The null space of the partial derivativeNeumann operator
(2004) In Annales de l'Institut Fourier 54(5). p.13051305 Abstract
 Let Q be a complex analytic manifold of dimension n with a hermitian metric and Cinfinity boundary, and let rectangle = deltadelta* + delta* delta be the selfadjoint deltaNeumann operator on the space L0,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 Cinfinity boundary, and let rectangle = deltadelta* + delta* delta be the selfadjoint deltaNeumann operator on the space L0,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 Cn is a spherical shell and q = n  1. Then Ker rectangle has n independent multipliers; this is only true for shells Omega subset of Cn 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)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/250374
 author
 Hörmander, Lars ^{LU}
 organization
 publishing date
 2004
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 partial derivativeNeumann 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
 03730956
 language
 English
 LU publication?
 yes
 id
 801917f9e6984a4e902943b843c85853 (old id 250374)
 alternative location
 http://aif.cedram.org/aifbin/item?id=AIF_2004__54_5_1305_0
 date added to LUP
 20071030 16:14:49
 date last changed
 20160415 20:11:58
@article{801917f9e6984a4e902943b843c85853, abstract = {Let Q be a complex analytic manifold of dimension n with a hermitian metric and Cinfinity boundary, and let rectangle = deltadelta* + delta* delta be the selfadjoint deltaNeumann operator on the space L0,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 Cn is a spherical shell and q = n  1. Then Ker rectangle has n independent multipliers; this is only true for shells Omega subset of Cn 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 = {03730956}, keyword = {partial derivativeNeumann operator,reproducing kernel}, language = {eng}, number = {5}, pages = {13051305}, publisher = {ANNALES DE L INSTITUT FOURIER}, series = {Annales de l'Institut Fourier}, title = {The null space of the partial derivativeNeumann operator}, volume = {54}, year = {2004}, }