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

Harrington, Phillip S.; Raich, Andrew S.

doi:10.5802/aif.2972

Closed Range for

\bar{\partial}

and

{\bar{\partial}}_{b}

on Bounded Hypersurfaces in Stein Manifolds

Harrington, Phillip S.; Raich, Andrew S.¹

¹ Department of Mathematical Sciences 1 University of Arkansas SCEN 309 Fayetteville, AR 7201 (USA)

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

Abstract
Résumé

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 $\bar{\partial}$ . Under the hypothesis of weak $Z (q)$ , we also show (i) that harmonic $(0, q)$ -forms are trivial and (ii) if $\partial Ω$ satisfies weak $Z (q)$ and weak $Z (n - 1 - q)$ , then ${\bar{\partial}}_{b}$ has closed range on $(0, q)$ -forms on $\partial Ω$ . We provide examples to show that our condition contains examples that are excluded from $(q - 1)$ -pseudoconvexity and the authors’ previous notion of weak $Z (q)$ .

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 $\bar{\partial}$ 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 $\partial Ω$ satisfait une $Z (q)$ faible et une $Z (n - 1 - q)$ faible, alors ${\bar{\partial}}_{b}$ a une image fermée sur les $(0, q)$ -formes sur $\partial Ω$ . 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.

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

Author's affiliations:

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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     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},
     pages = {1711--1754},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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, Volume 65 (2015) no. 4, pp. 1711-1754. doi : 10.5802/aif.2972. https://aif.centre-mersenne.org/articles/10.5802/aif.2972/

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