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 $\Omega$ be an open subset of $\mathbb{H}$, which intersects $\mathbb{R}$ and is invariant under rotations of $\mathbb{H}$ around $\mathbb{R}$. A function $f:\Omega\to\mathbb{H}$ is slice regular if it is of class $\mathscr{C}^1$ and, for all complex planes $\mathbb{C}_I$ spanned by $1$ and a quaternionic imaginary unit $I$, the restriction $f_I$ of $f$ to $\Omega_I=\Omega\cap\mathbb{C}_I$ satisfies the Cauchy-Riemann equations associated to $I$, i.e., $\overline{\partial}_I f_I=0$ on $\Omega_I$, 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 $f:\Omega\to\mathbb{H}$ is called slice polyanalytic of order $n$ if it is of class $\mathscr{C}^n$ and $\overline{\partial}_I^{\,n} f_I=0$ on $\Omega_I$ for all $I$. We define global slice polyanalytic functions of order $n$ as the functions $f:\Omega\to\mathbb{H}$, 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 $n\geq2$, 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 $\Omega_I$ and as solutions of their global version $\overline{\vartheta}^nf=0$ on $\Omega\setminus\mathbb{R}$. Our quaternionic results extend to the monogenic case.