The verification of the Nirenberg-Treves conjecture

The verification of the Nirenberg-Treves conjecture
[La vérification de la conjecture de Nirenberg-Treves]

Lerner, Nicolas

Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 960, pp. 211-235.

Résumé
Abstract

Dans une série de preprints récents, Nils Dencker démontre que la condition $(Ψ)$ implique la résolubilité locale des opérateurs pseudodifférentiels de type principal (complexe) avec une perte de deux dérivées, établissant la dernière partie de la conjecture de Nirenberg-Treves, formulée en 1971. L’origine de cette question remonte au contre-exemple de Hans Lewy, publié en 1957. Nous suivrons dans notre exposé une partie du développement de l’analyse microlocale afférente et mettrons en évidence les nouvelles idées géométriques apportées par Dencker.

In a series of recent papers, Nils Dencker proves that condition $(ψ)$ implies the local solvability of principal type pseudodifferential operators (with loss of $\frac{3}{2} + ϵ$ derivatives for all positive $ϵ$ ), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of $\frac{3}{2}$ derivatives.

EuDML MR Zbl

Classification : 35S05, 47G30
Keywords: résolubilité, opérateurs pseudodifférentiels, estimations d'énergie, opérateurs non autoadjoints
Mot clés : solvability, pseudodifferential operators, energy estimates, nonselfadjoint operators

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Lerner, Nicolas. The verification of the Nirenberg-Treves conjecture, dans Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 960, pp. 211-235. http://www.numdam.org/item/SB_2005-2006__48__211_0/

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