The null space of the ¯-Neumann operator
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1305-1369
Let Ω be a complex analytic manifold of dimension n with a hermitian metric and C boundary, and let = ¯ ¯ * + ¯ * ¯ be the self-adjoint ¯-Neumann operator on the space L 0,q 2 (Ω) of forms of type (0,q). If the Levi form of Ω has everywhere at least n-q positive or at least q+1 negative eigenvalues, it is well known that Ker has finite dimension and that the range of is the orthogonal complement. In this paper it is proved that dim Ker = if the range of is closed and the Levi form of Ω has signature n-q-1,q at some boundary point. The starting point for the proof is an explicit determination of Ker when Ω n is a spherical shell and q=n-1. Then Ker has n independent multipliers; this is only true for shells Ω n bounded by two confocal ellipsoids. These models lead to asymptotics in a weak sense for the kernel of the orthogonal projection on Ker when the range of is closed, at points on Ω where the Levi form is negative definite, q=n-1. Crude bounds are also given when the signature is n-q-1,q with 1q<n-1.
Soit Ω une variété complexe de dimension n avec une métrique hermitienne et une frontière C , et soit = ¯ ¯ * + ¯ * ¯ l'opérateur autoadjoint ¯-Neumann dans l'espace L 0,q 2 (Ω) des (0,q) formes. Si la forme de Levi a au moins n-q valeurs propres positives ou au moins q+1 valeurs propres négatives en chaque point de Ω, il est bien connu que dim Ker < et que l'image de est l'espace orthogonal. Ici nous démontrons que dim Ker = si l'image de est fermée et si la signature de la forme de Levi est n-q-1,q en un point de Ω. Le point de départ de la démonstration est une formule explicite pour Ker quand Ω n est borné par deux sphères concentriques et q=n-1. Alors Ker a n multiplicateurs indépendants ; ceci est vrai si et seulement si Ω n est borné par deux ellipsoïdes confocaux. Ces modèles conduisent à une asymptotique faible pour le noyau de la projection orthogonale sur Ker quand l'image de est fermée, aux points de Ω où la forme de Levi est définie négative q=n-1. Des bornes grossières sont aussi données quand la signature est n-q-1,q avec 1q<n-1.
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     pages = {1305-1369},
     doi = {10.5802/aif.2051},
     mrnumber = {2127850},
     zbl = {1083.32033},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2004__54_5_1305_0}
}
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. https://aif.centre-mersenne.org/item/AIF_2004__54_5_1305_0/

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