Closed Range for $\overline{\partial }$ and ${\overline{\partial }}_b$ on Bounded Hypersurfaces in Stein Manifolds

Harrington, Phillip S.; Raich, Andrew S.

Annales de l'Institut Fourier, Tome 65 (2015) no. 4, pp. 1711-1754.

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 $(0,q)$-formes harmoniques sont triviales et (ii) si $\partial \Omega$ satisfait une $Z\left(q\right)$ faible et une $Z(n-1-q)$ faible, alors ${\overline{\partial }}_b$ a une image fermée sur les $(0,q)$-formes sur $\partial \Omega$. 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 $Z\left(q\right)$ faible.

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 $(0,q)$-forms are trivial and (ii) if $\partial \Omega$ satisfies weak $Z\left(q\right)$ and weak $Z(n-1-q)$, then ${\overline{\partial }}_b$ has closed range on $(0,q)$-forms on $\partial \Omega$. We provide examples to show that our condition contains examples that are excluded from $(q-1)$-pseudoconvexity and the authors’ previous notion of weak $Z\left(q\right)$.

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

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