Separate real analiticity and CR extendibility

Abstract

In $\C^2=\R^2+i\R^2$ with coordinates $z=(z_1,z_2), z=x+iy$, we consider a function $f$ continuous on a domain $Ω$ of $\R^2$ separately real analytic in $x_1$ and CR extendible to $y_2$ (resp. CR extendible to $y_2>0$). This means that $f(\cdot,x_2)$ extends holomorphically for $|y_1|<ε_{x_2}$ and $f(x_1,\cdot)$ for $| y_2|<ε$ (resp. $0\leq y_2<ε$ continuous up to $y_2=0$) with $ε$ independent of $x_1$. We prove in Theorem 3.4 that $f$ is then real analytic (resp. in Theorem 3.5 that it extends holomorphically to a &#34;wedge&#34; $W= Ω+iΓ_ε$ where $Γ_ε$ is an open cone trumcated by $|y|<ε$ and containing the ray $0<y_2<ε)$.