# Global boundary regularity for the $\overline{partial}$-equation on $q$-pseudo-convex domains

• Volume: 16, Issue: 1, page 5-9
• ISSN: 1120-6330

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## Abstract

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For a bounded domain $D$ of ${\mathbb{C}}^{n}$, we introduce a notion of «$q$-pseudoconvexity» of new type and prove that for a given $\overline{\partial }$-closed $\left(p,r\right)$-form $f$ that is smooth up to the boundary on $D$, and for $r\ge q$, there exists a $\left(p,r-1\right)$-form $u$ smooth up to the boundary on $D$ which is a solution of the equation $\overline{\partial }u=f$

## How to cite

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Ahn, Heungju. "Global boundary regularity for the $\overline{\partial}$-equation on $q$-pseudo-convex domains." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 16.1 (2005): 5-9. <http://eudml.org/doc/252429>.

@article{Ahn2005,
abstract = {For a bounded domain $D$ of $\mathbb\{C\}^\{n\}$, we introduce a notion of «$q$-pseudoconvexity» of new type and prove that for a given $\overline\{\partial\}$-closed $(p,r)$-form $f$ that is smooth up to the boundary on $D$, and for $r \ge q$, there exists a $(p,r-1)$-form $u$ smooth up to the boundary on $D$ which is a solution of the equation $\overline\{\partial\} u = f$},
author = {Ahn, Heungju},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {$\overline\{\partial \}$ ∂ ¯ -equation; $q$ q -pseudoconvexity; Cauchy-Riemann system; -equation; -pseudoconvexity},
language = {eng},
month = {3},
number = {1},
pages = {5-9},
publisher = {Accademia Nazionale dei Lincei},
title = {Global boundary regularity for the $\overline\{\partial\}$-equation on $q$-pseudo-convex domains},
url = {http://eudml.org/doc/252429},
volume = {16},
year = {2005},
}

TY - JOUR
AU - Ahn, Heungju
TI - Global boundary regularity for the $\overline{\partial}$-equation on $q$-pseudo-convex domains
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2005/3//
PB - Accademia Nazionale dei Lincei
VL - 16
IS - 1
SP - 5
EP - 9
AB - For a bounded domain $D$ of $\mathbb{C}^{n}$, we introduce a notion of «$q$-pseudoconvexity» of new type and prove that for a given $\overline{\partial}$-closed $(p,r)$-form $f$ that is smooth up to the boundary on $D$, and for $r \ge q$, there exists a $(p,r-1)$-form $u$ smooth up to the boundary on $D$ which is a solution of the equation $\overline{\partial} u = f$
LA - eng
KW - $\overline{\partial }$ ∂ ¯ -equation; $q$ q -pseudoconvexity; Cauchy-Riemann system; -equation; -pseudoconvexity
UR - http://eudml.org/doc/252429
ER -

## References

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1. CHEN, S.-C. - SHAW, M.-C., Partial differential equations in several complex variables. AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI2001. MR 2001m:32071 Zbl0963.32001MR1800297
2. HO, L.-H., $\overline{\partial }$-problem on weakly $q$-convex domains. Math. Ann., 290, no. 1, 1991, 3-18. MR 92j:32052 Zbl0714.32006MR1107660DOI10.1007/BF01459235
3. KOHN, J.J., Global regularity for $\overline{\partial }$ on weakly pseudo-convex manifolds. Trans. Amer. Math. Soc., 181, 1973, 273-292. MR 49 #9442 Zbl0276.35071MR344703
4. KOHN, J.J., Methods of partial differential equations in complex analysis. Amer. Math. Soc. Proc. Sympos. Pure Math., XXX, Part. 1, Providence, RI1977, 215-237. Zbl0635.32011MR477156
5. MICHEL, V., Sur la régularité ${C}^{\mathrm{\infty }}$ du $\overline{\partial }$ au bord d'un domaine de ${\mathbb{C}}^{n}$ dont la forme de Levi a exactement $s$ valeurs propres strictement négatives. Math. Ann., 295, 1993, no. 1, 135-161. MR 93k:32030 Zbl0788.32010MR1198845DOI10.1007/BF01444880
6. ZAMPIERI, G., $q$-pseudoconvexity and regularity at the boundary for solutions of the $\overline{\partial }$-problem. Compositio Math., 121, 2000, no. 2, 155-162. MR 2001a:32048 Zbl0953.32030MR1757879DOI10.1023/A:1001811318865

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