Levi geometry and the tangential Cauchy-Riemann equations on a real analytic submanifold of

Author:
Al Boggess

Journal:
Trans. Amer. Math. Soc. **272** (1982), 351-374

MSC:
Primary 32F20; Secondary 32F25, 35N15

DOI:
https://doi.org/10.1090/S0002-9947-1982-0656494-3

MathSciNet review:
656494

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Abstract: The relationship between the Levi geometry of a submanifold of and the tangential Cauchy-Riemann equations is studied. On a real analytic codimension two submanifold of , we find conditions on the Levi algebra which allow us to locally solve the tangential Cauchy-Riemann equations (in most bidegrees) with kernels. Under the same conditions, we show that, locally, any CR-function is the boundary value jump of a holomorphic function defined on some suitable open set in . This boundary value jump result is the best possible result because we also show that there is no one-sided extension theory for such submanifolds of . In fact, we show that if is a real analytic, generic, submanifold of (any codimension) where the excess dimension of the Levi algebra is *less* than the real codimension, then is not extendible to *any* open set in .

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DOI:
https://doi.org/10.1090/S0002-9947-1982-0656494-3

Article copyright:
© Copyright 1982
American Mathematical Society