  Let $\Omega \subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain
with smooth boundary such that $\partial \Omega$ has only nondegenerate
elliptic CR singularities, and let $f \colon \partial \Omega \to {\mathbb C}$
be a smooth function that is CR at CR points of $\partial \Omega$ (when $n=1$
we require separate holomorphic extensions for each real parameter). Then $f$
extends to a smooth CR function on $\bar{\Omega}$, that is, an analogue of
Hartogs-Bochner holds. In addition, if $f$ and $\partial \Omega$ are
real-analytic, then $f$ is the restriction of a function that is holomorphic on
a neighborhood of $\bar{\Omega}$ in ${\mathbb C}^{n+1}$. An immediate
application is a (possibly singular) solution of the Levi-flat Plateau problem
for codimension 2 submanifolds that are CR images of $\partial \Omega$ as
above. The extension also holds locally near nondegenerate, holomorphically
flat, elliptic CR singularities.
