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A quantitative analysis of Oka’s lemma

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Abstract

In this paper, we will examine a strong form of Oka’s lemma which provides sufficient conditions for compact and subelliptic estimates for the \({{\overline\partial}}\) -Neumann operator on Lipschitz domains. On smooth domains, the condition for subellipticity is equivalent to D’Angelo finite type and the condition for compactness is equivalent to Catlin’s condition (P). As an application, we will prove regularity for the \({{\overline\partial}}\) -Neumann operator in the Sobolev space W s, \({0\leq s < \frac{1}{2}}\) , on C 2 domains.

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References

  1. Barrett D. (1992): Behavior of the Bergman projection on the Diederich–Fornaess worm. Acta Math. 168: 1–10

    Article  MATH  MathSciNet  Google Scholar 

  2. Berndtsson B. (1987): \({\overline\partial_b}\) and Carleson type inequalities. Complex Analysis II, Lecture Notes in Mathematics, vol. 1276, pp. 42–54. Springer, Berlin Heidelberg New York

  3. Berndtsson B., Charpentier P. (2000): A Sobolev mapping property of the Bergman kernel. Math. Z. 235, 1–10

    Article  MATH  MathSciNet  Google Scholar 

  4. Boas H.P., Straube E.J. (1990): Equivalence of regularity for the Bergman projection and the \({\overline\partial}\) -Neumann operator. Manuscripta Math. 67, 25–33

    MATH  MathSciNet  Google Scholar 

  5. Boas, H.P., Straube, E.J.: Global regularity of the \({\overline\partial}\) -Neumann problem: a suvey of the L 2-Sobolev theory, several complex variables. Mathematical Science Research Institute Publication, vol. 37. Cambridge University Press, pp. 79–111 (1999) Cambridge

  6. Cao, J., Shaw, M.-C., Wang, L.: Estimates for the \({\overline\partial}\) -Neumann problem and nonexistence of C 2 Levi-flat hypersurfaces in \({\mathbb{C}{P}^n}\) . Math. Z. 248, 183–221 (2004)

  7. Catlin, D.: Global regularity of the \({\overline\partial}\) -Neumann problem. In: Proceedings of the Symposium on Pure Mathematics, vol. 41. Amer. Math. Soc. Providence, RI, pp. 39–49 (1984)

  8. Catlin D. (1987): Subelliptic estimates for the \({\overline\partial}\) -Neumann problem on pseudoconvex domains. Ann. Math. 126, 131–191

    Article  MathSciNet  Google Scholar 

  9. Chen, S.-C., Shaw, M.-C.: Partial differential equations in several complex variables. Studies in Advanced Mathematics, vol. 19. American Mathematical Society-International Press, Providence, RI (2001)

  10. D’Angelo J. (1993): Several complex variables and the geometry of real hypersurfaces. CRC Press, Boca Raton

    MATH  Google Scholar 

  11. Demailly J.-P. (1982): Estimations L 2 pour l’opérateur \({\overline\partial}\) d’un fibré vectoriel holomorphe sémi-positif. Ann. Sci. Ec. Norm. Sup. 15, 457–511

    MATH  MathSciNet  Google Scholar 

  12. Demailly J.-P. (1987): Mesures de Monge–Ampère et mesures pluriharmoniques. Math. Z. 194, 519–564

    Article  MATH  MathSciNet  Google Scholar 

  13. Detraz J. (1981): Classes de Bergman de fonctions harmoniques. Bull. Soc. Math. France 109, 259–268

    MATH  MathSciNet  Google Scholar 

  14. Diederich K., Fornaess J.E. (1977): Pseudoconvex domains: an example with nontrivial Nebenhülle. Math. Ann. 225, 275–292

    Article  MATH  MathSciNet  Google Scholar 

  15. Diederich K., Fornaess J.E. (1977): Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39, 129–141

    Article  MATH  MathSciNet  Google Scholar 

  16. Donnelly H., Fefferman C. (1983): L 2 cohomology and index theorem for the Bergman metric. Ann. Math. 118, 593–618

    Article  MathSciNet  Google Scholar 

  17. Donnelly H., Xavier F. (1984): On the differential form spectrum of negatively curved Riemannian manifolds. Am. J. Math 106, 169–185

    Article  MATH  MathSciNet  Google Scholar 

  18. Folland G.B., Kohn J.J. (1972): The Neumann problem for the Cauchy–Riemann complex. Annals of Mathematical Studies, vol. 75. Princeton University Press, Princeton

    MATH  Google Scholar 

  19. Fu S., Straube E.J. (1998): Compactness of the \({\overline\partial}\) -Neumann problem on convex domains. J. Funct. Anal. 159, 629–641

    Article  MATH  MathSciNet  Google Scholar 

  20. Fu, S., Straube, E.J.: Compactness in the \({\overline\partial}\) -Neumann problem. Complex Analysis and Geometry, Proceedings of a Conference at Ohio State University (Berlin), vol. 9. Walter De Gruyter, New York pp. 141–160 (2001)

  21. Grisvard P. (1985): Elliptic Problems in Nonsmooth Domains. Pitman, Boston

    MATH  Google Scholar 

  22. Harrington, P.S.: Compactness and subellipticity for the \({\overline\partial}\) -Neumann operator on domains with minimal smoothness. Ph.D. thesis, University of Notre Dame, 2004

  23. Henkin G.M., Iordan A. (1997): Compactness of the Neumann operator for hyperconvex domains with non-smooth b-regular boundary. Math. Ann. 307, 151–168

    Article  MATH  MathSciNet  Google Scholar 

  24. Henkin G.M., Iordan A., Kohn J.J. (1996): Estimations sous-elliptiques pour le problème \({\overline\partial}\) -Neumann dans un domain strictement pseudoconvexe à frontière lisse par morceaux. C. R. Acad. Sci. Paris Sèr. I Math. 323, 17–22

    MATH  MathSciNet  Google Scholar 

  25. Hörmander L. (1965): L 2 estimates and existence theorems for the \({\overline\partial}\) operator. Acta Math. 113, 89–152

    Article  MATH  MathSciNet  Google Scholar 

  26. Hörmander L. (1990): An Introduction to Complex Analysis in Several Complex Variables, 3rd edn. Van Nostrand, Princeton

    MATH  Google Scholar 

  27. Jerison D., Kenig C.E. (1995): The inhomogeneous Dirichlet problem in Lipschitz domains. J. Funct. Anal. 130, 161–219

    Article  MATH  MathSciNet  Google Scholar 

  28. Kerzman N., Rosay J.-P. (1981): Fonctions plurisousharmoniques d’exhaustion bornées et domaines taut. Math. Ann. 257, 171–184

    Article  MATH  MathSciNet  Google Scholar 

  29. Kohn J.J. (1963): Harmonic integrals on strongly pseudoconvex manifolds I. Ann. Math. 78, 112–148

    Article  MathSciNet  Google Scholar 

  30. Kohn J.J. (1964): Harmonic integrals on strongly pseudoconvex manifolds II. Ann. Math. 79, 450–472

    Article  MathSciNet  Google Scholar 

  31. Kohn J.J., Nirenberg L. (1965): Noncoercive boundary value problems. Commun. Pure Appl. Math. 18, 443–492

    MATH  MathSciNet  Google Scholar 

  32. McNeal J.D. (1996): On large values of L 2 holomorphic functions. Math. Res. Lett. 3, 247–259

    MATH  MathSciNet  Google Scholar 

  33. McNeal J.D. (2002): A sufficient condition for compactness of the \({\overline\partial}\) -Neumann operator. J Funct Anal. 195, 190–205

    Article  MATH  MathSciNet  Google Scholar 

  34. Michel J., Shaw M.-C. (1998): Subelliptic estimates for the \({\overline\partial}\) -Neumann operator on piecewise smooth strictly pseudoconvex domains. Duke Math. J. 93, 115–128

    Article  MATH  MathSciNet  Google Scholar 

  35. Morrey C.B. (1966): Multiple Integrals in the Calculus of Variations. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  36. Ohsawa T., Takegoshi K. (1987): On the extension of L 2 holomorphic functions. Math. Z. 195, 197–204

    Article  MATH  MathSciNet  Google Scholar 

  37. Sibony N. (1987): Une classe de domaines pseudoconvexes. Duke Math. J. 55, 299–319

    Article  MATH  MathSciNet  Google Scholar 

  38. Siu, Y.-T.: The Fujita conjecture and the extension theorem of Ohsawa-Takegoshi. Geometric Complex Analysis. World Scientific, Hayama (Singapore) pp. 577–592 (1996)

  39. Straube E.J. (1997): Plurisubharmonic functions and subellipticity of the \({\overline\partial}\) -Neumann problem on nonsmooth domains. Math. Res. Lett. 4, 459–467

    MATH  MathSciNet  Google Scholar 

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  1. University of South Dakota, 414 East Clark Street, Vermillion, SD, 57069, USA

    Phillip S. Harrington

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Harrington, P.S. A quantitative analysis of Oka’s lemma. Math. Z. 256, 113–138 (2007). https://doi.org/10.1007/s00209-006-0062-7

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