Algebra of slice regular functions
on non-symmetric domains in several quaternionic variables

Slice analysis extends the theory of holomorphic functions in one complex variable to higher dimensions, offering a bridge to multifaceted mathematical disciplines. Initially introduced in the quaternions context by Gentili and Struppa in 2006 [Gentili2006001, Gentili2007001], this theory has since expanded, revealing that power series within non-commutative quaternion algebras qualify as slice regular functions, thus broadening the scope of slice analysis.

For those interested in the diverse applications and advancements in slice analysis, we recommend various sources: for geometric function theory, see [Ren2017001, Ren2017002, Wang2017001]; for quaternionic Schur analysis, refer to [Alpay2012001]; for insights into quaternionic operator theory, consult [Alpay2015001, MR3887616, MR3967697, Gantner2020001, MR4496722]. Further extensions of slice analysis include its application to real Clifford algebras [Colombo2009002], octonions [Gentili2010001], real alternative $*$-algebras [Ghiloni2011001], and $2n$-dimensional Euclidean spaces [Dou2023002].

Two principal methodologies have emerged for exploring the several-variable aspect of slice analysis, both originating from the theories in one variable. The first, strongly slice regular functions, focuses on real alternative $*$-algebras in one variable [Ghiloni2011001] and is characterized by its reliance on intrinsic stem functions and its applicability to axially symmetric domains. For several-variable extensions, see [Colombo2012002] for the quaternionic context, and [Ghiloni2012001] for real Clifford algebras and [Ghiloni2020001] for real alternative $*$-algebras. The second approach, weakly slice regular functions, involves functions satisfying the Cauchy-Riemann equations within each complex plane slice, a concept originating from the initial definition of slice regular functions [Gentili2007001]. Recent advancements in this area, such as the introduction of slice topology, have facilitated the study of non-axially symmetric domains, leading to comprehensive global theories [Dou2023001]. For a broader understanding, refer to [Dou2023001, Dou2021001] for general settings, and [Dou2021001] for the octonions context.

This paper delves into the algebraic structure of weakly slice regular functions across multiple quaternionic variables, with a focus on identifying a suitable multiplication method that maintains slice regularity. Historically, two approaches to defining the $*$-product have been proposed: one based on the unique slice regular extension of the $*$-product between restricted functions on specific complex planes [Gentili2008001, Colombo2009001], and another defined via a corresponding multiplication on their stem functions [Ghiloni2011001]. Both approaches, traditionally confined to axially symmetric slice domains, are equivalent in such contexts. However, the conventional extension theorem [Colombo2009001] proves insufficient for non-axially symmetric slice domains. Consequently, we propose utilizing the second method for defining the $*$-product, fulfilling the necessary conditions. In [Dou2023003], we introduced a $*$-product that preserves the path-slice property, thus encompassing $*$-products of slice regular functions, as each slice regular function on a slice-open set is inherently path-slice. It’s important to note that the criteria for a function to qualify as a stem function of a slice function are straightforward in axially symmetric slice domains. In these domains, the stem function is uniquely representable through the slice function, as per the representation formula. However, this becomes significantly more complex in non-axially symmetric slice domains. We addressed this challenge in [Dou2023003], offering auxiliary conditions to facilitate the process.

The primary focus of this paper is the slice regularity of the $*$-product of weakly slice regular functions. Central to this discussion is the holomorphy of stem functions. To apply differential operators to stem functions, we define an appropriate open neighborhood of a path $\gamma$ in the path space $\mathscr{P}(\mathbb{C}^{n})$. In weak slice analysis, the corresponding stem function of a path-slice function is defined along paths, and the compatibility of stem function holomorphy with weak slice regular functions is a crucial consideration.

The paper is structured as follows: Section 2 revisits key results from [Dou2023003], including the relationship between path-slice functions and their stem functions, and the $*$-product between path-slice functions. Section 3 introduces holomorphic stem functions, defining a suitable open neighborhood for paths in the path space $\mathscr{P}(\mathbb{C}^{n})$ and establishing the holomorphy of stem functions corresponding to slice regular functions. Finally, Section 4 demonstrates that the $*$-product of two slice regular functions retains slice regularity, thereby confirming that the space of weakly slice regular functions, when equipped with the $*$-product, forms an algebra.

This section revisits certain key concepts in the study of slice regular functions and elaborates on the $*$-product of path-slice functions as presented in [Dou2023003]. The quaternion algebra is denoted by $\mathbb{H}$. The set of imaginary units of quaternions is represented as:

For $n\in\mathbb{N}$, the $n$-dimensional weakly slice cone is defined as:

with slice topology

where $\mathbb{C}_{I}=\mathbb{R}+\mathbb{R}I$ and $\mathbb{C}_{I}^{n}=(\mathbb{C}_{I})^{n}$

Open sets, connected sets, and paths in $\tau_{s}$ are termed slice-open sets, slice-connected sets, and slice-paths in $\Omega$, respectively.

For $I\in\mathbb{S}$, define:

The set of continuous paths with real-valued start points in $\mathbb{C}^{n}$ is denoted as:

For a subset $\Omega\subset\mathbb{H}_{s}^{n}$, we define:

where $\delta^{I}:=\Psi_{i}^{I}(\delta)$ is a path in $\Omega_{I}$. For a fixed path $\gamma\in\mathscr{P}(\mathbb{C}^{n})$, we define:

According to [Dou2023002, Corollary 5.9], weakly slice regular functions qualify as path-slice.

Given a subset $\Omega\subset\mathbb{H}_{s}^{n}$, denote

Based on [1, Proposition 3.7], the subsequent definition is deemed well-defined:

Define the function $\mathfrak{I}:\mathbb{H}_{s}^{n}\rightarrow\mathbb{S}\cup{0}$ for $q=(q_{1},...,q_{n})\in\mathbb{H}_{s}^{n}$ by

where $\imath\in\{1,...,n\}$ is the smallest index such that $q_{\imath}\notin\mathbb{R}$.

From [Dou2023003, Proposition 3.11], we define:

For $p:=(p_{1},p_{2})^{T},q:=(q_{1},q_{2})^{T}\in\mathbb{H}^{2\times 1}$, denote the product $p*q$ as

where $\mathbb{I}$, $\sigma$, and $e_{1}$ are defined as

For functions $F,G:\mathscr{P}(\mathbb{C}^{n},\Omega)\rightarrow\mathbb{H}^{2\times 1}$ for a domain $\Omega\subset\mathbb{H}_{s}^{n}$, define

In this section, we explore the concept of holomorphy for path-dependent stem functions.

We discover that a path-slice stem function $F$ derived from a path-slice function $f$ is not inherently holomorphic. However, the specific restriction $F_{\Omega_{1}}^{f}$ of $F$ exhibits holomorphy.

For any two points $z,w$ in the complex space $\mathbb{C}^{n}$, we define a standard path connecting $z$ to $w$. This is a function $\mathcal{L}_{z}^{w}:[0,1]\to\mathbb{C}^{n}$ given by the formula

For a path $\gamma\in\mathscr{P}(\mathbb{C}^{n})$ and a positive radius $r$, $B_{\mathbb{C}^{n}}(\gamma(1),r)$ represents a ball in $\mathbb{C}^{n}$ centered at the endpoint of the path $\gamma(1)$ with radius $r$.

The ball in the path space $\mathscr{P}(\mathbb{C}^{n})$ is defined as the set of all paths one can form by taking the standard path $\mathcal{L}_{\gamma(1)}^{z}$ from the end point of $\gamma$ to the point $z$ inside the ball $B_{\mathbb{C}^{n}}(\gamma(1),r)$ in $\mathbb{C}^{n}$. Essentially, this process creates new paths in $\mathscr{P}(\mathbb{C}^{n})$ by extending $\gamma$ to reach any point within a radius $r$ from its endpoint, and each of these new paths starts from the origin of $\gamma$ and ends at some point within this spherical region in $\mathbb{C}^{n}$. This concept is mathematically expressed as

Let $\Omega\subset\mathbb{H}_{s}^{n}$ and $\gamma\in\mathscr{P}(\mathbb{C}^{n})$. We consider a subset $\mathbb{S}^{\prime}$ of $\mathbb{S}(\Omega,\gamma)$, which could be specific selections or categories within a larger set that are related to the region $\Omega$ and the path $\gamma$.

We need to introduce some concepts which are about finding the largest possible balls in different spaces, and how these balls fit within certain regions.

• •

  $r_{\gamma,\Omega}$: This represents the supremum of radii $r$ for which the ball in the path space centered at $\gamma$ with radius $r$ is entirely contained within the path space related to $\Omega$. Geometrically, this is finding the largest possible radius of a ball in the path space, centered at $\gamma$, that still fits entirely within the region defined by $\Omega$.

• •

  $r_{\gamma,\scriptscriptstyle\Omega}$: This is the supremum of radii $r$ for which the ball centered at the endpoint of $\gamma$ (in the image space of $\gamma$) with radius $r$ is contained within $\Omega_{I}$ for all selections $I$ in $\mathbb{S}^{\prime}$. Geometrically, it is about finding the largest radius of a ball in the image space that fits within the regions $\Omega_{I}$ for every category in $\mathbb{S}^{\prime}$.

• •

  $r_{\gamma,\scriptscriptstyle\Omega}^{2}$: This represents the supremum of $r_{\gamma,\scriptscriptstyle\Omega}^{\mathbb{S}^{\prime\prime}}$ over all subsets $\mathbb{S}^{{}^{\prime\prime}}$ of $\mathbb{S}(\Omega,\gamma)$ that have at least two elements. Geometrically, this is finding the largest ball (in terms of radius) that fits within the $\Omega_{I}$ regions for every possible subset of $\mathbb{S}(\Omega,\gamma)$ that contains two or more elements.

These concepts can be expressed as

Under certain conditions, these constants remain positive, which proves to be valuable for practical applications.

For any $\gamma\in\mathscr{P}(\mathbb{C}^{n})$ and $r>0$, we introduce a map

defined by

The transformation $\gamma\circ\mathcal{L}_{\gamma(1)}^{z}$ geometrically represents the extension of the original path to reach the point $z$. It illustrates how a point-wise approach (selecting a point $z$) can lead to a new understanding or interpretation in a path-dependent framework (creating a new path in $\mathscr{P}(\mathbb{C}^{n})$). This highlights the interconnectedness and complementary nature of point-wise and path-dependent perspectives in slice analysis.

We are now in a position to define the notion of a function being holomorphic over a space of paths.

In this section, we show that the $*$-product of slice regular functions is still slice regular. It implies that the class of slice regular functions is an associative unitary real algebra. For this purpose, we will require the assistance of several results.

Let $\Omega\subset\mathbb{H}_{s}^{n}$, $\gamma\in\mathscr{P}(\mathbb{C}^{n},\Omega)$ and $I\in\mathbb{S}(\Omega,\gamma)$. Denote

Geometrically, it represents the largest radius $r$ for which a ball $B_{I}(\gamma^{I}(1),r)$, centered at the point $\gamma^{I}(1)$ in the slice $I$ of $\Omega$, remains entirely inside the subset $\Omega_{I}$ of $\Omega$ associated with the slice $I$. In simpler terms, it is the largest ball you can draw around the endpoint of the path $\gamma^{I}$ without leaving the space $\Omega_{I}$.

Let $c\in\mathbb{H}$ and $I\in\mathbb{S}$. Then