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Uniform estimates with weights for the\(\bar \partial - equation\)

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Abstract

This paper concernsL ∞-variants of Hörmanders weightedL 2-estimates for the\(\bar \partial - equation\). In particular, we discuss a conjecture concerning suchL ∞-estimates which is related to the corona problem in the ball, and show a weaker version of this conjecture. The proof uses a refinedL 2-estimate for the canonical solution to the\(\bar \partial - equation\). An alternative approach based on von Neumann’s Minimax theorem is also given.

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References

  1. Andersson, M., and Carlsson, H. Hp-estimates for holomorphic division formulas.Pacific J. Math. 173 (2) (1996), 307–335.

    MathSciNet  MATH  Google Scholar 

  2. Berndtsson, B.\(\bar \partial _b \) and Carleson type estimates.Complex Analysis, II (C. A. Berenstein, ed.), Lecture Notes in Math., vol. 1276, pp. 42–54. Springer-Verlag, New York, 1987.

    Chapter  Google Scholar 

  3. Berndtsson, B. Weighted estimates for the\(\bar \partial - equation\) in domains in ℂ.Duke Math. J. (1992), 239–255.

  4. Berndtsson, B. A smoothly bounded pseudoconvex domain in ℂ2 where L∞ estimates for\(\bar \partial \) don’t hold.Arkiv. för Matemalik 31 (1993), 209–218.

    Article  MathSciNet  MATH  Google Scholar 

  5. Berndtsson, B. Some recent results on estimates for the\(\bar \partial - equation\). InContributions to Complex Analysis and Analytic Geometry (H. Skoda and J-M. Trepreau, eds.), pp. 27–42. Vieweg, Braunschweig, 1994.

  6. Berndtsson, B.\(\bar \partial \) and Schrödinger operators.Math. Z. 221 (3) (1996), 401–413.

    Article  MathSciNet  MATH  Google Scholar 

  7. Carleson, L. Interpolation by bounded analytic functions and the Corona problem.Ann. of Math. 76 (1962), 547–559.

    Article  MathSciNet  Google Scholar 

  8. Christ, M. On the\(\bar \partial - equation\) in ℂ1 with weights.J. Geom. Anal. 1 (1991), 193–230.

    MathSciNet  MATH  Google Scholar 

  9. Dautov, S. A., and Henkin, G. M. Zeros of holomorphic functions of finite order and weighted estimates for solutions of the\(\bar \partial - equation\).Mat. Sb. 107 (1978), 163–174.

    MathSciNet  Google Scholar 

  10. Fornaess, J., and Sibony, N. Lp-estimates for\(\bar \partial \).Proc. Symp. Pure Math. A M S52.3 (1990), 129–163.

    Google Scholar 

  11. Fornaess, J., and Sibony, N. Pseudoconvex domains in ℂ2, where the Corona Theorem and Lp-estimates for\(\bar \partial \) don’t hold.Complex Analysis and Geometry, Univ. Ser. Math., pp. 209–222. Plenum, New York, 1993.

    Google Scholar 

  12. Gamelin, T.Uniform Algebras. Prentice Hall, Englewood Cliffs, NJ, 1966.

    Google Scholar 

  13. Gamelin, T. Wolff’s proof of the Corona Theorem.Israel J. Math. 37 (1980), 113–119.

    Article  MathSciNet  MATH  Google Scholar 

  14. Henkin, G. M., and Leiterer, J.Theory of Functions on Complex Manifolds. Akademie-Verlag, Berlin, 1984.

    Google Scholar 

  15. Hörmander, L. Generators for some rings of analytic functions.B.A.M. S. 73, 943–949 (1967).

    Article  MATH  Google Scholar 

  16. Hörmander, L. L2-estimates and existence theorems for the\(\bar \partial - operator\).Acta Math. 113 (1965), 89–152.

    Article  MathSciNet  MATH  Google Scholar 

  17. Kohn, J., and Folland, G.The Neumann Problem for the Cauchy-Riemann complex. Annals of Mathematical Studies. Princeton University Press, Princeton, NJ, 1972.

    Google Scholar 

  18. Michel, V. Private communication.

  19. Rudin, W.Function Theory in the Unit Ball of ℂn. Springer-Verlag, Berlin, 1980.

    Google Scholar 

  20. Sibony, N. Prolongement analytique des fonctions holomorph bornées et métrique de Caratheodory.Inv. Math. 29 (1975), 205–230.

    Article  MathSciNet  MATH  Google Scholar 

  21. Sibony, N. Un example de domaine pseudoconvexe regulier ou l’equation\(\bar \partial u = f\) n’admet pas de solution bornée pourf bornée.Inv. Math. 62 (1980), 235–242.

    Article  MathSciNet  MATH  Google Scholar 

  22. Sibony, N. Problème de la Couronne pour des domaines pseudoconvex.Ann. of Math. 126 (1987), 675–689.

    Article  MathSciNet  Google Scholar 

  23. Varopoulos, N. Th. BMO functions and the\(\bar \partial - equation\).Pacific J. Math. 71 (1977), 221–273.

    MathSciNet  MATH  Google Scholar 

  24. Wu, H. The Bochner technique.Proceedings of the 1980 Bei**g Symposium on Differential Geometry and Differential Equations, vols. 1, 2, 3 (Bei**g, 1980), pp. 929–1071. Science Press, Bei**g, 1982.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, CTH S-412 96, Göteborg, Sweden

    Bo Berndtsson

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  1. Bo Berndtsson
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Correspondence to Bo Berndtsson.

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Communicated by David Jerison

Supported by the Natural Science Research Council.

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Berndtsson, B. Uniform estimates with weights for the\(\bar \partial - equation\) . J Geom Anal 7, 195–215 (1997). https://doi.org/10.1007/BF02921720

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  • DOI: https://doi.org/10.1007/BF02921720

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