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

Hörmander, Lars

The null space of the

\bar{\partial}

-Neumann operator
[Le noyau de l’opérateur

\bar{\partial}

-Neumann]

Hörmander, Lars

Annales de l'Institut Fourier, Tome 54 (2004), p. 1305-1369 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

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 $1 \leq q < n - 1$ .

Publié le : 2004-01-01

MR 2127850 | Zbl 1083.32033 | 1 citation dans Numdam

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

@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},
     volume = {54},
     year = {2004},
     pages = {1305-1369},
     doi = {10.5802/aif.2051},
     mrnumber = {2127850},
     zbl = {1083.32033},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2004__54_5_1305_0}
}

Hörmander, Lars. The null space of the $\bar{\partial }$-Neumann operator. Annales de l'Institut Fourier, Tome 54 (2004) pp. 1305-1369. doi : 10.5802/aif.2051. http://gdmltest.u-ga.fr/item/AIF_2004__54_5_1305_0/

Bibliographie

[BS] H.P. Boas & M.-C. Shaw, Sobolev estimates for the Lewy operator on weakly pseudo-convex boundaries, Math. Ann 274 (1986) p. 221-231 | MR 838466 | Zbl 0588.32023

[BMS] L. Boutet De Monvel & J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Astérisque 34-35 (1976) p. 123-164 | Numdam | MR 590106 | Zbl 0344.32010

[CM] S.-S. Chern & J.-K. Moser, Real hypersurfaces in complex manifolds, Acta Math 133 (1974) p. 219-271 | MR 425155 | Zbl 0302.32015

[CS] S.-C. Chen & M.-C. Shaw, Partial differential equations in several complex variables, AMS/IP Studies in advanced mathematics Vol. 19, Amer. Math. Soc, 2001 | MR 1800297 | Zbl 0963.32001

[H1] L. Hörmander, $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator, Acta Math. 113 (1965) p. 89-152 | MR 179443 | Zbl 0158.11002

[H2] L. Hörmander, The multinomial distribution and some Bergman kernels, Geometric analysis of PDE and several complex variables. Contemporary Mathematics Proceedings (to appear) | MR 2126474 | Zbl 1102.41028

[H3] L. Hörmander, The analysis of linear partial differential operators I, Springer Verlag, 1983 | Zbl 0521.35001

[K] J.-J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. I., Ann. of Math. 78 (1963) p. 112-148 | MR 153030 | Zbl 0161.09302

[KN] J.-J. Kohn & L. Nirenberg, Non-coercive boundary problems, Comm. Pure Appl. Math 18 (1965) p. 443-492 | MR 181815 | Zbl 0125.33302

[KS] J.-J. Kohn & D.C. Spencer, Complex Neumann problems, Ann. of Math 66 (1957) p. 89-140 | MR 87879 | Zbl 0099.30605

[S1] M.-C. Shaw, Global solvability and regularity for $\bar{\partial}$ on an annulus between two weakly pseudo-convex domains, Trans. Amer. Math. Soc 291 (1985) p. 255-267 | MR 797058 | Zbl 0594.35010

[S2] M.-C. Shaw, $L^{2}$ estimates and existence theorems for the tangential Cauchy-Riemann complex, Invent. Math 82 (1985) p. 133-150 | MR 808113 | Zbl 0581.35057

[H3] L. Hörmander, The analysis of linear partial differential operators III, Springer-Verlag, 1985 | MR 781536 | Zbl 0601.35001

[K] J.J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. Math. (2) 79 (1964) p. 450-472 | MR 208200 | Zbl 0178.11305