Slice Hyperholomorphic Functions

• Slice hyperholomorphic functions are quaternionic functions defined on axially symmetric domains that satisfy Cauchy–Riemann equations on every complex slice.
• They generalize classical holomorphic functions to higher-dimensional non-commutative settings, underpinning quaternionic function theory and operator calculus.
• Key applications include solving non-commutative differential equations and advancing research in quantum mechanics, PDEs, and Clifford analysis.

A slice hyperholomorphic function is a quaternionic (or more generally, Clifford algebra–valued) function defined on an axially symmetric domain whose restriction to every complex slice parametrized by an imaginary unit satisfies the Cauchy–Riemann equations. These functions generalize holomorphic functions of one complex variable to higher-dimensional, non-commutative settings, providing the underpinning for quaternionic function theory, operator theory, and extensions to Clifford analysis. The structure, extension properties, Cauchy theory, and associated operator calculus for slice hyperholomorphic functions constitute a rich area of current research (Colombo et al., 2018).

1. Fundamental Definitions and Structure

Let $H$ be the real algebra of quaternions with the standard basis $\{1,e_1,e_2,e_3\}$. The unit sphere of imaginary units is

$$S = \{q = e_1 x_1 + e_2 x_2 + e_3 x_3 : x_1^2 + x_2^2 + x_3^2 = 1\}.$$

Any nonreal $q \in H$ can be written in the form $q = u + jv$ for $u,v \in \mathbb{R}$ and $j \in S$. The associated 2-sphere is $[q]=\{u + I v:I\in S\}$. An open set $U \subset H$ is axially symmetric if with $q \in U$, the entire $[q] \subset U$.

Fixing a two-sided quaternionic Banach space $X$, a function $f:U \to X$ is called a left slice function if

$f(u + j v) := f_0(u, v) + j f_1(u, v)$

with $f_0$, $f_1$ real-differentiable, $X$-valued, and $f_0(u,-v)=f_0(u,v)$, $f_1(u,-v)=-f_1(u,v)$. The function is slice hyperholomorphic (or slice regular) if, in addition, $f_0$, $f_1$ satisfy the Cauchy–Riemann equations: $$\partial_u f_0 - \partial_v f_1 = 0,\quad \partial_v f_0 + \partial_u f_1 = 0.$$ The space of such functions is denoted $\mathrm{SH}_L(U, X)$; right slice hyperholomorphic functions are analogously defined by $f(u+jv) = f_0(u,v) + f_1(u,v)j$.

A core structural fact is the representation formula: $$f(u+jv) = \tfrac12(1 - jI) f(u + Iv) + \tfrac12(1 + jI) f(u - Iv)$$ for any $I \in S$ and $f \in \mathrm{SH}_L(U, X)$. Thus, knowledge of $f$ on one slice determines $f$ everywhere in $U$.

2. The Quaternionic Cauchy Theory and Cauchy Transform

Given an axially symmetric set $U \subset H$ bounded by a piecewise-$C^1$ axially symmetric hypersurface, the slice versions of the Cauchy kernel are given by

$$S_L^{-1}(s, q) := - (q^2 - 2\mathrm{Re}(s)q + |s|^2)^{-1}(q - \bar{s}),$$

which is right slice hyperholomorphic in $s$ and left slice hyperholomorphic in $q$. The central Cauchy formula is: $$f(q) = \frac{1}{2\pi} \int_{\partial U_I} S_L^{-1}(s, q)\, ds_I\, f(s)$$ for $f \in \mathrm{SH}_L(U, X)$ and $q \in U$, independent of $I \in S$ and of homologous deformations of $\partial U_I$.

Given $f:\partial U \to X$ continuous and left-slice, the left Cauchy transform is

$$(T_L f)(p) := \frac{1}{2\pi} \int_{\partial U_I} S_L^{-1}(s, p)\, ds_I\, f(s)$$

for $p \in H \setminus \partial U$, and is left slice hyperholomorphic in $p$ on both $U$ and $H \setminus \overline{U}$. The additive splitting theorem asserts that for $p$ on $U_+ = U$ and $U_- = H\setminus \overline{U}$,

$$f_{+}(p) := (T_L f)(p) \big|_{p \in U_+},\quad f_{-}(p) := - (T_L f)(p) \big|_{p \in U_-}$$

define two slice-hyperholomorphic functions continuous up to the boundary, with $f(s) = f_+(s) + f_-(s)$ for $s \in \partial U$, and this splitting is unique among functions vanishing at infinity.

If $f$ is Hölder continuous on $\partial U$, each $f_\pm$ extends with precise boundary behavior: $\|f_\pm(p)\| \leq C \|f\|_{C^\alpha} d^{\alpha-1}$ for $p$ at distance $d$ from $\partial U$, and both $f_\pm$ are Hölder continuous up to their respective boundaries.

3. Fundamental Solution of the Global Slice Operator

A fundamental operator in quaternionic analysis, biologically tied to slice regularity, is

$$G_L f(q) := |q|^2 \, \partial_{q^c} \partial_q f(q)$$

which, in real coordinates $q = (q_0, \ldots, q_3)$ and $e_j$ the basis, reads

$$G_L f(q) = |q|^2 \sum_{j=0}^3 e_j\, \partial_{q_j} f(q)$$

with $$\partial_q = \frac{1}{2}(\partial_{x_0} + e_1\partial_{x_1} + e_2\partial_{x_2} + e_3\partial_{x_3})$$.

Slice-hyperholomorphic functions lie in the kernel of $G_L$. The fundamental solution is given by $p \mapsto S_L^{-1}(s, p)$, with

$$G_L\left(S_L^{-1}(s, p)\right) = 2\pi I |s|^2 \delta_{p=s}$$

in the sense of distributions (for any $I \in S$), identifying $S_L^{-1}(s, p)$ as the fundamental solution (modulo scalar factors) of $G_L$ in the variable $p$.

A consequence is an explicit solution method for inhomogeneous equations: $$f(p) = \frac{1}{2\pi} \int_U S_L^{-1}(s, p) V(s)\, d\mu(s)$$ yields $G_L f(p) = |p|^2 V(p)$ almost everywhere; for continuous $V$ and a slice domain $U$, slice-continuous global solutions to $G_L f = |q|^2 V$ exist via a sheaf-theoretic argument.

4. Spherical Laurent Expansions and Mittag-Leffler Theorem

The natural “monomials” for expansions around $q_0$ are polynomials in the