Closed Range for $\overline{\partial }$ and ${\overline{\partial }}_{b}$ on Bounded Hypersurfaces in Stein Manifolds
Annales de l'Institut Fourier, Volume 65 (2015) no. 4, p. 1711-1754

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

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

DOI : https://doi.org/10.5802/aif.2972
Classification:  32W05,  32W10,  32Q28,  35N15
Keywords: Stein manifold, ${\overline{\partial }}_{b}$, tangential Cauchy-Riemann operator, closed range, $\overline{\partial }$-Neumann, weak $Z\left(q\right)$, $q$-pseudoconvexity
@article{AIF_2015__65_4_1711_0,
author = {Harrington, Phillip S. and Raich, Andrew S.},
title = {Closed Range for $\bar{\partial }$ and $\bar{\partial }\_b$ on Bounded Hypersurfaces in Stein Manifolds},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {65},
number = {4},
year = {2015},
pages = {1711-1754},
doi = {10.5802/aif.2972},
language = {en},
url = {http://www.numdam.org/item/AIF_2015__65_4_1711_0}
}

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, Volume 65 (2015) no. 4, pp. 1711-1754. doi : 10.5802/aif.2972. http://www.numdam.org/item/AIF_2015__65_4_1711_0/

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