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

Hörmander, Lars. "The null space of the $\bar{\partial }$-Neumann operator." Annales de l’institut Fourier 54.5 (2004): 1305-1369. <http://eudml.org/doc/116144>.

@article{Hörmander2004,
abstract = {Let $\Omega $ be a complex analytic manifold of dimension $n$ with a hermitian metric and $C^\{\infty \}$ boundary, and let $\Box =\bar\{\partial \}\,\bar\{\partial \}^*+\bar\{\partial \}^*\,\bar\{\partial \}$ be the self-adjoint $\bar\{\partial \}$-Neumann operator on the space $L^2_\{0,q\}(\Omega )$ of forms of type $(0,q)$. 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 $\{\rm Ker\}$$\Box $ has finite dimension and that the range of $\Box $ is the orthogonal complement. In this paper it is proved that dim $\{\rm Ker\}$$\Box =\infty $ if the range of $\Box $ 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 $\{\rm Ker\} $$\Box $ when $\Omega \subset \{\mathbb \{C\}\}^n$ is a spherical shell and $q=n-1$. Then $\{\rm Ker\}$$\Box $ has $n$ independent multipliers; this is only true for shells $\Omega \subset \{\mathbb \{C\}\}^n$ bounded by two confocal ellipsoids. These models lead to asymptotics in a weak sense for the kernel of the orthogonal projection on $\{\rm Ker\} $$\Box $ when the range of $\Box $ 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 $1\le q&lt;n-1$.},
affiliation = {University of Lund, Department of Mathematics, Box 118, 221 00 Lund, (Sweden)},
author = {Hörmander, Lars},
journal = {Annales de l’institut Fourier},
keywords = {$\bar\{\partial \}$-Neumann operator; reproducing kernel; -Neumann Laplacian},
language = {eng},
number = {5},
pages = {1305-1369},
publisher = {Association des Annales de l'Institut Fourier},
title = {The null space of the $\bar\{\partial \}$-Neumann operator},
url = {http://eudml.org/doc/116144},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Hörmander, Lars
TI - The null space of the $\bar{\partial }$-Neumann operator
JO - Annales de l’institut Fourier
PY - 2004
PB - Association des Annales de l'Institut Fourier
VL - 54
IS - 5
SP - 1305
EP - 1369
AB - Let $\Omega $ be a complex analytic manifold of dimension $n$ with a hermitian metric and $C^{\infty }$ boundary, and let $\Box =\bar{\partial }\,\bar{\partial }^*+\bar{\partial }^*\,\bar{\partial }$ be the self-adjoint $\bar{\partial }$-Neumann operator on the space $L^2_{0,q}(\Omega )$ of forms of type $(0,q)$. 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 ${\rm Ker}$$\Box $ has finite dimension and that the range of $\Box $ is the orthogonal complement. In this paper it is proved that dim ${\rm Ker}$$\Box =\infty $ if the range of $\Box $ 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 ${\rm Ker} $$\Box $ when $\Omega \subset {\mathbb {C}}^n$ is a spherical shell and $q=n-1$. Then ${\rm Ker}$$\Box $ has $n$ independent multipliers; this is only true for shells $\Omega \subset {\mathbb {C}}^n$ bounded by two confocal ellipsoids. These models lead to asymptotics in a weak sense for the kernel of the orthogonal projection on ${\rm Ker} $$\Box $ when the range of $\Box $ 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 $1\le q&lt;n-1$.
LA - eng
KW - $\bar{\partial }$-Neumann operator; reproducing kernel; -Neumann Laplacian
UR - http://eudml.org/doc/116144
ER -