The resolution of the NirenbergTreves conjecture
(2006) In Annals of Mathematics 163(2). p.405444 Abstract
 We give a proof of the NirenbergTreves conjecture: that local solvability of principaltype pseudodifferential 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 BealsFefferman) 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 NirenbergTreves conjecture: that local solvability of principaltype pseudodifferential 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 BealsFefferman) 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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/399268
 author
 Dencker, Nils ^{LU}
 organization
 publishing date
 2006
 type
 Contribution to journal
 publication status
 published
 subject
 keywords
 principal type, pseudodifferential operators, NirenbergTreves conjecture, solvability
 in
 Annals of Mathematics
 volume
 163
 issue
 2
 pages
 405  444
 publisher
 Annals of Mathematics
 external identifiers

 wos:000239506200002
 scopus:33746158191
 ISSN
 0003486X
 language
 English
 LU publication?
 yes
 id
 77855c68d5174d9bbf02fbc2b9937035 (old id 399268)
 alternative location
 http://annals.math.princeton.edu/issues/2006/March2006/Dencker.pdf
 date added to LUP
 20160401 15:19:25
 date last changed
 20201222 01:25:30
@article{77855c68d5174d9bbf02fbc2b9937035, abstract = {We give a proof of the NirenbergTreves conjecture: that local solvability of principaltype pseudodifferential 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 BealsFefferman) 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 = {0003486X}, language = {eng}, number = {2}, pages = {405444}, publisher = {Annals of Mathematics}, series = {Annals of Mathematics}, title = {The resolution of the NirenbergTreves conjecture}, url = {http://annals.math.princeton.edu/issues/2006/March2006/Dencker.pdf}, volume = {163}, year = {2006}, }