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The resolution of the Nirenberg-Treves conjecture

Dencker, Nils LU (2006) In Annals of Mathematics 163(2). p.405-444
Abstract
We give a proof of the Nirenberg-Treves conjecture: that local solvability of principal-type pseudo-differential operators is equivalent to condition (Psi). This condition rules out sign changes from - to + of the imaginary part of the principal symbol along the oriented bicharacteristics of the real part. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus which makes it possible to reduce to the case when the gradient of the imaginary part is nonvanishing, so that the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures... (More)
We give a proof of the Nirenberg-Treves conjecture: that local solvability of principal-type pseudo-differential operators is equivalent to condition (Psi). This condition rules out sign changes from - to + of the imaginary part of the principal symbol along the oriented bicharacteristics of the real part. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus which makes it possible to reduce to the case when the gradient of the imaginary part is nonvanishing, so that the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the changes of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of the zeroes. By using condition (Psi) and the weight, we can construct a multiplier giving the estimate. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
principal type, pseudodifferential operators, Nirenberg-Treves conjecture, solvability
in
Annals of Mathematics
volume
163
issue
2
pages
405 - 444
publisher
Annals of Mathematics
external identifiers
  • wos:000239506200002
  • scopus:33746158191
ISSN
0003-486X
language
English
LU publication?
yes
id
77855c68-d517-4d9b-bf02-fbc2b9937035 (old id 399268)
alternative location
http://annals.math.princeton.edu/issues/2006/March2006/Dencker.pdf
date added to LUP
2016-04-01 15:19:25
date last changed
2020-12-22 01:25:30
@article{77855c68-d517-4d9b-bf02-fbc2b9937035,
  abstract     = {We give a proof of the Nirenberg-Treves conjecture: that local solvability of principal-type pseudo-differential operators is equivalent to condition (Psi). This condition rules out sign changes from - to + of the imaginary part of the principal symbol along the oriented bicharacteristics of the real part. We obtain local solvability by proving a localizable a priori estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman) calculus which makes it possible to reduce to the case when the gradient of the imaginary part is nonvanishing, so that the zeroes form a smooth submanifold. The estimate uses a new type of weight, which measures the changes of the distance to the zeroes of the imaginary part along the bicharacteristics of the real part between the minima of the curvature of the zeroes. By using condition (Psi) and the weight, we can construct a multiplier giving the estimate.},
  author       = {Dencker, Nils},
  issn         = {0003-486X},
  language     = {eng},
  number       = {2},
  pages        = {405--444},
  publisher    = {Annals of Mathematics},
  series       = {Annals of Mathematics},
  title        = {The resolution of the Nirenberg-Treves conjecture},
  url          = {http://annals.math.princeton.edu/issues/2006/March2006/Dencker.pdf},
  volume       = {163},
  year         = {2006},
}