Slice Fueter-Regular Functions

Abstract

Slice Fueter-regular functions, originally called slice Dirac-regular functions, are generalized holomorphic functions defined over the octonion algebra ${\mathbb {O}}$, recently introduced by M. Jin, G. Ren and I. Sabadini. A function $f:\Omega _D\subset {\mathbb {O}}\rightarrow {\mathbb {O}}$ is called (quaternionic) slice Fueter-regular if, given any quaternionic subalgebra ${\mathbb {H}}_{\mathbb {I}}$ of ${\mathbb {O}}$ generated by a pair ${\mathbb {I}}=(I,J)$ of orthogonal imaginary units I and J (${\mathbb {H}}_{\mathbb {I}}$ is a ‘quaternionic slice’ of ${\mathbb {O}}$), the restriction of f to ${\mathbb {H}}_{\mathbb {I}}$ belongs to the kernel of the corresponding Cauchy–Riemann–Fueter operator $\frac{\partial }{\partial x_0}+I\frac{\partial }{\partial x_1}+J\frac{\partial }{\partial x_2}+(IJ)\frac{\partial }{\partial x_3}$. The goal of this paper is to show that slice Fueter-regular functions are standard (complex) slice functions, whose stem functions satisfy a Vekua system having exactly the same form of the one characterizing axially monogenic functions of degree zero. The mentioned standard sliceness of slice Fueter-regular functions is able to reveal their ‘holomorphic nature’: slice Fueter-regular functions have Cauchy integral formulas, Taylor and Laurent series expansions, and a version of Maximum Modulus Principle, and each of these properties is global in the sense that it is true on genuine 8-dimesional domains of ${\mathbb {O}}$. Slice Fueter-regular functions are real analytic. Furthermore, we introduce the global concepts of spherical Dirac operator $\Gamma$ and of slice Fueter operator ${\overline{\vartheta }}_F$ over octonions, which allow to characterize the slice Fueter-regular functions as the ${\mathscr {C}}^2$-functions in the kernel of ${\overline{\vartheta }}_F$ satisfying a second order differential system associated with $\Gamma$.