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Compactness of the complex Green operator on CR-manifolds of hypersurface type. (English) Zbl 1238.32032

A surface \(S\subset \mathbb{R}^k\) satisfies property \((CR-P_q)\) if for every \(A>0,\) there exists a function \(\lambda\) and a neighborhood \(U\supset S\) so that \(0\leq \lambda \leq 1\) and \(\lambda\) is CR-plurisubharmonic on \((0,q)\)-forms on \(U\) with plurisubharmonicity constant \(A.\) The main result is the following: Let \(M\subset \mathbb{C}^N\) be a smooth, compact, orientable weakly pseudoconvex CR-manifold of hypersurface type of real dimension \((2n-1)\) that satisfies \((CR-P_q)\) and \((CR-P_{n-1-q}).\) If \(1\leq q \leq n-2\) and \(s\geq 0,\) then \(\overline \partial_b\) and \(\overline \partial_b^*\) acting on the Sobolev space \(H^s_{(0,q)}\) have closed range, the complex Green operator \(G_q\) exists and is a compact operator on \(H^s_{(0,q)},\) and the space of harmonic forms is finite dimensional.

MSC:

32W10 \(\overline\partial_b\) and \(\overline\partial_b\)-Neumann operators
35N10 Overdetermined systems of PDEs with variable coefficients
32V20 Analysis on CR manifolds
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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References:

[1] Boggess A.: CR Manifolds and the Tangential Cauchy-Riemann Complex. Studies in Advanced Mathematics. CRC Press, Boca Raton (1991) · Zbl 0760.32001
[2] Boas H., Shaw M.-C.: Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries. Math. Ann. 274, 221–231 (1986) · Zbl 0588.32023
[3] 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) · Zbl 0747.32007
[4] Catlin D.: Necessary conditions for subellipticity of the \({\overline\partial}\) -Neumann problem. Ann. Math. 117, 147–171 (1983) · Zbl 0552.32017
[5] 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)
[6] Catlin D.: Subelliptic estimates for the \({\overline\partial}\) -Neumann problem on pseudoconvex domains. Ann. Math. 126, 131–191 (1987) · Zbl 0627.32013
[7] D’Angelo J.: Inequalities from Complex Analysis. Number 28 in the Carus Mathematical Monographs. The Mathematical Association of America, Washingon (2002)
[8] Diaz R.: Necessary conditions for subellipticity of \(\square_{b}\) on pseudoconvex domains. Comm. Partial Differ. Equ. 11(1), 1–61 (1986) · Zbl 0586.32025
[9] 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) · Zbl 0247.35093
[10] Fu S., Straube E.: Compactness of the \({\overline\partial}\) -Neumann problem on convex domains. J. Funct. Anal. 159(2), 629–641 (1998) · Zbl 0959.32042
[11] 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)
[12] Hörmander L.: L 2 estimates and existence theorems for the \({\bar \partial}\) operator. Acta Math. 113, 89–152 (1965) · Zbl 0158.11002
[13] Koenig K.: On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian. Am. J. Math. 124, 129–197 (2002) · Zbl 1014.32031
[14] Koenig K.: A parametrix for the \({\overline\partial}\) -Neumann problem on pseudoconvex domains of finite type. J. Funct. Anal. 216(1), 243–302 (2004) · Zbl 1072.32029
[15] Kohn J.J., Nirenberg L.: Non-coercive boundary value problems. Comm. Pure Appl. Math. 18, 443–492 (1965) · Zbl 0125.33302
[16] 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) · Zbl 1109.32032
[17] 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)
[18] Kohn J.J.: The range of the tangential Cauchy-Riemann operator. Duke Math. J. 53, 525–545 (1986) · Zbl 0609.32015
[19] Kohn J.J.: Superlogarithmic estimates on pseudoconvex domains and CR manifolds. Ann. Math. 156, 213–248 (2002) · Zbl 1037.32032
[20] Lax P., Nirenberg L.: On stability for difference schemes: a sharp form of Gårding’s inequality. Comm. Pure Appl. Math. 19, 473–492 (1966) · Zbl 0185.22801
[21] Nicoara, A.: Equivalence of types and Catlin boundary systems. arXiv:0711.0429
[22] Nicoara A.: Global regularity for \({\bar\partial_b}\) on weakly pseudoconvex CR manifolds. Adv. Math. 199, 356–447 (2006) · Zbl 1091.32017
[23] Raich A., Straube E.: Compactness of the complex Green operator. Math. Res. Lett. 15(4), 761–778 (2008) · Zbl 1157.32032
[24] 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) · Zbl 0594.35010
[25] Shaw M.-C.: L 2-estimates and existence theorems for the tangential Cauchy-Riemann complex. Invent. Math. 82, 133–150 (1985) · Zbl 0581.35057
[26] Straube, E.: Lectures on the \({\mathcal {L}^2}\) -Sobolev Theory of the \({\bar\partial}\) -Neumann Problem. European Mathematical Society (EMS), Zürich
[27] Straube E.: Plurisubharmonic functions and subellipticity of the \({\overline\partial}\) -Neumann problem on non-smooth domains. Math. Res. Lett. 4, 459–467 (1997) · Zbl 0887.32005
[28] 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) · Zbl 1106.32028
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