A pure smoothness condition for Radó's theorem for $\alpha$-analytic functions

Daghighi, Abtin ; Wikström, Frank

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 1, pp. 57-62.

Résumé

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$ be a bounded, simply connected $\mathbb{C}$-convex domain. Let $\alpha \in {\mathbb{Z}}_{+}^n$ and let $f$ be a function on $\mathrm{\Omega }$ which is separately $C^{2{\alpha }_j-1}$-smooth with respect to $z_j$ (by which we mean jointly $C^{2{\alpha }_j-1}$-smooth with respect to $\mathrm{Re}{z}_j$, $\mathrm{Im}{z}_j$). If $f$ is $\alpha$-analytic on $\mathrm{\Omega }\setminus f^{-1}(0)$, then $f$ is $\alpha$-analytic on $\mathrm{\Omega }$. The result is well-known for the case ${\alpha }_i=1$, $1\le i\le n$, even when $f$ a priori is only known to be continuous.

Détail

DOI : 10.1007/s10587-016-0238-1

Classification : 30C15, 32A99, 32U15, 35G05
Mots-clés : $\alpha$-analytic function; polyanalytic function; zero set; Radó's theorem