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

TL;DR

This paper extends the Lewy extension theorem to smooth hypersurfaces in complex Euclidean space times real line, analyzing conditions under which CR functions extend across CR singular manifolds with quadratic parts.

Contribution

It generalizes the Lewy extension theorem to CR singular manifolds with nondegenerate quadratic parts, identifying eigenvalue conditions for extension to either side.

Findings

CR functions extend under positive Levi-form eigenvalues

Extension to neighborhoods when Levi-form has eigenvalues of both signs

Eigenvalue conditions determine extension side for CR singular manifolds

Abstract

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.