Axially Monogenic Functions in Clifford Analysis

• Axially monogenic functions are solutions to a first-order Dirac-type operator on axially symmetric domains, defined by their dependence on the real part and the modulus of the imaginary part.
• They are generated via the Fueter mapping theorem from holomorphic or slice-monogenic functions, effectively bridging classical complex analysis and higher-dimensional Clifford frameworks.
• Their practical applications include noncommutative functional calculi, operator theory, and the resolution of boundary value and spectral problems in quaternionic and Clifford settings.

An axially monogenic function is a Clifford or quaternionic- (or more generally, real normed algebra-) valued solution to a first-order Dirac-type (Cauchy–Riemann–Fueter) operator on an axially symmetric domain, characterized by its dependence only on the modulus and (possibly) direction of the imaginary (vector) part. The class is pivotal for quaternionic and Clifford analysis, extending holomorphic function theory to higher dimensions via the Fueter theorem and its generalizations. Surjective correspondences with one-variable holomorphic and slice-monogenic functions, explicit formulas for functional calculi, and connections to generalized analytic (Vekua-type) systems illustrate their utility in operator theory, boundary value problems, and spectral analysis.

1. Definition, Characterization, and Axial Symmetry

An axially monogenic function is a null solution of the Euclidean Dirac operator (Cauchy–Riemann–Fueter operator), typically written for $q \in \mathbb{H}$ (quaternions) or $x \in \mathbb{R}^{n+1}$ (Clifford algebra context) as

$$\mathcal{D} = \frac{\partial}{\partial x_0} + \sum_{j=1}^n e_j \frac{\partial}{\partial x_j}$$

where the $e_j$ satisfy $e_ie_j + e_je_i = -2\delta_{ij}$, and $x = x_0 + \underline{x}$ is a paravector.

A $C^1$-function $f \colon U \to \mathbb{R}_n$ or $f \colon U \to \mathbb{H}$ on an axially symmetric domain $U$ is (left-)axially monogenic if

$\mathcal{D}f(x) = 0$

and $f$ has the form

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

where $A$, $B$ are real analytic in their arguments. Axial symmetry requires that for all $x \in U$, the entire sphere $x_0 + r S^{n-1}$ is contained in $U$.

Axially monogenic functions are invariant under rotations fixing the "real axis": $$f(x_0, R \underline{x}) = A(x_0, | \underline{x} | ) + R\omega B(x_0, | \underline{x} |), \quad R \in SO(n)$$ so they are solutions respecting a physically or geometrically meaningful symmetry (Dong et al., 2018, Martino et al., 2022, Peña et al., 2014).

2. The Fueter Mapping, Slice Monogenic Extension, and Surjectivity

The Fueter mapping theorem provides an explicit method to generate axially monogenic functions from holomorphic functions in one complex variable. For $f_0(z) = u(x_0,r) + i v(x_0,r)$ holomorphic, consider the slice extension (intrinsic slice-hyperholomorphic function)

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

Then apply the fractional Laplacian (in Clifford analysis, $(-\Delta)^{(n-1)/2}$, defined pointwise for odd $n$ and as a Fourier multiplier for even $n$), yielding

$\beta(f_0)(x) = (-\Delta)^{(n-1)/2}[\vec{f}_0(x)]$

The resulting image is axially monogenic by explicit computation. For every axially monogenic function, the surjectivity theorem guarantees the existence of a unique holomorphic $f_0$ such that $f = \beta(f_0)$ (Dong et al., 2018, Martino et al., 2022).

This yields the commutative diagram: $$O(D) \xrightarrow{T_F} SH(\mathcal{D}) \xrightarrow{(-\Delta)^{(n-1)/2}} AM(\mathcal{D})$$ where $SH$ denotes slice-hyperholomorphic functions, and $AM$ axially monogenic functions.

3. Functional Calculus, Integral Representation, and Operator Theory

Axially monogenic functions underpin the development of noncommutative functional calculi for quaternionic or Clifford operators. For a bounded quaternionic operator $T = T_0 + \sum e_j T_j$ with joint commutative $S$-spectrum $\sigma_S(T)$, the polyanalytic functional calculus of order $2$ is given by

$$f^\circ(T) = \frac{1}{2\pi} \int_{\partial ( U \cap \mathbb{C}_J )} P_L(s, T)\, ds_J\, f(s)$$

where $f \in SH_L( \text{domain}(T) )$, $ds_J = ds \cdot (-J)$, and $P_L(s,T)$ is the left polyanalytic resolvent kernel: $$P_L(s,T) = \sum_{j=0}^{1} (-1)^{j+1} F_L(s,T) s^{1-j}$$ with $F_L(s,T)$ as in section 4 below.

Importantly, this calculus is independent of the choice of imaginary unit $J$ and the domain $U$ containing $\sigma_S(T)$ (Martino et al., 2022).

4. Axially Polyanalytic and Two-Sided Monogenic Functions

The Laplacian admits the factorization $\Delta = \overline{\mathcal{D}} \mathcal{D}$, which motivates considering null solutions of higher powers $\mathcal{D}^2 f = 0$ (order-2 axially polyanalytic functions), intermediate between slice-hyperholomorphic and axially monogenic functions: $$SH(\mathcal{D}) \xrightarrow{\mathcal{D}} AP_2(\mathcal{D}) \xrightarrow{\mathcal{D}} AM(\mathcal{D})$$ A function $f$ of the form $f(q) = a(u,v) + J b(u,v)$, $q = u + Jv$, is axially polyanalytic of order $2$ if $\mathcal{D}^2 f(q) = 0$.

The integral representation (De Martino–Pinton, Thm 4.4) for a left slice-hyperholomorphic $f$ is

$$f^{\circ}(q) = \frac{1}{2\pi} \sum_{k=0}^1 (-1)^k \int_{\partial ( U \cap \mathbb{C}_J )} F_L(s, q) s^{1-k} ds_J f(s)$$

with the $F$-kernel: $$F_L(s, q) = -4 (s-q) (s^2 - 2 \text{Re}(q) s + |q|^2)^{-2}$$ (Martino et al., 2022), and similarly for the right version.

Two-sided monogenic functions solve both $\mathcal{D}f = 0$ and $f \mathcal{D} = 0$, and can be represented as

$$F(x) = A(x_0, r) P_{k,l}(\underline{x}) + B(x_0, r)\, \underline{x} P_{k,l}(\underline{x}) + B(x_0, r) P_{k,l}(\underline{x})\, \underline{x} + D(x_0, r) \underline{x} P_{k,l}(\underline{x})\, \underline{x}$$

with $(A,B,D)$ subject to a Vekua-type system; solutions involve Bessel functions $J_\nu, Y_\nu$ and separability in $(x_0,r)$ (Peña et al., 2010, Peña et al., 2016). These functions feature in explicit polynomial and plane-wave integral constructions.

5. Analytic, Spectral, and Boundary Value Properties

The Vekua system satisfied by $A,B$ for axially monogenic functions is

$$\begin{cases} \partial_{x_0} A - \partial_r B = \frac{n-1}{r} B \ \partial_{x_0} B + \partial_r A = 0 \end{cases}$$

This ties axially monogenic functions to generalized analytic functions and permits reductions of certain boundary value problems for the Dirac operator to classical Riemann–Hilbert or Schwarz-type problems in the complex plane (Huang et al., 2022). The boundary data map to Robin-type or integral equations, solvable by Cauchy integrals with Schwarz factors and vanishing moment conditions in the index (Huang et al., 2022).

In quaternionic operator theory, the Fueter mapping relates eigenfunctions of slice derivatives to axially monogenic eigenfunctions of $\overline D$; e.g.,

$\overline D f = \lambda f$

arises from $\partial_s f = \lambda f$ via $f_{monogenic} = \Delta_4 f_{slice-regular}$, leading to representations of solutions for time-harmonic Helmholtz and stationary Klein–Gordon equations in terms of series in monogenic polynomials (Krausshar et al., 2021).

6. Series Expansions, Clifford–Appell Polynomials, and Function Spaces

Monogenic and axially monogenic functions admit series expansions in Clifford–Appell polynomials $P_k(x)$, satisfying

$$\partial_x P_k(x) = k P_{k-1}(x), \quad P_k(x_0,0) = x_0^k$$

and expressing generalized Taylor expansions (Martino et al., 2023). The Fueter–Sce map and generalized CK-extension coincide on elementary functions, mapping $e^x$ to a series in $P_k(x)$.

Weighted Hardy, Fock, and Bergman modules (Hilbert spaces) are defined for axially monogenic functions:

• Fock space: $\sum_{k=0}^\infty k! |a_k|^2 < \infty$ for $f(x) = \sum_{k=0}^\infty P_k(x) a_k$
• Hardy space: $\sum_{k=0}^\infty |a_k|^2 < \infty$ for unit ball expansion

The kernel and range of the Fueter–Sce map are explicitly described: $$\ker( (-\Delta)^{(n-1)/2} ) = \left\{ f(x) = \sum_{k=0}^{n-2} x^k A_k \right\}$$ while its image covers axially monogenic functions with suitable Hilbert norm parameters. The $\odot$-product, defined via CK-extension, renders Clifford–Appell polynomials a closed algebraic system: $P_k \odot P_\ell = P_{k+\ell}$ (Martino et al., 2023).

7. Harmonic Intermediates, CK Extensions, and Integral Methods

Axially harmonic functions form an intermediate class $\mathrm{AH}(\Omega)$, satisfying $\Delta_{x_0,\underline{x}} u = 0$ and admitting power series and Bessel-function representations for initial boundary data (Martino et al., 21 Jan 2025). The generalized harmonic Cauchy–Kovalevskaya extension constructs $u$ with prescribed values and derivatives at the origin, allowing decomposition into plane-wave integrals over the sphere and explicit bases for Riesz potentials. This process relates slice monogenic, axially harmonic, and axially monogenic classes via factorization of the Laplacian and operator calculus; the commutative diagram is

$$\mathrm{SM}(\Omega) \xrightarrow{\Delta^{\tfrac{m-3}{2}}\mathcal{D}} \mathrm{AH}(\Omega) \xrightarrow{\overline{\mathcal{D}}} \mathrm{AM}(\Omega)$$

Integral representations for monogenic functions via plane-wave and Funk–Hecke formulae allow explicit construction of polynomial and exponential-type solutions (Martino et al., 21 Jan 2025, Peña et al., 2016, Peña et al., 2014).

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The axially monogenic framework, centered on functional calculi, integral formulas, basis polynomials, and operator-theoretic mapping theorems, underlies advanced results in Clifford analysis, quaternionic operator theory, boundary value and spectral theory, and generalizations to polyanalytic and harmonic intermediates.