Czechoslovak Mathematical Journal

, Volume 67, Issue 2, pp 515–523

Ck-regularity for the \(\bar \partial \) -equation with a support condition



Let D be a Cdq-convex intersection, d ≥ 2, 0 ≤ qn − 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, Ck-estimates, k = 2, 3,...,∞, for solutions to the \(\bar \partial \) -equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when nqsn. In addition, we solve the \(\bar \partial \) -equation with a support condition in Ck-spaces. More precisely, we prove that for a \(\bar \partial \) -closed form f in C0,qk(XD,E), 1 ≤ qn − 2, n ≥ 3, with compact support and for ε with 0 < ε < 1 there exists a form u in C0,q−1kε(XD,E) with compact support such that \(\bar \partial u = f\) in \(X\backslash \bar D\) . Applications are given for a separation theorem of Andreotti-Vesentini type in Ck-setting and for the solvability of the \(\bar \partial \) -equation for currents.


\(\bar \partial \) -equation q-convexity Ck-estimate 

MSC 2010

32F10 32W05 


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  1. [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.MathSciNetMATHGoogle Scholar
  2. [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.MathSciNetMATHGoogle Scholar
  3. [3]
    M.-Y. Barkatou, S. Khidr: Global solution with C k-estimates for \(\bar \partial \)-equation on q-convex intersections. Math. Nachr. 284 (2011), 2024–2031.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    J. Brinkschulte: The \(\bar \partial \)-problem with support conditions on some weakly pseudoconvex domains. Ark. Mat. 42 (2004), 259–282.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    H. Grauert: Kantenkohomologie. Compos. Math. 44 (1981), 79–101. (In German.)MathSciNetMATHGoogle Scholar
  6. [6]
    G. M. Henkin, J. Leiterer: Andreotti-Grauert Theory by Integral Formulas. Progress in Mathematics 74, Birkhäuser, Boston, 1988.Google Scholar
  7. [7]
    S. Khidr, M.-Y. Barkatou: Global solutions with C k-estimates for \(\bar \partial \)-equations on q-concave intersections. Electron. J. Differ. Equ. 2013 (2013), Paper No. 62, 10 pages.Google Scholar
  8. [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, pp. 416–439.Google Scholar
  9. [9]
    I. Lieb, R. M. Range: Lösungsoperatoren für den Cauchy-Riemann-Komplex mit C k-Abschätzungen. Math. Ann. 253 (1980), 145–164. (In German.)MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    J. Michel: Randregularität des \(\bar \partial \)-Problems für stückweise streng pseudokonvexe Gebiete in Cn. Math. Ann. 280 (1988), 45–68. (In German.)MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    J. Michel, A. Perotti: C k-regularity for the \(\bar \partial \)-equation on strictly pseudoconvex domains with piecewise smooth boundaries. Math. Z. 203 (1990), 415–427.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    H. Ricard: Estimations 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.)MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    S. Sambou: Résolution du \(\bar \partial \) pour les courants prolongeables. Math. Nachr. 235 (2002), 179–190. (In French.)MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    S. Sambou: Résolution du \(\bar \partial \) pour les courants prolongeables définis dans un anneau. Ann. Fac. Sci. Toulouse, VI. Sér., Math. 11 (2002), 105–129. (In French.)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2017

Authors and Affiliations

  1. 1.Mathematics Department, Faculty of ScienceUniversity of JeddahJeddahSaudi Arabia
  2. 2.Mathematics Department, Faculty of ScienceBeni-Suef UniversityBeni-SuefEgypt
  3. 3.Mathematics Department, Faculty of ScienceMinia UniversityMiniaEgypt

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