Manuscripta Mathematica

, Volume 144, Issue 3–4, pp 517–534 | Cite as

Existence and compactness for the \({\overline{\partial}}\) -Neumann operator on q-convex domains

  • Mau Hai Le
  • Quang Dieu NguyenEmail author
  • Xuan Hong Nguyen


The aim of this paper is to give a sufficient condition for existence and compactness of the \({\overline{\partial}}\) -Neumann operator N q on \({L^2_{(0,q)}(\Omega)}\) in the case Ω is an arbitrary q-convex domain in \({\mathbb{C}^n}\).

Mathematics Subject Classification (2010)

Primary 32W05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahn H., Dieu N.Q.: The Donnelly–Fefferman theorem on q-pseudoconvex domains. Osaka J. Math. 46, 599–610 (2009)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Catlin, D.W.: Global regularity of the \({\overline{\partial}}\) -Neumann problem. In: Complex Analysis of Several Variables (Madison, 1982). Proceedings Symposia in Pure Mathematics, 41, pp. 39–49 (1984)Google Scholar
  3. 3.
    Chen S.C., Shaw, M.C.: Partial Differential Equations in Several Complex Variables, Studied in Advanced Mathematics, vol. 19. American Mathematical Society, Providence, RI (2001)Google Scholar
  4. 4.
    Demailly, J.: Complex analytic and differential geometry.
  5. 5.
    Gansberger, K.: On the weighted \({\overline{\partial}}\) -Neumann problem on unbounded domains. arXiv:0912.0841
  6. 6.
    Gansberger, K., Haslinger, F.: Compactness estimates for the \({\overline{\partial}}\) -Neumann problem in weighted L 2-spaces on \({\mathbb{C}^n}\), to appear in Proccedings of the Conference on Complex Analysis 2008, in honour of Linda Rothschild. arXiv:0903.1783
  7. 7.
    Hai L.M., Dieu N.Q., Hong N.X.: L 2-Approximation of differential forms by \({\overline{\partial}}\) -closed ones on smooth hypersurfaces. J. Math. Anal. Appl. 383, 379–390 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ho L.H.: \({\overline\partial}\) -Problem on weakly q-convex domains. Math. Ann. 290, 3–18 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Sibony N.: Une classes des domaines pseudoconvexe. Duke Math. J. 55, 299–319 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Straube E.J.: Plurisubharmonic functions and subellipticity of the \({\overline{\partial}}\) -Neumann problem on non-smooth domains. Math. Res. Lett. 4, 459–467 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Straube, E.J.: Lectures on the L 2-Sobolev theory of the \({\overline{\partial}}\) -Neumann problem. In: ESI Lectures in Mathematics and Physics. European Mathematical Society, Zürich (2010)Google Scholar
  12. 12.
    Walsh J.: Continuity of envelopes of plurisubharmonic functions. J. Math. Mech. 18, 143–148 (1968)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Mau Hai Le
    • 1
  • Quang Dieu Nguyen
    • 1
    Email author
  • Xuan Hong Nguyen
    • 1
  1. 1.Department of MathematicsHanoi National University of Education (Dai hoc Su Pham Ha Noi)HanoiVietnam

Personalised recommendations