On Radó's theorem for polyanalytic functions

Abstract

We prove versions of Radó'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 $p^{\prime }$-Laplace equation ($p^{\prime \prime }$-Laplace equation respectively) on $\Omega \setminus g^{-1}(0)$, for some $p^{\prime },p^{\prime \prime }>1$. 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.