Slice regular holomorphic Cliffordian functions of order $k$

TL;DR

The article analyzes holomorphic Cliffordian functions of order $k$ in odd Clifford algebras by examining the kernel of $\overline{\partial}\Delta^{k}_{m+1}$ on slice regular functions. It proves a sharp dichotomy: for $k<\gamma_m=(m-1)/2$, the kernel consists precisely of slice regular polynomials of degree at most $2k$, whereas for $k\ge\gamma_m$ the Fueter–Sce theorem implies all slice regular functions are holomorphic Cliffordian of order $k$. This establishes a clear structural boundary between polynomial versus general slice-regular behavior under higher-order differential operators and situates the Fueter–Sce bridge as the threshold where the entire slice-regular class lies in the holomorphic Cliffordian category. The results provide a finite-and-polynomial description of the kernel in the subcritical range and reinforce the unifying role of the Fueter–Sce construction in hypercomplex analysis on paravectors.

Abstract

Holomorphic Cliffordian functions of order $k$ are functions in the kernel of the differential operator $\overline{\partial}Δ^k$. When $\overline{\partial}Δ^k$ is applied to functions defined on the paravector space of some Clifford Algebra $\mathbb{R}_m$ with an odd number of imaginary units, the Fueter-Sce construction establish a critical index $k=\frac{m-1}{2}$ (sometimes called Fueter-Sce exponent) for which the class of slice regular functions is contained in the one of holomorphic Cliffordian functions of order $\frac{m-1}{2}$. In this paper we analyze the case $k<\frac{m-1}{2}$ and we find that the polynomials of degree at most $2k$ are the only slice regular holomorphic Cliffordian functions of order $k$.

Theorems & Definitions (24)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Remark 1
• Proposition 2.1
• Theorem 2.2: Fueter-Sce theorem
• Definition 2.4
• Definition 3.1
• Theorem 3.1
• Remark 2