no. 5
The null space of the ∂ ¯-Neumann operator
[Le noyau de l’opérateur ∂ ¯-Neumann]
Hörmander, Lars1
1 University of Lund, Department of Mathematics, Box 118, 221 00 Lund, (Sweden)
Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1305-1369.

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⊂ℂ 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⊂ℂ 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 1≤q<n-1.

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⊂ℂ n is a spherical shell and q=n-1. Then Ker ⊓⊔ has n independent multipliers; this is only true for shells Ω⊂ℂ 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 1≤q<n-1.

DOI : 10.5802/aif.2051
Classification : 32W05, 32A25

Hörmander, Lars 1

1 University of Lund, Department of Mathematics, Box 118, 221 00 Lund, (Sweden) 
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Hörmander, Lars. The null space of the $\bar{\partial }$-Neumann operator. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1305-1369. doi : 10.5802/aif.2051. https://aif.centre-mersenne.org/articles/10.5802/aif.2051/

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