Projections in several complex variables
Mémoires de la Société Mathématique de France, no. 123 (2010) , 144 p.
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This work consists two parts. In the first part, we completely study the heat equation method of Menikoff-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 Szegő projection for (0,q) forms when the Levi form is non-degenerate. This generalizes a result of Boutet de Monvel and Sjöstrand for (0,0) forms. Our main tools are Fourier integral operators with complex valued phase Melin and Sjöstrand functions. 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 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 of Menikoff and Sjöstrand to this operator. We obtain a description of a new Szegő projection up to smoothing operators. Finally, we get our main result by using the Poisson operator.

Ce travail comporte deux parties. Dans la première, nous appliquons la méthode de Menikoff-Sjöstrand au laplacien de Kohn, défini sur une varieté CR compacte orientée connexe et nous obtenons un développement asymptotique complet du projecteur de Szegő pour les (0,q) formes quand la forme de Levi est non-dégénérée. Cela généralise un résultat de Boutet de Monvel et Sjöstrand pour les (0,0) formes. Nous utilisons des opérateurs intégraux de Fourier à phases complexes de Melin et Sjöstrand. Dans la deuxième partie, nous obtenons un développement asymptotique de la singularité du noyau de Bergman pour les (0,q) formes quand la forme de Levi est non-dégénérée. Cela généralise un résultat de Boutet de Monvel et Sjöstrand pour les (0,0) formes. Nous introduisons un nouvel opérateur analogue au laplacien de Kohn défini sur le bord du domaine, et nous y appliquons la méthode de Menikoff-Sjöstrand. Cela donne une description modulo les opérateurs régularisants d’un nouvel projecteur de Szegő. Enfin, nous obtenons notre résultat principal en utilisant l’opérateur de Poisson.

DOI : https://doi.org/10.24033/msmf.435
Classification:  32A25,  32V05,  32V20,  32W30,  58A14
Keywords: CR manifold, Szegő kernel, Bergman kernel, heat equation, microlocal analysis
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     author = {Hsiao, Chin-Yu},
     title = {Projections in several complex variables},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {123},
     year = {2010},
     doi = {10.24033/msmf.435},
     zbl = {1229.32002},
     mrnumber = {2780123},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2010_2_123__1_0}
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Hsiao, Chin-Yu. Projections in several complex variables. Mémoires de la Société Mathématique de France, Serie 2, , no. 123 (2010), 144 p. doi : 10.24033/msmf.435. http://www.numdam.org/item/MSMF_2010_2_123__1_0/

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