Let be a complex analytic manifold of dimension with a hermitian metric and boundary, and let be the self-adjoint -Neumann operator on the space of forms of type . If the Levi form of has everywhere at least positive or at least negative eigenvalues, it is well known that has finite dimension and that the range of is the orthogonal complement. In this paper it is proved that dim if the range of is closed and the Levi form of has signature at some boundary point. The starting point for the proof is an explicit determination of when is a spherical shell and . Then has independent multipliers; this is only true for shells bounded by two confocal ellipsoids. These models lead to asymptotics in a weak sense for the kernel of the orthogonal projection on when the range of is closed, at points on where the Levi form is negative definite, . Crude bounds are also given when the signature is with .
Soit une variété complexe de dimension avec une métrique hermitienne et une frontière , et soit l'opérateur autoadjoint -Neumann dans l'espace des formes. Si la forme de Levi a au moins valeurs propres positives ou au moins valeurs propres négatives en chaque point de , il est bien connu que dim et que l'image de est l'espace orthogonal. Ici nous démontrons que dim si l'image de est fermée et si la signature de la forme de Levi est en un point de . Le point de départ de la démonstration est une formule explicite pour quand est borné par deux sphères concentriques et . Alors a multiplicateurs indépendants ; ceci est vrai si et seulement si est borné par deux ellipsoïdes confocaux. Ces modèles conduisent à une asymptotique faible pour le noyau de la projection orthogonale sur quand l'image de est fermée, aux points de où la forme de Levi est définie négative . Des bornes grossières sont aussi données quand la signature est avec .
@article{AIF_2004__54_5_1305_0, author = {H\"ormander, Lars}, title = {The null space of the $\bar{\partial }${-Neumann} operator}, journal = {Annales de l'Institut Fourier}, pages = {1305--1369}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {54}, number = {5}, year = {2004}, doi = {10.5802/aif.2051}, mrnumber = {2127850}, zbl = {1083.32033}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2051/} }
TY - JOUR AU - Hörmander, Lars TI - The null space of the $\bar{\partial }$-Neumann operator JO - Annales de l'Institut Fourier PY - 2004 SP - 1305 EP - 1369 VL - 54 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2051/ DO - 10.5802/aif.2051 LA - en ID - AIF_2004__54_5_1305_0 ER -
%0 Journal Article %A Hörmander, Lars %T The null space of the $\bar{\partial }$-Neumann operator %J Annales de l'Institut Fourier %D 2004 %P 1305-1369 %V 54 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2051/ %R 10.5802/aif.2051 %G en %F AIF_2004__54_5_1305_0
Hörmander, Lars. The null space of the $\bar{\partial }$-Neumann operator. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1305-1369. doi : 10.5802/aif.2051. http://archive.numdam.org/articles/10.5802/aif.2051/
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