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The Proof Of The Nirenberg-Treves Conjecture According To N. Dencker And N. Lerner

In: Unpublished Manuscripts

Author

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  • Lars Hörmander

    (Lund University)

Abstract

Zusammenfassung 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.

Suggested Citation

  • Lars Hörmander, 2018. "The Proof Of The Nirenberg-Treves Conjecture According To N. Dencker And N. Lerner," Springer Books, in: Unpublished Manuscripts, chapter 0, pages 216-256, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-69850-2_23
    DOI: 10.1007/978-3-319-69850-2_23
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