Abstract: A natural question is when does a function defined on a real
submanifold, extend to a holomorphic function on a neighborhood. An
answer for hypersurfaces in $\mathbb{C}^n$ is the classical Lewy
extension theorem: A CR function extends to a side if the Levi-form
has a nonzero eigenvalue. In $\mathbb{C}^n \times \mathbb{R}$, we
encounter CR singularities, but still we find a Lewy-extension-like
theorem: A CR function extends near a CR singularity if a certain
analogue of the Levi-form has at least two eigenvalues of the same
sign. Joint work with Alan Noell and Sivaguru Ravisankar.
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