Projections in several complex variables

Abstract

This thesis consists two parts. In the first part, we completely study the heat equation method ofMenikoff-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 deMonvel 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 ofMenikoff 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.