## Abstract and Applied Analysis

### Global Regularity for the ${\stackrel{-}{\partial }}_{b}$-Equation on $CR$ Manifolds of Arbitrary Codimension

#### Abstract

Let $M$ be a ${\mathcal{C}}^{\mathrm{\infty }}$ compact $CR$ manifold of $CR$-codimension $\mathcal{l}\ge 1$ and $CR$-dimension $n-\mathcal{l}$ in a complex manifold $X$ of complex dimension $n\ge 3$. In this paper, assuming that $M$ satisfies condition $Y(s)$ for some $s$ with $1\le s\le n-\mathcal{l}-1$, we prove an ${L}^{2}$-existence theorem and global regularity for the solutions of the tangential Cauchy-Riemann equation for $(0,s)$-forms on $M$.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 326434, 11 pages.

Dates
First available in Project Euclid: 2 October 2014

https://projecteuclid.org/euclid.aaa/1412277348

Digital Object Identifier
doi:10.1155/2014/326434

Mathematical Reviews number (MathSciNet)
MR3224308

Zentralblatt MATH identifier
07022176

#### Citation

Khidr, Shaban; Abdelkader, Osama. Global Regularity for the ${\stackrel{-}{\partial }}_{b}$ -Equation on $CR$ Manifolds of Arbitrary Codimension. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 326434, 11 pages. doi:10.1155/2014/326434. https://projecteuclid.org/euclid.aaa/1412277348

#### References

• 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.
• M.-C. Shaw, “${L}^{2}$-estimates and existence theorems for the tangential Cauchy-Riemann complex,” Inventiones Mathematicae, vol. 82, no. 1, pp. 133–150, 1985.
• 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.
• J. J. Kohn, “The range of the tangential Cauchy-Riemann operator,” Duke Mathematical Journal, vol. 53, no. 2, pp. 525–545, 1986.
• A. C. Nicoara, “Global regularity for ${\overline{\partial }}_{b}$ on weakly pseudoconvex CR manifolds,” Advances in Mathematics, vol. 199, no. 2, pp. 356–447, 2006.
• J. J. Kohn and A. C. Nicoara, “The ${\overline{\partial }}_{b}$ equation on weakly pseudo-convex CR manifolds of dimension 3,” Journal of Functional Analysis, vol. 230, no. 2, pp. 251–272, 2006.
• P. S. Harrington and A. Raich, “Regularity results for ${\overline{\partial }}_{b}$ on CR-manifolds of hypersurface type,” Communications in Partial Differential Equations, vol. 36, no. 1, pp. 134–161, 2011.
• S. Khidr and O. Abdelkader, “Global regularity and ${L}^{p}$-estimates for $\overline{\partial }$ 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.
• S. Khidr and O. Abdelkader, “The ${\overline{\partial }}$-equation on an annulus between two strictly $q$-convex domains with smooth boundaries,” Complex Analysis and Operator Theory, 2013.
• M.-C. Shaw and L. Wang, “Hölder and ${L}^{p}$ estimates for ${{\square}}_{b}$ on CR manifolds of arbitrary codimension,” Mathematische Annalen, vol. 331, no. 2, pp. 297–343, 2005.
• 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.
• A. Boggess, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1991.
• J. J. Kohn, “Hypoellipticity and loss of derivatives,” Annals of Mathematics: Second Series, vol. 162, no. 2, pp. 943–986, 2005.
• 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.
• L. Hörmander, “${L}^{2}$ estimates and existence theorems for the $\overline{\partial }$ operator,” Acta Mathematica, vol. 113, pp. 89–152, 1965.
• E. J. Straube, “The complex Green operator on CR-submanifolds of ${C}^{n}$ of hypersurface type: compactness,” Transactions of the American Mathematical Society, vol. 364, no. 8, pp. 4107–4125, 2012.
• A. Raich, “Compactness of the complex Green operator on CR-manifolds of hypersurface type,” Mathematische Annalen, vol. 348, no. 1, pp. 81–117, 2010.
• A. S. Raich and E. J. Straube, “Compactness of the complex Green operator,” Mathematical Research Letters, vol. 15, no. 4, pp. 761–778, 2008.
• 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.
• J. J. Kohn and L. Nirenberg, “Non-coercive boundary value problems,” Communications on Pure and Applied Mathematics, vol. 18, pp. 443–492, 1965.
• 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. \endinput