Abstract
The purpose of this article is to study compactness of the complex Green operator on CR manifolds of hypersurface type. We introduce (CR-P q ), a potential theoretic condition on (0, q)-forms that generalizes Catlin’s property (P q ) to CR manifolds of arbitrary codimension. We prove that if an embedded CR-manifold of hypersurface type of real dimension at least five satisfies (CR-P q ) and (CR-P n-1-q), then the complex Green operator is a compact operator on the Sobolev spaces \({H^s_{0,q}(M)}\) and \({H^s_{0,n-1-q}(M)}\) , if 1 ≤ q ≤ n−2 and s ≥ 0. We use CR-plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle.
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References
Boggess A.: CR Manifolds and the Tangential Cauchy-Riemann Complex. Studies in Advanced Mathematics. CRC Press, Boca Raton (1991)
Boas H., Shaw M.-C.: Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries. Math. Ann. 274, 221–231 (1986)
Boas H., Straube E.: Sobolev estimates for the complex Green operator on a class of weakly pseudoconvex boundaries. Comm. Partial Differ. Equ. 16, 1573–1582 (1991)
Catlin D.: Necessary conditions for subellipticity of the \({\overline\partial}\) -Neumann problem. Ann. Math. 117, 147–171 (1983)
Catlin, D.: Global regularity of the \({\bar\partial}\) -Neumann problem. In: Complex Analysis of Several Variables (Madison, Wis., 1982). Proceedings of Symposium Pure Mathematics, vol. 41, pp. 39–49. Amer. Math. Soc., Providence (1984)
Catlin D.: Subelliptic estimates for the \({\overline\partial}\) -Neumann problem on pseudoconvex domains. Ann. Math. 126, 131–191 (1987)
D’Angelo J.: Inequalities from Complex Analysis. Number 28 in the Carus Mathematical Monographs. The Mathematical Association of America, Washingon (2002)
Diaz R.: Necessary conditions for subellipticity of \(\square_{b}\) on pseudoconvex domains. Comm. Partial Differ. Equ. 11(1), 1–61 (1986)
Folland G.B., Kohn J.J.: The Neumann Problem for the Cauchy-Riemann Complex, vol. 75 of Annals of Mathematical Studies. Princeton University Press, Princeton (1972)
Fu S., Straube E.: Compactness of the \({\overline\partial}\) -Neumann problem on convex domains. J. Funct. Anal. 159(2), 629–641 (1998)
Fu, S., Straube, E.: Compactness in the \({\overline\partial}\) -Neumann problem. In: Complex Analysis and Geometry (Columbus, OH, 1999). Ohio State Univ. Math. Res. Inst. Publ., 9, pp. 141–160. de Gruyter, Berlin (2001)
Hörmander L.: L 2 estimates and existence theorems for the \({\bar \partial}\) operator. Acta Math. 113, 89–152 (1965)
Koenig K.: On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian. Am. J. Math. 124, 129–197 (2002)
Koenig K.: A parametrix for the \({\overline\partial}\) -Neumann problem on pseudoconvex domains of finite type. J. Funct. Anal. 216(1), 243–302 (2004)
Kohn J.J., Nirenberg L.: Non-coercive boundary value problems. Comm. Pure Appl. Math. 18, 443–492 (1965)
Kohn J.J., Nicoara A.: The \({\bar\partial_b}\) -equation on weakly pseudo-convex CR manifolds of dimension 3. J. Funct. Anal. 230, 251–272 (2006)
Kohn, J.J.: Boundary regularity of \({\bar \partial}\) . In: Recent Developments in Several Complex Variables (Proc. Conf., Princeton Univ., Princeton, N.J., 1979). Volume 100 of Ann. of Math. Stud., pp. 243–260. Princeton University Press, Princeton (1981)
Kohn J.J.: The range of the tangential Cauchy-Riemann operator. Duke Math. J. 53, 525–545 (1986)
Kohn J.J.: Superlogarithmic estimates on pseudoconvex domains and CR manifolds. Ann. Math. 156, 213–248 (2002)
Lax P., Nirenberg L.: On stability for difference schemes: a sharp form of Gårding’s inequality. Comm. Pure Appl. Math. 19, 473–492 (1966)
Nicoara, A.: Equivalence of types and Catlin boundary systems. arXiv:0711.0429
Nicoara A.: Global regularity for \({\bar\partial_b}\) on weakly pseudoconvex CR manifolds. Adv. Math. 199, 356–447 (2006)
Raich A., Straube E.: Compactness of the complex Green operator. Math. Res. Lett. 15(4), 761–778 (2008)
Shaw M.-C.: Global solvability and regularity for \({\bar\partial}\) on an annulus between two wekly pseudo-convex domains. Trans. Am. Math. Soc. 291, 255–267 (1985)
Shaw M.-C.: L 2-estimates and existence theorems for the tangential Cauchy-Riemann complex. Invent. Math. 82, 133–150 (1985)
Straube, E.: Lectures on the \({\mathcal {L}^2}\) -Sobolev Theory of the \({\bar\partial}\) -Neumann Problem. European Mathematical Society (EMS), Zürich
Straube E.: Plurisubharmonic functions and subellipticity of the \({\overline\partial}\) -Neumann problem on non-smooth domains. Math. Res. Lett. 4, 459–467 (1997)
Straube, E.: Aspects of the L 2-Sobolev theory of the \({\bar \partial}\) -Neumann problem. In: Proceedings of the International Congress of Mathematicians, Madrid 2006, vol. II, pp. 1453–1478. Eur. Math. Soc. (2006)
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Raich, A. Compactness of the complex Green operator on CR-manifolds of hypersurface type. Math. Ann. 348, 81–117 (2010). https://doi.org/10.1007/s00208-009-0470-1
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DOI: https://doi.org/10.1007/s00208-009-0470-1