Journal of Geometric Analysis

, Volume 23, Issue 4, pp 1729–1752 | Cite as

Fundamental Solutions to □ b on Certain Quadrics

Article
First Online:
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Abstract

The purpose of this article is to expand the number of examples for which the complex Green operator, that is, the fundamental solution to the Kohn Laplacian, can be computed. We use the Lie group structure of quadric submanifolds of ℂ n ×ℂ m and the group Fourier transform to reduce the □ b equation to ones that can be solved using modified Hermite functions. We use Mehler’s formula and investigate (1) quadric hypersurfaces, where the eigenvalues of the Levi form are not identical (including possibly zero eigenvalues), and (2) the canonical quadrics in ℂ4 of codimension two.

Keywords

Kohn Laplacian Complex Green operator Lie group Quadrics Heisenberg group Fundamental solution 

Mathematics Subject Classification (2000)

32W10 33C45 43A80 35H20 
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Copyright information

© Mathematica Josephina, Inc. 2012

Authors and Affiliations

  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of Mathematical Sciences1 University of ArkansasFayettevilleUSA

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