[Paper Review] Cullen-regular quaternionic functions in a Fueter operator framework

Daniel Alayón-Solarz|ArXiv.org|May 2, 2008

Algebraic and Geometric Analysis7 references3 citations

TL;DR

This paper establishes a Fueter operator framework to characterize Cullen-regular quaternionic functions, showing equivalence between Cullen-regularity and a differential identity involving the Fueter operator and the imaginary unit $ι$. It introduces a generalized notion of Cullen-regularity that extends beyond $C^1$ functions, enabling integral theorems for continuous functions with angular derivatives.

ABSTRACT

We show characterizations of the class of Cullen-regular functions in the sense of Gentili-Struppa for any domain $Ω$ in terms of the Fueter operator. We then state a Integral Theorem and discuss how it can be used to define a more general version of Cullen-regularity, that does not require the function to be of class $C^{1}$.

Motivation & Objective

• To provide a characterization of Cullen-regular functions using the left-Fueter operator in a spherical coordinate system.
• To extend the concept of Cullen-regularity beyond $C^1$ functions by leveraging angular derivatives and integral identities.
• To establish a generalized integral theorem for continuous functions that are $C^1$ in angular variables $\alpha$ and $\beta$, even when not $C^1$ in the standard sense.
• To define a broader class of Cullen-regular functions based on the validity of the integral theorem and its dual for $\iota f$.

Proposed method

• Express the left-Fueter operator $D_l$ in spherical coordinates $(t, r, \alpha, \beta)$, decomposing it into radial and angular components involving $\iota$ and $\partial/\partial_\iota$.
• Derive the key identity $D_l\iota f + \iota D_l f = -2f/r$ in $\Omega \setminus \mathbb{R}$, which characterizes left-Cullen-regular functions.
• Define $u = \frac{1}{2} \partial_\iota(\iota f)$ and $v = \frac{1}{2} \partial_\iota(f)$ to decompose $f = u + \iota v$, linking regularity to angular derivatives.
• Use Gauss’s theorem to derive an integral identity: $\int_K n(p)f(p)\,dS_K = \int_{K^*} (-2v/r)\,dV$ for hypersurfaces $K$ disjoint from the real axis.
• Generalize Cullen-regularity to continuous functions that are $C^1$ in $\alpha$ and $\beta$, provided the integral theorem and its $\iota f$ counterpart hold.
• Establish conditions under which hyperholomorphic functions (a subclass of Cullen-regular functions) preserve regularity under right-multiplication and satisfy Fueter’s theorem without power series assumptions.

Experimental results

Research questions

• RQ1 How can Cullen-regular functions be characterized using the Fueter operator in spherical coordinates?
• RQ2 What differential identity involving $D_l$, $\iota$, and $f$ is equivalent to left-Cullen-regularity in $\Omega \setminus \mathbb{R}$?
• RQ3 Can the concept of Cullen-regularity be extended to non-$C^1$ functions using angular derivatives and integral theorems?
• RQ4 Under what conditions does a generalized Cullen-regular function satisfy Fueter’s theorem $D_l\Delta f = 0$ without requiring power series representations?
• RQ5 How do the real-valued components $u$ and $v$ of $f = u + \iota v$ constrain the structure of hyperholomorphic functions?

Key findings

• Cullen-regular functions are characterized by the identity $D_l\iota f + \iota D_l f = -2f/r$ in $\Omega \setminus \mathbb{R}$, which is equivalent to standard $C^1$ left-Cullen-regularity.
• The decomposition $f = u + \iota v$ with $u = \frac{1}{2} \partial_\iota(\iota f)$, $v = \frac{1}{2} \partial_\iota(f)$ yields $D_l f = -2v/r$ and $D_l(\iota f) = -2u/r$ in $\Omega \setminus \mathbb{R}$.
• The generalized integral theorem $\int_K n(p)f(p)\,dS_K = \int_{K^*} (-2v/r)\,dV$ holds for continuous functions that are $C^1$ in $\alpha$ and $\beta$, even if not $C^1$ in $t,x,y,z$.
• The generalized Cullen-regularity condition is defined by the validity of the integral theorem and its $\iota f$ counterpart for all smooth, simple closed hypersurfaces disjoint from the real axis.
• Hyperholomorphic functions (where $u$ and $v$ satisfy the angular equations (1) and (2)) are preserved under right-multiplication by quaternionic constants and include power series with right quaternionic coefficients.
• If a hyperholomorphic function $f$ is $C^3$, then $D_l\Delta f = 0$ holds without requiring power series or Fueter’s original hypotheses.