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The null space of the partial derivative-Neumann operator

Hörmander, Lars LU (2004) In Annales de l'Institut Fourier 54(5). p.1305-1305
Abstract
Let Q be a complex analytic manifold of dimension n with a hermitian metric and C-infinity boundary, and let rectangle = deltadelta* + delta* delta be the self-adjoint delta-Neumann operator on the space L-0,q(2) (Omega) of forms of type (0, q). If the Levi form of deltaOmega has everywhere at least n - q positive or at least q+ I negative eigenvalues, it is well known that Ker rectangle has finite dimension and that the range of rectangle is the orthogonal complement. In this paper it is proved that dim Ker rectangle = infinity if the range of rectangle is closed and the Levi form of deltaOmega has signature n - q - 1, q at some boundary point. The starting point for the proof is an explicit determination of Ker rectangle when Omega... (More)
Let Q be a complex analytic manifold of dimension n with a hermitian metric and C-infinity boundary, and let rectangle = deltadelta* + delta* delta be the self-adjoint delta-Neumann operator on the space L-0,q(2) (Omega) of forms of type (0, q). If the Levi form of deltaOmega has everywhere at least n - q positive or at least q+ I negative eigenvalues, it is well known that Ker rectangle has finite dimension and that the range of rectangle is the orthogonal complement. In this paper it is proved that dim Ker rectangle = infinity if the range of rectangle is closed and the Levi form of deltaOmega has signature n - q - 1, q at some boundary point. The starting point for the proof is an explicit determination of Ker rectangle when Omega subset of C-n is a spherical shell and q = n - 1. Then Ker rectangle has n independent multipliers; this is only true for shells Omega subset of 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 rectangle when the range of 0 is closed, at points on deltaOmega 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 less than or equal to q less than or equal to n - 1. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
partial derivative-Neumann operator, reproducing kernel
in
Annales de l'Institut Fourier
volume
54
issue
5
pages
1305 - 1305
publisher
ANNALES DE L INSTITUT FOURIER
external identifiers
  • wos:000227440500004
ISSN
0373-0956
language
English
LU publication?
yes
id
801917f9-e698-4a4e-9029-43b843c85853 (old id 250374)
alternative location
http://aif.cedram.org/aif-bin/item?id=AIF_2004__54_5_1305_0
date added to LUP
2007-10-30 16:14:49
date last changed
2016-04-15 20:11:58
@article{801917f9-e698-4a4e-9029-43b843c85853,
  abstract     = {Let Q be a complex analytic manifold of dimension n with a hermitian metric and C-infinity boundary, and let rectangle = deltadelta* + delta* delta be the self-adjoint delta-Neumann operator on the space L-0,q(2) (Omega) of forms of type (0, q). If the Levi form of deltaOmega has everywhere at least n - q positive or at least q+ I negative eigenvalues, it is well known that Ker rectangle has finite dimension and that the range of rectangle is the orthogonal complement. In this paper it is proved that dim Ker rectangle = infinity if the range of rectangle is closed and the Levi form of deltaOmega has signature n - q - 1, q at some boundary point. The starting point for the proof is an explicit determination of Ker rectangle when Omega subset of C-n is a spherical shell and q = n - 1. Then Ker rectangle has n independent multipliers; this is only true for shells Omega subset of 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 rectangle when the range of 0 is closed, at points on deltaOmega 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 less than or equal to q less than or equal to n - 1.},
  author       = {Hörmander, Lars},
  issn         = {0373-0956},
  keyword      = {partial derivative-Neumann operator,reproducing kernel},
  language     = {eng},
  number       = {5},
  pages        = {1305--1305},
  publisher    = {ANNALES DE L INSTITUT FOURIER},
  series       = {Annales de l'Institut Fourier},
  title        = {The null space of the partial derivative-Neumann operator},
  volume       = {54},
  year         = {2004},
}