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

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