Global regularity and $ {L}^{p}$-estimates for $ \overline{\partial }$ on an annulus between two strictly pseudoconvex domains in a Stein manifold

Khidr, Shaban; Abdelkader, Osama

doi:10.1016/j.crma.2013.10.020

Complex analysis

Global regularity and

L^{p}

-estimates for

\bar{\partial}

on an annulus between two strictly pseudoconvex domains in a Stein manifold
[Régularité globale et estimations

L^{p}

pour

\bar{\partial}

sur une couronne entre deux domaines strictement pseudo-convexes dans une variété de Stein]

Présenté par : Jean-Pierre Demailly

Khidr, Shaban ¹ ; Abdelkader, Osama ²

¹ Mathematics Department, Faculty of Science, King Abdelaziz University, North Jeddah, Jeddah, Saudi Arabia
² Mathematics Department, Faculty of Science, El-minia University, El-minia, Egypt

Comptes Rendus. Mathématique, Tome 351 (2013) no. 23-24, pp. 883-888.

Résumé
Abstract

Dans cette Note, nous démontrons un théorème dʼexistence $L^{2}$ pour lʼopérateur de Neumann $\bar{\partial}$ et la régularité globale au bord de lʼéquation $\bar{\partial}$ sur de domaine de type couronne $D = D_{1} ∖ {\bar{D}}_{2}$ où $D_{1}$ et $D_{2}$ sont des domaines strictement pseudo-convexes dont les bords sont réguliés dans une variété de Stein X de dimension complexe $n ⩾ 3$ , tels que ${\bar{D}}_{2} \subset D_{1} ⋐ X$ . De plus, nous obtenons des estimations de Hölder et $L^{p}$ , $1 ⩽ p ⩽ \infty$ , pour $\bar{\partial}$ sur des domaines strictement pseudo-concaves de frontière $C^{3}$ dans X.

In this note, we prove an $L^{2}$ -existence theorem for the $\bar{\partial}$ -Neumann operator and the regularity for the $\bar{\partial}$ -equation on an annulus type domain $D = D_{1} ∖ \bar{D_{2}}$ , where $D_{1}$ and $D_{2}$ are strictly pseudoconvex domains with smooth boundaries in a Stein manifold X of complex dimension $n ⩾ 3$ , such that $\bar{D_{2}} \subset D_{1} ⋐ X$ . Moreover, we obtain Hölder and $L^{p}$ estimates for the $\bar{\partial}$ -equation on strictly pseudoconcave domains with smooth $C^{3}$ -boundaries in X.

Reçu le : 2013-03-14
Accepté le : 2013-10-16
Publié le : 2013-11-07

DOI : 10.1016/j.crma.2013.10.020

Affiliations des auteurs :

Khidr, Shaban ¹ ; Abdelkader, Osama ²

¹ Mathematics Department, Faculty of Science, King Abdelaziz University, North Jeddah, Jeddah, Saudi Arabia
² Mathematics Department, Faculty of Science, El-minia University, El-minia, Egypt

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     author = {Khidr, Shaban and Abdelkader, Osama},
     title = {Global regularity and $ {L}^{p}$-estimates for $ \overline{\partial }$ on an annulus between two strictly pseudoconvex domains in a {Stein} manifold},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {883--888},
     publisher = {Elsevier},
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     doi = {10.1016/j.crma.2013.10.020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.10.020/}
}

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Khidr, Shaban; Abdelkader, Osama. Global regularity and $ {L}^{p}$-estimates for $ \overline{\partial }$ on an annulus between two strictly pseudoconvex domains in a Stein manifold. Comptes Rendus. Mathématique, Tome 351 (2013) no. 23-24, pp. 883-888. doi : 10.1016/j.crma.2013.10.020. http://www.numdam.org/articles/10.1016/j.crma.2013.10.020/

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