Cutting the loss of derivatives for solvability under condition (Ψ)
Bulletin de la Société Mathématique de France, Volume 134 (2006) no. 4, p. 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 : https://doi.org/10.24033/bsmf.2522
Classification:  35S05,  47G30
Keywords: solvability, a priori estimates, pseudodifferential operators
@article{BSMF_2006__134_4_559_0,
     author = {Lerner, Nicolas},
     title = {Cutting the loss of derivatives for solvability under condition $(\Psi )$},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {134},
     number = {4},
     year = {2006},
     pages = {559-631},
     doi = {10.24033/bsmf.2522},
     zbl = {1181.35355},
     mrnumber = {2364944},
     language = {en},
     url = {http://www.numdam.org/item/BSMF_2006__134_4_559_0}
}
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://www.numdam.org/item/BSMF_2006__134_4_559_0/

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