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Erschienen in:
2018 | OriginalPaper | Buchkapitel
The Proof Of The Nirenberg-Treves Conjecture According To N. Dencker And N. Lerner
verfasst von : Lars Hörmander
Erschienen in: Unpublished Manuscripts
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Since 1980 the remaining part of this conjecture has been to prove that if P is a pseudodifferential operator of principal type, then the equation P u = f has a distribution solution locally for every f ∈ C∞ if the principal symbol p of P satisfies a condition called (Ψ). It means roughly speaking that Im p does not change sign from − to + when one moves in the positive direction along a bicharacteristic of Re p, where Re p = 0. (See Definition 26.4.6 in [H1] for a precise formulation and Theorem 26.4.7 for a proof of the necessity.) When P is a differential operator and in a number of other cases a positive answer is known, stating that if f is in the Sobolev space H(s) and P is of order m then a local solution in H(s+m−1) exists for these operators.