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The \(\partial \)-Neumann operator on strongly pseudoconvex domain with piecewise smooth boundary. (English) Zbl 1108.35027

The paper deals with the \(\bar \partial \)-Neumann operator \(N\:L^2_{(r,q)}(D)\to L^2_{(r,q)}(D)\), if \(D\subset \mathbb C^n\) is a bounded strongly pseudoconvex domain with piecewise smooth boundary. The authors prove that it can be extended as a bounded operator from the Sobolev space \(H^{-1/2}_{(r,q)}(D)\) into \(H^{1/2}_{(r,q)}(D)\). In particular, \(N\) is a compact operator on \(L^2_{(r,q)}(D)\) and \(H^{-1/2}_{(r,q)}(D)\).

MSC:

35F15 Boundary value problems for linear first-order PDEs
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
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References:

[1] BOAS H. P.-STRAUBE E. J.: Equivalence of regularity for the Bergman projection and the \(\overline\partial\)-Neumann operator. Manuscripta Math. 67 (1990), 25-33. · Zbl 0695.32011 · doi:10.1007/BF02568420
[2] BOAS H. P.-STRAUBE E. J.: Sobolev estimates for the \(\overline\partial\)-Neumann operator on domains in Cn admitting a defining function that is plurisubharmonic on the boundary. Math. Z. 206 (1991), 81-88. · Zbl 0696.32008 · doi:10.1007/BF02571327
[3] BOAS H. P.-STRAUBE E. J.: Global regularity of the \(\overline\partial\)-Neumann problem: A Survey of the \(L^2\)-Sobolev theory. Several Complex Variables (M. Schneider et al., Math. Sci. Res. Inst. Publ. 37, Cambridge University Press, Cambridge, 1999, pp. 79-111. · Zbl 0967.32033
[4] BONAMI A.-CHARPENTIER P.: Boundary values for the canonical solution to \(\overline\partial\)-equation and \(W^{1/2} -estimates. Preprint, Bordeaux, 1990.\)
[5] CHEN S.-C-SHAW M.-C: Partial Differential Equations in Several Complex Variables. Stud. Adv. Math. 19, AMS-International Press, Providence, RI, 2001.
[6] EHSANI D.: Analysis of the \(\overline\partial\)-Neumann problem along a straight edge. Preprint Math. CV/0309169. · Zbl 1057.32015 · doi:10.1007/s00208-004-0540-3
[7] EHSANI D.: Solution of the d-bar-Neumann problem on a bi-disc. Math. Res. Lett. 10 (2003), 523-533. · Zbl 1055.32023 · doi:10.4310/MRL.2003.v10.n4.a11
[8] EHSANI D.: Solution of the d-bar-Neumann problem on a non-smooth domain. Indiana Univ. Math. J. 52 (2003), 629-666. · Zbl 1056.32024 · doi:10.1512/iumj.2003.52.2261
[9] ENGLIŠ M.: Pseudolocal estimates for \(\overline\partial\) on general pseudoconvex domains. Indiana Univ. Math. J. 50 (2001), 1593-1607. · Zbl 1044.32029 · doi:10.1512/iumj.2001.50.2085
[10] FOLLAND G. B.-KOHN J. J.: The Neumann Problem for the Cauchy-Riemann Complex. Princeton University Press, Princeton, 1972. · Zbl 0247.35093
[11] GRISVARD P.: Elliptic Problems in Nonsmooth Domains. Monogr. and Stud, in Math. 24. Pitman Advanced Publishing Program, Pitman Publishing Inc., Boston-London-Melbourne, 1985. · Zbl 0695.35060
[12] HENKIN G.-IORDAN A.-KOHN J. J.: Estimations sous-elliptiques pour le problem \(\overline\partial\)-Neumann dans un domaine strictement pseudoconvexe a frontiere lisse par morceaux. C. R. Acad. Sci. Paris Ser. I Math. 323 (1996), 17-22. · Zbl 0861.35066
[13] HÖRMANDER L.: \(L^2\) -estimates and existence theorems for the \(\overline\partial\)-operator. Acta Math. 113 (1965), 89-152. · Zbl 0158.11002 · doi:10.1007/BF02391775
[14] KOHN J. J.: Harmonic integrals on strongly pseudo-convex manifolds I. Ann. Math. (2) 78 (1963), 112-148. · Zbl 0161.09302 · doi:10.2307/1970506
[15] KOHN J. J.: Global regularity for \(\overline\partial\) on weakly pseudoconvex manifolds. Trans. Amer. Math. Soc 181 (1973), 273-292. · Zbl 0276.35071 · doi:10.2307/1996633
[16] KOHN J. J.: Subellipticity of the \(\overline\partial\)-Neumann problem on pseudoconvex domains: Sufficient conditions. Acta Math. 142 (1979), 79-122. · Zbl 0395.35069 · doi:10.1007/BF02395058
[17] KOHN J. J.: A survey of the \(\overline\partial\) -Neumann problem. Complex Analysis of Several Variables (Yum-Tong Siu, Proc Sympos. Pure Math. 41, Amer. Math. Soc, Providence, RI, 1984, pp. 137-145.
[18] KRANTZ S. G.: Partial Differential Equations and Complex Analysis. CRC Press, Boca Raton, 1992. · Zbl 0852.35001
[19] MICHEL J.-SHAW M.-C.: Subelliptic estimates for the \(\overline\partial\)-Neumann operator on piecewise smooth strictly pseudoconvex domains. Duke Math. J. 93 (1998), 115-128. · Zbl 0953.32027 · doi:10.1215/S0012-7094-98-09304-8
[20] MICHEL J.-SHAW M.-C: The \(\overline\partial\)-Neumann operator on Lipschitz pseudoconvex domains with plurisubharmonic defining functions. Duke Math. J. 108 (2001), 421-447. · Zbl 1020.32030 · doi:10.1215/S0012-7094-01-10832-6
[21] SHAW M.-C.: Local existence theorems with estimates for \(\overline\partial_b\) on weakly pseudo-convex \(CR\) manifolds. Math. Ann. 294 (1992), 677-700. · Zbl 0766.32022 · doi:10.1007/BF01934348
[22] STEIN E. M.: Singular Integrals and Differentiability Properties of Functions. Princeton Math. Ser. 30, Princeton Univ. Press, Princeton, NJ, 1970. · Zbl 0207.13501
[23] STRAUBE E. : Plurisubharmonic functions and subellipticity of the \(\overline\partial\)-Neumann problem on nonsmooth domains. Math. Res. Lett. 4 (1997), 459-467. · Zbl 0887.32005 · doi:10.4310/MRL.1997.v4.n4.a2
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