On Rad\'o's theorem for polyanalytic functions

Abstract

We prove versions of Rad\'o's theorem for polyanalytic functions in one variable and also on simply connected $\mathbb{C}$-convex domains in $\mathbb{C}^n$. Let $\Omega\subset \mathbb{C}$ be a bounded, simply connected domain and let $q\in \mathbb{Z}_+.$ Suppose at least one of the following conditions holds true: (i) $g\in C^{q}(\Omega).$ (ii) $g\in C^\kappa(\Omega),$ for $\kappa=\min\{1,q-1\},$ such that $g$ is $q$-analytic on $\Omega\setminus g^{-1}(0)$ and such that Re$g$ (Im$g$ respectively) is a solutions to the $p'$-Laplace equation ($p''$-Laplace equation respectively) on $\Omega\setminus g^{-1}(0)$, for some $p',p''>1$. Then $g$ agrees (Lebesgue) a.e.\ with a function that is $q$-analytic on $\Omega.$ In the process we give a simple proof of the fact that: If $f\in C^q(\Omega)$ is $q$-analytic on $\Omega\setminus f^{-1}(0)$ then $f$ is $q$-analytic on $\Omega.$ The extensions of the results to several complex variables are straightforward using known techniques.