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

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- by Jiří Lebl, Alan Noell and Sivaguru Ravisankar PDF
- Trans. Amer. Math. Soc.
**371**(2019), 6581-6603 Request permission

## 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.## References

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## Additional Information

**Jiří Lebl**- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
- MR Author ID: 813143
- ORCID: 0000-0002-9320-0823
- Email: lebl@math.okstate.edu
**Alan Noell**- Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
- MR Author ID: 131880
- Email: noell@math.okstate.edu
**Sivaguru Ravisankar**- Affiliation: Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Bengaluru 560065, India
- MR Author ID: 1054138
- Email: sivaguru@tifrbng.res.in
- Received by editor(s): August 18, 2017
- Received by editor(s) in revised form: February 1, 2018
- Published electronically: October 24, 2018
- Additional Notes: The first author was supported in part by NSF grant DMS-1362337.
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**371**(2019), 6581-6603 - MSC (2010): Primary 32V40; Secondary 32V25
- DOI: https://doi.org/10.1090/tran/7605
- MathSciNet review: 3937338