Slice-by-slice and global smoothness of slice regular and polyanalytic functions

Abstract

The concept of slice regular function over the real algebra $\mathbb {H}$ of quaternions is a generalization of the notion of holomorphic function of a complex variable. Let $\varOmega \subset \mathbb {H}$ be a domain, i.e., a non-empty connected open subset of $\mathbb {H}=\mathbb {R}^4$. Suppose that $\varOmega$ intersects $\mathbb {R}$ and is invariant under rotations of $\mathbb {H}$ around $\mathbb {R}$. A function $f:\varOmega \rightarrow \mathbb {H}$ is slice regular if it is of class $\mathcal {C}^1$ and, for all complex planes $\mathbb {C}_I$ spanned by 1 and a quaternionic imaginary unit I ($\mathbb {C}_I$ is a ‘complex slice’ of $\mathbb {H}$), the restriction $f_I$ of f to $\varOmega _I=\varOmega \cap \mathbb {C}_I$ satisfies the Cauchy–Riemann equations associated with I, i.e., $\overline{\partial }_If_I=0$ on $\varOmega _I$, where $\overline{\partial }_I=\frac{1}{2}\big (\frac{\partial }{\partial \alpha }+I\frac{\partial }{\partial \beta }\big )$. Given any positive natural number n, a function $f:\varOmega \rightarrow \mathbb {H}$ is called slice polyanalytic of order n if it is of class $\mathcal {C}^n$ and $\overline{\partial }_I^{\,n}f_I=0$ on $\varOmega _I$ for all I. We define global slice polyanalytic functions of order n as the functions $f:\varOmega \rightarrow \mathbb {H}$, which admit a decomposition of the form $f(x)=\sum _{h=0}^{n-1}\overline{x}^hf_h(x)$ for some slice regular functions $f_0,\ldots ,f_{n-1}$. Global slice polyanalytic functions of any order n are slice polyanalytic of the same order n. The converse is not true: for each $n\ge 2$, we give examples of slice polyanalytic functions of order n, which are not global. The aim of this paper is to study the continuity and the differential regularity of slice regular and global slice polyanalytic functions viewed as solutions of the slice-by-slice differential equations $\overline{\partial }_I^{\,n}f_I=0$ on $\varOmega _I$ and as solutions of their global version ${\overline{\vartheta }\,}^nf=0$ on $\varOmega \setminus \mathbb {R}$. Our quaternionic results extend to the slice monogenic case.