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 $\Omega$ 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 $|y_1|<\epsilon_{x_2}$ and $f(x_1,\cdot)$ for $| y_2|<\epsilon$ (resp. $0\leq y_2<\epsilon$ continuous up to $y_2=0$) with $\epsilon$ 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 "wedge" $W= \Omega+i\Gamma_\epsilon$ where $\Gamma_\epsilon$ is an open cone trumcated by $|y|<\epsilon$ and containing the ray $0<y_2<\epsilon)$.