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The proof of the Nirenberg-Treves conjecture. (English) Zbl 1055.58011

Proceedings of the conference on partial differential equations, Forges-les-Eaux, France, June 2–6, 2003. Exp. Nos. I-XV. Nantes: Université de Nantes (ISBN 2-86939-207-9/pbk). Exp. No. V, 25 p. (2003).
The author studies the question of local solvability of a classical pseudodifferential operator \(P\) on a \(C^\infty\)-manifold \(M\). He proves the Nirenberg-Treves conjecture: that for principal type pseudo-differential operators local solvability is equivalent to condition \((\psi)\), that rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. The author obtains local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives. To this end he uses a new metric on the Weyl (or Beals-Fefferman) calculus.
For the entire collection see [Zbl 1027.00017].

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
47G30 Pseudodifferential operators
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