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

Lars Hörmander^{[1]}

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

Annales de l’institut Fourier (2004)

- Volume: 54, Issue: 5, page 1305-1369
- ISSN: 0373-0956

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topHörmander, 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 $1\le q<n-1$.},

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 $1\le q<n-1$.

LA - eng

KW - $\bar{\partial }$-Neumann operator; reproducing kernel; -Neumann Laplacian

UR - http://eudml.org/doc/116144

ER -

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