Closed Range for ¯ and ¯ b on Bounded Hypersurfaces in Stein Manifolds
[Image fermée pour ¯ et ¯ b sur les hypersurfaces bornées dans les variétés de Stein]
Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1711-1754.

Nous définissons Z(q) faible, une généralisation de Z(q) sur les domaines bornés Ω dans une variété de Stein M n qui suffit à prouver que l’image de ¯ est fermée. Sous l’hypothèse d’une Z(q) faible, nous montrons également que (i) les (0,q)-formes harmoniques sont triviales et (ii) si Ω satisfait une Z(q) faible et une Z(n-1-q) faible, alors ¯ b a une image fermée sur les (0,q)-formes sur Ω. Nous fournissons des exemples pour montrer que notre condition contient des exemples qui sont exclus de la (q-1)-pseudoconvexité et la notion précédente des auteurs de Z(q) faible.

We define weak Z(q), a generalization of Z(q) on bounded domains Ω in a Stein manifold M n that suffices to prove closed range of ¯. Under the hypothesis of weak Z(q), we also show (i) that harmonic (0,q)-forms are trivial and (ii) if Ω satisfies weak Z(q) and weak Z(n-1-q), then ¯ b has closed range on (0,q)-forms on Ω. We provide examples to show that our condition contains examples that are excluded from (q-1)-pseudoconvexity and the authors’ previous notion of weak Z(q).

DOI : 10.5802/aif.2972
Classification : 32W05, 32W10, 32Q28, 35N15
Keywords: Stein manifold, $\bar{\partial }_b$, tangential Cauchy-Riemann operator, closed range, $\bar{\partial }$-Neumann, weak $Z(q)$, $q$-pseudoconvexity
Mot clés : variété de Stein, $\bar{\partial }_b$, tangentielle opérateur de Cauchy-Riemann, image fermée, $\bar{\partial }$-Neumann, faible $Z(q)$, $q$-pseudoconvexité
Harrington, Phillip S.  ; Raich, Andrew S. 1

1 Department of Mathematical Sciences 1 University of Arkansas SCEN 309 Fayetteville, AR 7201 (USA)
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Harrington, Phillip S.; Raich, Andrew S. Closed Range for $\bar{\partial }$ and $\bar{\partial }_b$ on Bounded Hypersurfaces in Stein Manifolds. Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1711-1754. doi : 10.5802/aif.2972. http://www.numdam.org/articles/10.5802/aif.2972/

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