Lewy extension theorem

Let M𝐂n be a smooth real hypersurface. Let ρ be a defining function for M near p. That is, for some neighbourhood of p, the submanifoldMathworldPlanetmath M is defined by ρ=0. For a neighbourhood Un, define the set U+ to be the set U{ρ>0}. We will say that M has at least one negative eigenvalue if the Levi form defined by ρ has at least one negative eigenvalue. That is, if

j,k=1n2ρ(p)zjz¯kwjw¯k<0 for some wn such that j=1nwjρ(p)zj=0.

Let f be a smooth CR function on M. Suppose that near pM the Levi form of M has at least one positive eigenvalue at p. Then there exists a neighbourhood U of p, such that for every smooth CR function f on M, there exists a function F holomorphic in U+ and C1 up to M, such that F|UM=f|UM.

By considering -ρ instead of ρ as a defining function, we get the corresponding result for at least one negative eigenvalue. If the Levi form of M has both positive and negative eigenvalues at a point, then f extends to both sides of M and is then a restriction of a holomorphic function.

A point is the fact that U is fixed and does not depend on f. To see why this is necessary, imagine a Levi flat example. Let M be defined in 2 in coordinates (z,w) by Imw=0. The domains Uϵ:={|Imw|<ϵ}, for ϵ>0, are pseudoconvex and hence there exist functions holomorphic on Ωϵ (and hence CR on M) that do not extend past any point of the boundary. No neighbourhood of a point on M fits in all Uϵ. 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[4], 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
Canonical name LewyExtensionTheorem
Date of creation 2013-03-22 17:39:44
Last modified on 2013-03-22 17:39:44
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 4
Author jirka (4157)
Entry type Theorem
Classification msc 32V25
Synonym Lewy extension