On Lewy extension for smooth hypersurfaces in $\mathbb {C}^n \times \mathbb {R}$

We prove an analogue of the Lewy extension theorem for a real dimension $2n$ smooth submanifold $M \subset \mathbb {C}^{n}\times \mathbb {R}$, $n \geq 2$. A theorem of Hill and Taiani implies that if $M$ is CR and the Levi-form has a positive eigenvalue restricted to the leaves of $\mathbb {C}^n \times \mathbb {R}$, then every smooth CR function $f$ extends smoothly as a CR function to one side of $M$. If the Levi-form has eigenvalues of both signs, then $f$ extends to a neighborhood of $M$. Our main result concerns CR singular manifolds with a nondegenerate quadratic part $Q$. A smooth CR $f$ extends to one side if the Hermitian part of $Q$ has at least two positive eigenvalues, and $f$ extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.