Cutting the loss of derivatives for solvability under condition $\left(\Psi \right)$
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 $\left(\psi \right)$ 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 $\left(\psi \right)$ 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 ($\Psi$) 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 ($\Psi$) 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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