{r} v(x_0, r)$$

from $f_0(s+it) = u(s, t) + iv(s, t)$. The classical Fueter mapping for $n \geq 1$ is

$$\beta(f_0)(x) = (-\Delta)^{\frac{n-1}{2}}\,\vec{f}_0(x)$$

where $(-\Delta)^{\frac{n-1}{2}}$ is a pointwise differential operator for odd $n$ and the Fourier multiplier with symbol $|\xi|^{n-1}$ for even $n$ (Dong et al., 2018).

This construction generates axially monogenic functions, with $\beta(f_0)$ characterized by the axial form.

3. Structural Theorems: Axial Form and Surjectivity

Axial Form Theorem

Statement: For $n\geq 2$ even and intrinsic holomorphic $f_0$,

$$\beta(f_0)(x) = A(x_0, r) + \frac{\vec{x}}{r} B(x_0, r)$$

with $A, B$ scalar functions; thus, $\beta(f_0)$ is axially monogenic and of axial type (Dong et al., 2018).

Proof proceeds via computation on monomials:

• For $0 \leq l \leq n-2$, $(−\Delta)^{(n−1)/2}(x^l) \equiv 0$
• For $l \geq n-1$, $(−\Delta)^{(n−1)/2}(x^l)$ produces functions axially dependent only on $(x_0, r)$

Surjectivity Theorem

For any axially symmetric domain $\Omega$ and any axially monogenic $f(x) = A(x_0,r) + (\vec{x}/r) B(x_0,r)$, there exists a holomorphic intrinsic $f_0$ such that $\beta(f_0) = f$ on $\Omega$. The construction uses Cauchy-type integrals along axial slices with explicit holomorphic kernels (Dong et al., 2018).

4. Action on Monomials and Kelvin Inversion

For monomials $f_0^{(l)}(z) = z^l$, the mapping $\tau$ defined via the Fourier transform and Kelvin inversion coincides with the Fueter mapping:

$$\tau(z^l)(x) \equiv \beta(z^l)(x) = (-\Delta)^{\frac{n-1}{2}}(x^l)$$

For negative $l$, the Fourier multiplier suffices; for large positive $l$, Kelvin inversion is required. This is captured by the Monomial Theorem, demonstrating that on Laurent series, the extended $\tau$ matches the Fueter mapping via differential or Fourier multiplier techniques (Dong et al., 2018).

5. Connection to Generalized CK Extension and Plane Wave Decomposition

There is an explicit link between the Fueter mapping (and its Sce extension) and the generalized Cauchy–Kovalevskaya (CK) extension, which characterizes axially monogenic functions in terms of restrictions to the real line (Martino et al., 2023, Martino et al., 2022).

On intrinsic holomorphic data $f_0$, the correspondence

$$\mathrm{Fueter}(f_0) = c_n\,\mathrm{GCK}[f_0^{(n-1)}]$$

holds for an explicit normalization constant $c_n$.

The harmonic CK extension produces harmonic and monogenic bases via Bessel series and sphere-integrals; polynomials such as Clifford-Appell polynomials serve as building blocks for axially monogenic expansions (Martino et al., 21 Jan 2025).

Plane-wave and dual Radon transforms allow integral representations; CK extension and Fueter mapping fit into commutative diagrams linking function spaces.

6. Vekua Systems, Special Functions, and Explicit Solution Families

Axially monogenic functions correspond to solutions of Vekua-type systems in $(x_0, r)$. For two-sided monogenic functions, the reduction yields systems of the form

$$\begin{cases} \partial_{x_0}A_1 - r\partial_r A_2 = c_1 A_2 \ \partial_{x_0}A_2 + \frac{1}{r}A_1 = c_2 A_3 \ \partial_{x_0}A_2 - r\partial_r A_3 = c_3 A_3 \ \partial_{x_0}A_3 + \frac{1}{r}A_2 = 0 \end{cases}$$

which admit separable solutions in terms of Bessel functions $J_\nu(r)$ and $Y_\nu(r)$, demanding regularity at $r=0$ and boundary conditions for physical applications (Peña et al., 2010).

Cauchy–Kovalevskaya, plane-wave integration (via Funk–Hecke formula), and primitivation methods produce all polynomial two-sided axial monogenics and harmonics (Peña et al., 2016).

7. Functional Calculus, Boundary Problems, and Generalizations

Fueter's theorem admits generalizations to polyanalytic setting (order-$m$), leading to new functional calculi on the $S$-spectrum (Martino et al., 2022). Integral formulas via Fueter kernels (e.g., $F_L(s,q)$), connections to slice regularity, and polyanalytic modules expand the function theory.

Axially symmetric monogenic functions are central in boundary value problems; Riemann–Hilbert problems for axially monogenic data in Clifford modules are reduced to complex analytic RHPs via the Vekua system (Huang et al., 2022).

Further, extensions to biaxial monogenicity—functions invariant under $SO(p)\times SO(q)$—are constructed by composing Fueter’s map and Clifford-Funk–Hecke radialization, yielding explicit solutions in higher product group invariant settings (Peña et al., 2014).

Table: Fundamental Operators and Correspondence

Name	Operator / Construction	Correspondence/Nature
Dirac	$D = \partial_{x_0} + \sum e_i\partial_{x_i}$	Cauchy–Riemann analog
Fueter Mapping	$(-\Delta)^{\frac{n-1}{2}}$ or Fourier multipliers	Intrinsic holomorphic $\to$ axially monogenic
Slice Extension	Series expansion on paravectors	$\mathbb{C}$-analytic $\to$ $\mathbb{R}^{n+1}$ slice monogenic
CK Extension	Iterated directional derivatives, Bessel integrals	Real analytic data $\to$ axially monogenic
Kelvin Inversion	$I$ acting on Fourier image	Relates negative and positive monomials
Funk–Hecke	Radial-integration over spheres	Enforces axial/biaxial symmetry

Impact and Scope

The Fueter-Sce theorem, its CK and Fourier analytic generalizations, and associated operator factorizations establish a comprehensive framework for understanding and building monogenic function spaces in arbitrary dimension, directly generalizing holomorphic function theory. Surjectivity results guarantee that all axially monogenic functions arise via Fueter mapping, and explicit formulas for polynomial and functional bases enable algebraic, analytic, and spectral analysis, with wide applications in boundary value problems, harmonic analysis, and mathematical physics (Dong et al., 2018, Martino et al., 2023, Peña et al., 2010, Martino et al., 2022, Martino et al., 21 Jan 2025).

Subsequent developments—including polyanalytic functional calculi (Martino et al., 2022), Riemann–Hilbert problems (Huang et al., 2022), and further radialization constructs (Peña et al., 2014)—continue to expand the reach and depth of the Fueter mapping in modern analysis and geometry.