Czechoslovak Mathematical Journal, online first, 9 pp.

$\mathcal C^k$-regularity for the $\bar\partial$-equation with a support condition

Shaban Khidr, Osama Abdelkader

Received January 27, 2016.   First published March 20, 2017.

Shaban Khidr, Mathematics Department, Faculty of Science, University of Jeddah, Asfan St., Jeddah 21589, Saudi Arabia, and Mathematics Department, Faculty of Science, Beni-Suef University, Salah Salem St., Beni-Suef 62511, Egypt, e-mail:; Osama Abdelkader, Mathematics Department, Faculty of Science, Minia University, Main Road St., Minia 61915, Egypt, e-mail:

Abstract: Let $D$ be a $\mathcal{C}^d$ $q$-convex intersection, $d\geq2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, $\mathcal C^k$-estimates, $k=2, 3, \dots, \infty$, for solutions to the $\bar\partial$-equation with small loss of smoothness are obtained for $E$-valued $(0, s)$-forms on $D$ when $ n-q\le s\le n$. In addition, we solve the $\bar\partial$-equation with a support condition in $\mathcal C^k$-spaces. More precisely, we prove that for a $\bar\partial$-closed form $f$ in $\mathcal C_{0,q}^k(X\setminus D, E)$, $1\le q\le n-2$, $n\ge3$, with compact support and for $\varepsilon$ with $0<\varepsilon<1$ there exists a form $u$ in $\mathcal C_{0,q-1}^{k-\varepsilon}(X\setminus D, E)$ with compact support such that $\bar{\partial}u=f$ in $X\setminus\ol D$. Applications are given for a separation theorem of Andreotti-Vesentini type in $\mathcal C^k$-setting and for the solvability of the $\bar\partial$-equation for currents.

Keywords: $\bar\partial$-equation; $q$-convexity; $\mathcal C^k$-estimate

Classification (MSC 2010): 32F10, 32W05

DOI: 10.21136/CMJ.2017.0039-16

Full text available as PDF.

  [1] A. Andreotti, C. D. Hill: E. E. Levi convexity and the Hans Lewy problem I: Reduction to vanishing theorems. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 26 (1972), 325-363. MR 0460725 | Zbl 0256.32007
  [2] A. Andreotti, C. D. Hill: E. E. Levi convexity and the Hans Lewy problem II: Vanishing theorems. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 26 (1972), 747-806. MR 0477150 | Zbl 0283.32013
  [3] M.-Y. Barkatou, S. Khidr: Global solution with $\mathcal C^k$-estimates for $\overline\partial$-equation on $q$-convex intersections. Math. Nachr. 284 (2011), 2024-2031. DOI 10.1002/mana.200910063 | MR 2844676 | Zbl 1227.32016
  [4] J. Brinkschulte: The $\overline\partial$-problem with support conditions on some weakly pseudoconvex domains. Ark. Mat. 42 (2004), 259-282. DOI 10.1007/BF02385479 | MR 2101387 | Zbl 1078.32023
  [5] H. Grauert: Kantenkohomologie. Compos. Math. 44 (1981), 79-101. (In German.) MR 0662457 | Zbl 0512.32011
  [6] G. M. Henkin, J. Leiterer: Andreotti-Grauert Theory by Integral Formulas. Progress in Mathematics 74, Birkhäuser, Boston (1988). DOI 10.1007/978-1-4899-6724-4 | MR 0986248 | Zbl 0654.32002
  [7] S. Khidr, M.-Y. Barkatou: Global solutions with $\mathcal C^k$-estimates for $\bar\partial$-equations on $q$-concave intersections. Electron. J. Differ. Equ. 2013 (2013), Paper No. 62, 10 pages. MR 3040639 | Zbl 1287.32002
  [8] C. Laurent-Thiébaut, J. Leiterer: The Andreotti-Vesentini separation theorem with $C^k$ estimates and extension of CR-forms. Several Complex Variables, Proc. Mittag-Leffler Inst., Stockholm, 1987/1988 Math. Notes 38, Princeton Univ. Press, Princeton (1993), 416-439. MR 1207871 | Zbl 0776.32012
  [9] I. Lieb, R. M. Range: Lösungsoperatoren für den Cauchy-Riemann-Komplex mit $\mathcal C^k$-Abschätzungen. Math. Ann. 253 (1980), 145-164. (In German.) DOI 10.1007/BF01578911 | MR 0597825 | Zbl 0441.32007
  [10] J. Michel: Randregularität des $\overline\partial$-Problems für stückweise streng pseudokonvexe Gebiete in $\mathbb C^n$. Math. Ann. 280 (1988), 45-68. (In German.) DOI 10.1007/BF01474180 | MR 0928297 | Zbl 0617.32032
  [11] J. Michel, A. Perotti: $C^k$-regularity for the $\overline\partial$-equation on strictly pseudoconvex domains with piecewise smooth boundaries. Math. Z. 203 (1990), 415-427. DOI 10.1007/BF02570747 | MR 1038709 | Zbl 0673.32019
  [12] H. Ricard: Estimations $\mathcal C^k$ pour l'opérateur de Cauchy-Riemann sur des domaines à coins $q$-convexes et $q$-concaves. Math. Z. 244 (2003), 349-398. (In French.) DOI 10.1007/s00209-003-0504-4 | MR 1992543 | Zbl 1036.32012
  [13] S. Sambou: Résolution du $\overline\partial$ pour les courants prolongeables. Math. Nachr. 235 (2002), 179-190. (In French.) DOI 10.1002/1522-2616(200202)235:1<179::AID-MANA179>3.0.CO;2-8 | MR 1889284 | Zbl 1007.32012
  [14] S. Sambou: Résolution du $\overline\partial$ pour les courants prolongeables définis dans un anneau. Ann. Fac. Sci. Toulouse, VI. Sér., Math. 11 (2002), 105-129. (In French.) DOI 10.5802/afst.1020 | MR 1986385 | Zbl 1080.32502

Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at
Subscribers of Springer need to access the articles on their site, which is

[List of online first articles] [Contents of Czechoslovak Mathematical Journal]