On Rad\'o's theorem for polyanalytic functions

TL;DR

This paper extends Radó's theorem to polyanalytic functions in one and several complex variables, establishing conditions under which functions that are polyanalytic outside their zero set are globally polyanalytic.

Contribution

It provides new versions of Radó's theorem for polyanalytic functions, including simple proofs and extensions to higher dimensions and complex convex domains.

Findings

Polyanalytic functions agreeing a.e. with globally polyanalytic functions under certain regularity conditions.

Extension of Radó's theorem to functions satisfying Laplace-type equations.

Simplified proof of polyanalyticity for functions with certain regularity and zero set properties.

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.