The null space of the $\bar{\partial }$-Neumann operator

Hörmander, Lars

doi:10.5802/aif.2051

The null space of the

\bar{\partial}

-Neumann operator
[Le noyau de l’opérateur

\bar{\partial}

-Neumann]

Hörmander, Lars ¹

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.

Résumé
Abstract

Soit $Ω$ une variété complexe de dimension $n$ avec une métrique hermitienne et une frontière $C^{\infty}$ , et soit $⊓ ⊔ = \bar{\partial} {\bar{\partial}}^{*} + {\bar{\partial}}^{*} \bar{\partial}$ l'opérateur autoadjoint $\bar{\partial}$ -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 $\partial Ω$ , il est bien connu que dim $Ker$ $⊓ ⊔ < \infty$ et que l'image de $⊓ ⊔$ est l'espace orthogonal. Ici nous démontrons que dim $Ker$ $⊓ ⊔ = \infty$ si l'image de $⊓ ⊔$ est fermée et si la signature de la forme de Levi est $n - q - 1, q$ en un point de $\partial Ω$ . Le point de départ de la démonstration est une formule explicite pour $Ker$ $⊓ ⊔$ quand $Ω \subset ℂ^{n}$ est borné par deux sphères concentriques et $q = n - 1$ . Alors $Ker$ $⊓ ⊔$ a $n$ multiplicateurs indépendants ; ceci est vrai si et seulement si $Ω \subset ℂ^{n}$ 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 $\partial Ω$ 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 $1 \leq q < n - 1$ .

Let $Ω$ be a complex analytic manifold of dimension $n$ with a hermitian metric and $C^{\infty}$ boundary, and let $⊓ ⊔ = \bar{\partial} {\bar{\partial}}^{*} + {\bar{\partial}}^{*} \bar{\partial}$ be the self-adjoint $\bar{\partial}$ -Neumann operator on the space $L_{0, q}^{2} (Ω)$ of forms of type $(0, q)$ . If the Levi form of $\partial Ω$ 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 $Ker$ $⊓ ⊔ = \infty$ if the range of $⊓ ⊔$ is closed and the Levi form of $\partial Ω$ has signature $n - q - 1, q$ at some boundary point. The starting point for the proof is an explicit determination of $Ker$ $⊓ ⊔$ when $Ω \subset ℂ^{n}$ is a spherical shell and $q = n - 1$ . Then $Ker$ $⊓ ⊔$ has $n$ independent multipliers; this is only true for shells $Ω \subset ℂ^{n}$ 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 $\partial Ω$ where the Levi form is negative definite, $q = n - 1$ . Crude bounds are also given when the signature is $n - q - 1, q$ with $1 \leq q < n - 1$ .

MR Zbl | 2 citations dans Numdam

DOI : 10.5802/aif.2051

Classification : 32W05, 32A25

Affiliations des auteurs :

Hörmander, Lars ¹

¹ University of Lund, Department of Mathematics, Box 118, 221 00 Lund, (Sweden)

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     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},
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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://www.numdam.org/articles/10.5802/aif.2051/

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