Semilocal Levi-flat extensions

Let $G\subset\mathbb C\times\mathbb R$ be a domain such that $G\times\mathbb R\subset\mathbb C^2$ is strictly pseudoconvex and let $U\subset bG$ be an open subset. We define the hull $\mathscr E(U)$ with respect to the algebra $\mathscr A(G\times\mathbb R)$ and study its properties. It is proved that every continuous function on $U$ can be extended to a continuous function on $\mathscr E(U)$ whose graph is locally foliated by holomorphic curves.