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Global regularity for the \(\overline{\partial}_b\)-equation on \(C R\) manifolds of arbitrary codimension. (English) Zbl 1470.32102

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

MSC:

32V05 CR structures, CR operators, and generalizations
32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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[1] Kohn, J. J.; Rossi, H., On the extension of holomorphic functions from the boundary of a complex manifold, Annals of Mathematics: Second Series, 81, 451-472 (1965) · Zbl 0166.33802
[2] Shaw, M.-C., \(L^2\)-estimates and existence theorems for the tangential Cauchy-Riemann complex, Inventiones Mathematicae, 82, 1, 133-150 (1985) · Zbl 0581.35057
[3] Boas, H. P.; Shaw, M.-C., Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries, Mathematische Annalen, 274, 2, 221-231 (1986) · Zbl 0588.32023
[4] Kohn, J. J., The range of the tangential Cauchy-Riemann operator, Duke Mathematical Journal, 53, 2, 525-545 (1986) · Zbl 0609.32015
[5] Nicoara, A. C., Global regularity for \(\overline{\partial}_b\) on weakly pseudoconvex CR manifolds, Advances in Mathematics, 199, 2, 356-447 (2006) · Zbl 1091.32017
[6] Kohn, J. J.; Nicoara, A. C., The \(\overline{\partial}_b\) equation on weakly pseudo-convex CR manifolds of dimension 3, Journal of Functional Analysis, 230, 2, 251-272 (2006) · Zbl 1109.32032
[7] Harrington, P. S.; Raich, A., Regularity results for \(\overline{\partial}_b\) on CR-manifolds of hypersurface type, Communications in Partial Differential Equations, 36, 1, 134-161 (2011) · Zbl 1216.32025
[8] Khidr, S.; Abdelkader, O., 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, 351, 23-24, 883-888 (2013) · Zbl 1288.32056
[9] Khidr, S.; Abdelkader, O., The \(\overset{-}{\partial} \)-equation on an annulus between two strictly \(q\)-convex domains with smooth boundaries, Complex Analysis and Operator Theory (2013) · Zbl 1306.32028
[10] Shaw, M.-C.; Wang, L., Hölder and \(L^p\) estimates for \(\square_b\) on CR manifolds of arbitrary codimension, Mathematische Annalen, 331, 2, 297-343 (2005) · Zbl 1076.32030
[11] Folland, G. B.; Kohn, J. J., The Neumann Problem for the Cauchy-Riemann Complex. The Neumann Problem for the Cauchy-Riemann Complex, Annals of Mathematics Studies, 75 (1972), Princeton, NJ, USA: Princeton University Press, Princeton, NJ, USA · Zbl 0247.35093
[12] Boggess, A., CR Manifolds and the Tangential Cauchy-Riemann Complex. CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics (1991), Boca Raton, Fla, USA: CRC Press, Boca Raton, Fla, USA · Zbl 0760.32001
[13] Kohn, J. J., Hypoellipticity and loss of derivatives, Annals of Mathematics: Second Series, 162, 2, 943-986 (2005) · Zbl 1107.35044
[14] Derridj, M., Subelliptic estimates for some systems of complex vector fields, Hyperbolic Problems and Regularity Questions. Hyperbolic Problems and Regularity Questions, Trends in Mathematics, 101-108 (2007), Basel, Switzerland: Birkhäuser, Basel, Switzerland · Zbl 1120.35036
[15] Hörmander, L., \(L^2\) estimates and existence theorems for the \(\overline{\partial}\) operator, Acta Mathematica, 113, 89-152 (1965) · Zbl 0158.11002
[16] Straube, E. J., The complex Green operator on CR-submanifolds of \(C^n\) of hypersurface type: compactness, Transactions of the American Mathematical Society, 364, 8, 4107-4125 (2012) · Zbl 1278.32027
[17] Raich, A., Compactness of the complex Green operator on CR-manifolds of hypersurface type, Mathematische Annalen, 348, 1, 81-117 (2010) · Zbl 1238.32032
[18] Raich, A. S.; Straube, E. J., Compactness of the complex Green operator, Mathematical Research Letters, 15, 4, 761-778 (2008) · Zbl 1157.32032
[19] Munasinghe, S.; Straube, E. J., Geometric sufficient conditions for compactness of the complex Green operator, Journal of Geometric Analysis, 22, 4, 1007-1026 (2012) · Zbl 1269.32022
[20] Kohn, J. J.; Nirenberg, L., Non-coercive boundary value problems, Communications on Pure and Applied Mathematics, 18, 443-492 (1965) · Zbl 0125.33302
[21] Kohn, J. J., Methods of partial differential equations in complex analysis, complex variables (Williamstown, Mass., 1975), Proceedings of Symposia in Pure Mathematics, Providence, RI, USA: American Mathematical Society, Providence, RI, USA
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