# ANNALES DE L'INSTITUT FOURIER

The null space of the $\overline{\partial }$-Neumann operator
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, p. 1305-1369

Let $\Omega$ be a complex analytic manifold of dimension $n$ with a hermitian metric and ${C}^{\infty }$ boundary, and let $\square =\overline{\partial }\phantom{\rule{0.166667em}{0ex}}{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\phantom{\rule{0.166667em}{0ex}}\overline{\partial }$ be the self-adjoint $\overline{\partial }$-Neumann operator on the space ${L}_{0,q}^{2}\left(\Omega \right)$ of forms of type $\left(0,q\right)$. If the Levi form of $\partial \Omega$ has everywhere at least $n-q$ positive or at least $q+1$ negative eigenvalues, it is well known that $\mathrm{Ker}$ $\square$ has finite dimension and that the range of $\square$ is the orthogonal complement. In this paper it is proved that dim $\mathrm{Ker}$ $\square =\infty$ if the range of $\square$ is closed and the Levi form of $\partial \Omega$ has signature $n-q-1,q$ at some boundary point. The starting point for the proof is an explicit determination of $\mathrm{Ker}$ $\square$ when $\Omega \subset {ℂ}^{n}$ is a spherical shell and $q=n-1$. Then $\mathrm{Ker}$ $\square$ has $n$ independent multipliers; this is only true for shells $\Omega \subset {ℂ}^{n}$ bounded by two confocal ellipsoids. These models lead to asymptotics in a weak sense for the kernel of the orthogonal projection on $\mathrm{Ker}$ $\square$ when the range of $\square$ is closed, at points on $\partial \Omega$ 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\le q.

Soit $\Omega$ une variété complexe de dimension $n$ avec une métrique hermitienne et une frontière ${C}^{\infty }$, et soit $\square =\overline{\partial }{\overline{\partial }}^{*}+{\overline{\partial }}^{*}\phantom{\rule{0.166667em}{0ex}}\overline{\partial }$ l’opérateur autoadjoint $\overline{\partial }$-Neumann dans l’espace ${L}_{0,q}^{2}\left(\Omega \right)$ des $\left(0,q\right)$ 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 $\partial \Omega$, il est bien connu que dim $\mathrm{Ker}$ $\square <\infty$ et que l’image de $\square$ est l’espace orthogonal. Ici nous démontrons que dim $\mathrm{Ker}$ $\square =\infty$ si l’image de $\square$ est fermée et si la signature de la forme de Levi est $n-q-1,q$ en un point de $\partial \Omega$. Le point de départ de la démonstration est une formule explicite pour $\mathrm{Ker}$ $\square$ quand $\Omega \subset {ℂ}^{n}$ est borné par deux sphères concentriques et $q=n-1$. Alors $\mathrm{Ker}$ $\square$ a $n$ multiplicateurs indépendants ; ceci est vrai si et seulement si $\Omega \subset {ℂ}^{n}$ est borné par deux ellipsoïdes confocaux. Ces modèles conduisent à une asymptotique faible pour le noyau de la projection orthogonale sur $\mathrm{Ker}$ $\square$ quand l’image de $\square$ est fermée, aux points de $\partial \Omega$ 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\le q.

DOI : https://doi.org/10.5802/aif.2051
Classification:  32W05,  32A25
Keywords: $\overline{\partial }$-Neumann operator, reproducing kernel
@article{AIF_2004__54_5_1305_0,
author = {H\"ormander, Lars},
title = {The null space of the $$-Neumann operator}, journal = {Annales de l'Institut Fourier}, publisher = {Association des Annales de l'institut Fourier}, volume = {54}, number = {5}, year = {2004}, pages = {1305-1369}, doi = {10.5802/aif.2051}, zbl = {1083.32033}, mrnumber = {2127850}, language = {en}, url = {aif.centre-mersenne.org/item/AIF_2004__54_5_1305_0} }  Hörmander, Lars. The null space of the$$-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/

L. Hörmander The analysis of linear partial differential operators III, Springer-Verlag, Berlin, Heidelberg, 1985 | MR 838466 | Zbl 0588.32023

J.J. Kohn Harmonic integrals on strongly pseudo-convex manifolds. II, Ann. Math. (2), Tome 79 (1964), pp. 450-472 | Numdam | MR 590106 | Zbl 0344.32010

[BMS] L. Boutet de Monvel; J. Sjöstrand Sur la singularité des noyaux de Bergman et de Szegö, Astérisque, Tome 34-35 (1976), pp. 123-164 | MR 425155 | Zbl 0302.32015

[BS] H.P. Boas; M.-C. Shaw Sobolev estimates for the Lewy operator on weakly pseudo-convex boundaries, Math. Ann, Tome 274 (1986), pp. 221-231 | MR 1800297 | Zbl 0963.32001

[CM] S.-S. Chern; J.-K. Moser Real hypersurfaces in complex manifolds, Acta Math, Tome 133 (1974), pp. 219-271 | MR 179443 | Zbl 0158.11002

[CS] S.-C. Chen; M.-C. Shaw Partial differential equations in several complex variables, AMS/IP Studies in advanced mathematics, Tome Vol. 19, Amer. Math. Soc, 2001 | MR 2126474 | Zbl 1102.41028

[H2] L. Hörmander The multinomial distribution and some Bergman kernels (Geometric analysis of PDE and several complex variables. Contemporary Mathematics Proceedings (to appear)) | Zbl 0521.35001

[H1] L. Hörmander $L^2$ estimates and existence theorems for the $\bar\partial$ operator, Acta Math., Tome 113 (1965), pp. 89-152 | MR 153030 | Zbl 0161.09302

[H3] L. Hörmander The analysis of linear partial differential operators I, Springer Verlag, Berlin, Heidelberg, 1983 | MR 181815 | Zbl 0125.33302

[K] J.-J. Kohn Harmonic integrals on strongly pseudo-convex manifolds. I., Ann. of Math., Tome 78 (1963), pp. 112-148 | MR 87879 | Zbl 0099.30605

[KN] J.-J. Kohn; L. Nirenberg Non-coercive boundary problems, Comm. Pure Appl. Math, Tome 18 (1965), pp. 443-492 | MR 797058 | Zbl 0594.35010

[KS] J.-J. Kohn; D.C. Spencer Complex Neumann problems, Ann. of Math, Tome 66 (1957), pp. 89-140 | MR 808113 | Zbl 0581.35057

[S1] M.-C. Shaw Global solvability and regularity for $\bar\partial$ on an annulus between two weakly pseudo-convex domains, Trans. Amer. Math. Soc, Tome 291 (1985), pp. 255-267 | MR 781536 | Zbl 0601.35001

[S2] M.-C. Shaw $L^2$ estimates and existence theorems for the tangential Cauchy-Riemann complex, Invent. Math, Tome 82 (1985), pp. 133-150 | MR 208200 | Zbl 0178.11305