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Theses

Projections in several complex variables

Abstract : This thesis consists two parts. In the first part, we completely study the heat equation method ofMenikoff-Sjöstrand and apply it to the Kohn Laplacian defined on a compact orientable connected CR manifold. We then get the full asymptotic expansion of the Szegö projection for (0,q) forms when the Levi formis nondegenerate. This generalizes a result of Boutet de Monvel and Sjöstrand for (0,0) forms. Our main tool is Fourier integral operators with complex valued phase functions of Melin and Sjöstrand. 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 Sjöstrand 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 Sjöstrand to this operator. We obtain a description of a new Szegö projection up to smoothing operators. Finally, by using the Poisson operator, we get our main result.
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Submitted on : Tuesday, October 21, 2008 - 4:59:21 PM
Last modification on : Wednesday, March 27, 2019 - 4:10:22 PM
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Chin-Yu Hsiao. Projections in several complex variables. Mathematics [math]. Ecole Polytechnique X, 2008. English. ⟨tel-00332787⟩

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