Projections in several complex variables

TL;DR

This paper develops asymptotic expansions for Szego and Bergman projections on CR manifolds with non-degenerate Levi form, generalizing classical results and introducing new operators using advanced Fourier integral operator techniques.

Contribution

It extends the heat equation method to analyze Szego and Bergman projections on CR manifolds, introducing a new boundary operator and generalizing prior results for (0,0) forms.

Findings

Full asymptotic expansion of Szego projection for (0,q) forms

Full asymptotic expansion of Bergman projection for (0,q) forms

Introduction of a new boundary operator analogous to the Kohn Laplacian

Abstract

This work consists two parts. In the first part, we completely study the heat equation method of Menikoff-Sjostrand and apply it to the Kohn Laplacian defined on a compact orientable connected CR manifold. We then get the full asymptotic expansion of the Szego projection for (0,q) forms when the Levi formis nondegenerate. This generalizes a result of Boutet de Monvel and Sjostrand for (0,0) forms. Our main tool is Fourier integral operators with complex valued phase functions of Melin and Sjostrand. In the second part, we obtain the full asymptotic expansion of the Bergman projection for (0,q) forms when the Levi form is non-degenerate. This also generalizes a result of Boutet de Monvel and Sjostrand for (0,0) forms. We introduce a new operator analogous to the Kohn Laplacian defined on the boundary of a domain and we apply the heat equation method of Menikoff and Sjostrand to this operator. We obtain a description of a new Szego projection up to smoothing operators. Finally, by using the Poisson operator, we get our main result.