This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The principal investigator will undertake a study of regularity properties and decay estimates of solutions to the weighted Cauchy-Riemann equations in several variables as well as the tangential Cauchy-Riemann equations on the boundaries of pseudoconvex domains. When the domain is unbounded, he will investigate the Kohn Laplacian via its heat equation and develop the necessary harmonic analysis and regularity theory to obtain pointwise estimates on the heat kernel and its derivatives. In the case that the CR-manifold is compact, the questions will depend on whether the CR-manifold is the boundary of a pseudoconvex domain or has at least two totally real directions to the tangent bundle. In the former case, the examination will focus on the relationship of compactness of the complex Green operator, the existence of Stein neighborhood bases, and a certain property known as the "P" sub "q" property. In the latter case, the principal investigator will undertake an analysis of the closed range properties of the tangential Cauchy-Riemann operator.
A fundamental problem in several complex variables is to solve the tangential Cauchy-Riemann equations. Perhaps the most challenging aspects to this problem are the lack of ellipticity and the influence of the geometry on the analysis. The study of the tangential Cauchy-Riemann equations, however, has led to significant advances in the understanding of partial differential equations, perhaps the most striking examples being the existence of a locally nonsolvable linear partial differential equation (i.e., Lewy's example) and the development of pseudodifferential operators. This project will contribute to the understanding of the effect of the geometry of a CR-manifold on the regularity theory of the tangential Cauchy-Riemann operators.
