Estimations pour $\overline{\partial }$ dans des domaines non pseudo-convexes
Annales de l'Institut Fourier, Volume 28 (1978) no. 4, p. 239-254
We study the domains in ${\mathbf{C}}^{n}$, $\Omega$, which satisfy (locally) the following estimate:$\sum _{i,k=1}^{n}\parallel \frac{\partial {u}_{j}}{\partial {\overline{z}}_{k}}\parallel \le C\left(\parallel \overline{\partial }u\parallel +\parallel {\overline{\partial }}^{*}u\parallel +\parallel u\parallel \right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall u\in {𝒟}^{0,1}\left(V\cap \overline{\Omega }\right)$where $V$ is a neighborhood of a point ${z}_{0}$ in the boundary $\partial \Omega$.The interest of this estimate is that it is used for proving subelliptic estimates. Remark that it is always satisfied by pseudoconvex domains so it is naturally related to the behavior in $V$ of the negative parts of the eigenvalues of the Levi form.
Nous étudions les domaines $\Omega$ de ${\mathbf{C}}^{n}$ qui satisfont (localement) à l’estimation suivante :$\sum _{i,k=1}^{n}\parallel \frac{\partial {u}_{j}}{\partial {\overline{z}}_{k}}\parallel \le C\left(\parallel \overline{\partial }u\parallel +\parallel {\overline{\partial }}^{*}u\parallel +\parallel u\parallel \right),\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall u\in {𝒟}^{0,1}\left(V\cap \overline{\Omega }\right)$$V$ est un voisinage d’un point ${z}_{0}$ du bord $\partial \Omega$.L’intérêt de cette estimation réside dans son utilisation pour montrer une estimation sous-elliptique. Remarquons qu’elle est toujours satisfaite par les domaines pseudo-convexes, ce qui rend naturel le fait qu’elle soit liée au comportement dans $V$ des parties négatives des valeurs propres de la forme de Levi.
@article{AIF_1978__28_4_239_0,
author = {Derridj, Maklouf},
title = {Estimations pour $\bar{\partial }$ dans des domaines non pseudo-convexes},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Durand},
volume = {28},
number = {4},
year = {1978},
pages = {239-254},
doi = {10.5802/aif.723},
zbl = {0377.35057},
mrnumber = {80b:32021},
language = {fr},
url = {http://www.numdam.org/item/AIF_1978__28_4_239_0}
}

Derridj, Maklouf. Estimations pour $\bar{\partial }$ dans des domaines non pseudo-convexes. Annales de l'Institut Fourier, Volume 28 (1978) no. 4, pp. 239-254. doi : 10.5802/aif.723. http://www.numdam.org/item/AIF_1978__28_4_239_0/

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