Cutting the loss of derivatives for solvability under condition (Ψ)
Bulletin de la Société Mathématique de France, Volume 134 (2006) no. 4, pp. 559-631.

For a principal type pseudodifferential operator, we prove that condition (ψ) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from ϵ+3/2 for any ϵ>0 (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition (ψ) does not imply local solvability with a loss of 1 derivative, so we have to content ourselves with a loss >1.

Pour un opérateur de type principal, nous démontrons que la condition (Ψ) implique la résolubilité locale avec perte de 3/2 dérivées. Nous utilisons beaucoup d’éléments de la démonstration par Dencker de la conjecture de Nirenberg-Treves et nous limitons la perte de dérivées à 3/2, améliorant le résultat le plus récent de Dencker (perte de ϵ+3/2 dérivées pour tout ϵ>0). La condition (Ψ) n’impliquant pas la résolubilité locale avec perte d’une dérivée, nous devons nous contenter d’une perte >1.

DOI: 10.24033/bsmf.2522
Classification: 35S05, 47G30
Keywords: solvability, a priori estimates, pseudodifferential operators
Mot clés : résolubilité, estimations a priori, op´erateurs pseudodifférentiels
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Lerner, Nicolas. Cutting the loss of derivatives for solvability under condition $(\Psi )$. Bulletin de la Société Mathématique de France, Volume 134 (2006) no. 4, pp. 559-631. doi : 10.24033/bsmf.2522. http://archive.numdam.org/articles/10.24033/bsmf.2522/

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