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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 326434, 11 pages
http://dx.doi.org/10.1155/2014/326434
Research Article

Global Regularity for the -Equation on Manifolds of Arbitrary Codimension

1Mathematics Department, Faculty of Science, King Abdulaziz University, North Jeddah, Jeddah 21589, Saudi Arabia
2Mathematics Department, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt
3Mathematics Department, Faculty of Science, Minia University, El-Minia 61915, Egypt

Received 9 April 2014; Accepted 12 May 2014; Published 12 June 2014

Academic Editor: Dumitru Baleanu

Copyright © 2014 Shaban Khidr and Osama Abdelkader. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. J. Kohn and H. Rossi, “On the extension of holomorphic functions from the boundary of a complex manifold,” Annals of Mathematics: Second Series, vol. 81, pp. 451–472, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. M.-C. Shaw, “L2-estimates and existence theorems for the tangential Cauchy-Riemann complex,” Inventiones Mathematicae, vol. 82, no. 1, pp. 133–150, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. P. Boas and M.-C. Shaw, “Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries,” Mathematische Annalen, vol. 274, no. 2, pp. 221–231, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. J. Kohn, “The range of the tangential Cauchy-Riemann operator,” Duke Mathematical Journal, vol. 53, no. 2, pp. 525–545, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. A. C. Nicoara, “Global regularity for ¯b on weakly pseudoconvex CR manifolds,” Advances in Mathematics, vol. 199, no. 2, pp. 356–447, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. J. Kohn and A. C. Nicoara, “The ¯b equation on weakly pseudo-convex CR manifolds of dimension 3,” Journal of Functional Analysis, vol. 230, no. 2, pp. 251–272, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. P. S. Harrington and A. Raich, “Regularity results for ¯b on CR-manifolds of hypersurface type,” Communications in Partial Differential Equations, vol. 36, no. 1, pp. 134–161, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. S. Khidr and O. Abdelkader, “Global regularity and Lp-estimates for ¯ on an annulus between two strictly pseudoconvex domains in a Stein manifold,” Comptes Rendus Mathématique. Académie des Sciences: Paris, vol. 351, no. 23-24, pp. 883–888, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Khidr and O. Abdelkader, “The --equation on an annulus between two strictly q-convex domains with smooth boundaries,” Complex Analysis and Operator Theory, 2013. View at Publisher · View at Google Scholar · View at Scopus
  10. M.-C. Shaw and L. Wang, “Hölder and Lp estimates for b on CR manifolds of arbitrary codimension,” Mathematische Annalen, vol. 331, no. 2, pp. 297–343, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, vol. 75 of Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, USA, 1972. View at MathSciNet
  12. A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1991. View at MathSciNet
  13. J. J. Kohn, “Hypoellipticity and loss of derivatives,” Annals of Mathematics: Second Series, vol. 162, no. 2, pp. 943–986, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Derridj, “Subelliptic estimates for some systems of complex vector fields,” in Hyperbolic Problems and Regularity Questions, Trends in Mathematics, pp. 101–108, Birkhäuser, Basel, Switzerland, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  15. L. Hörmander, “L2 estimates and existence theorems for the ¯ operator,” Acta Mathematica, vol. 113, pp. 89–152, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. E. J. Straube, “The complex Green operator on CR-submanifolds of Cn of hypersurface type: compactness,” Transactions of the American Mathematical Society, vol. 364, no. 8, pp. 4107–4125, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. A. Raich, “Compactness of the complex Green operator on CR-manifolds of hypersurface type,” Mathematische Annalen, vol. 348, no. 1, pp. 81–117, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. A. S. Raich and E. J. Straube, “Compactness of the complex Green operator,” Mathematical Research Letters, vol. 15, no. 4, pp. 761–778, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Munasinghe and E. J. Straube, “Geometric sufficient conditions for compactness of the complex Green operator,” Journal of Geometric Analysis, vol. 22, no. 4, pp. 1007–1026, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. J. J. Kohn and L. Nirenberg, “Non-coercive boundary value problems,” Communications on Pure and Applied Mathematics, vol. 18, pp. 443–492, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J. J. Kohn, “Methods of partial differential equations in complex analysis, complex variables (Williamstown, Mass., 1975),” in Proceedings of Symposia in Pure Mathematics, vol. 30, pp. 215–237, American Mathematical Society, 1977.