Lewy extension theorem
Let be a smooth real hypersurface. Let be a defining function for near That is, for some neighbourhood of the submanifold is defined by For a neighbourhood define the set to be the set We will say that has at least one negative eigenvalue if the Levi form defined by has at least one negative eigenvalue. That is, if
By considering instead of as a defining function, we get the corresponding result for at least one negative eigenvalue. If the Levi form of has both positive and negative eigenvalues at a point, then extends to both sides of and is then a restriction of a holomorphic function.
A point is the fact that is fixed and does not depend on To see why this is necessary, imagine a Levi flat example. Let be defined in in coordinates by The domains for are pseudoconvex and hence there exist functions holomorphic on (and hence CR on ) that do not extend past any point of the boundary. No neighbourhood of a point on fits in all So at least one nonzero eigenvalue of the Levi form is needed.
The statement of this theorem is not exactly the theorem that Lewy formulated, but this is generally called the Lewy extension. There have been many results in this direction since Lewy’s original paper, but this is the most result.
- 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
- 2 Albert Boggess. , CRC, 1991.
- 3 Lars Hörmander. , North-Holland Publishing Company, New York, New York, 1973.
- 4 Hans Lewy. Ann. of Math. (2) 64 (1956), 514–522.
|Title||Lewy extension theorem|
|Date of creation||2013-03-22 17:39:44|
|Last modified on||2013-03-22 17:39:44|
|Last modified by||jirka (4157)|