Zeros Sets of Hp Functions in Lineally Convex Domains of Finite Type in \( \mathbb {C}^{n}\)

Abstract

In this note, we extend N. Th. Varopoulos result on zero sets of Hp functions of strictly pseudo-convex domains in \(\mathbb {C}^{n}\) to lineally convex domains of finite type.

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References

  1. 1.

    Alexandre, W.: Zero sets of H p functions in convex domains of finite type. Math. Z. 287(1-2), 85–115 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Andersson, M., Carlsson, H.: On varopoulos’ theorem about zero sets of H p-functions. Bull. Sci. Math. 114(4), 463–484 (1990)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bruna, J., Charpentier, P., Dupain, Y.: Zeros varieties for the Nevanlinna class in convex domains of finite type in C n. Ann. Math. (2) 147(2), 391–415 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Bruna, J., Grellier, S.: Zero sets of H p functions in convex domains of strict finite type in C n. Complex Var. Theory Appl. 38, 243–261 (1999)

    MATH  Google Scholar 

  5. 5.

    Chang, D.C., Nagel, A., Stein, E.M.: Estimates for the \(\bar {\partial }\)-Neumann problem for pseudoconvex domains of finite type in C 2. Acta Math. 169(3–4), 153–228 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Charpentier, P., Dupain, Y.: Weighted and boundary l p estimates for solutions of the \(\overline {\partial }\)-equation on lineally convex domains of finite type and applications. To appear in Math Z. https://doi.org/10.1007/s00209-017-2015-8

  7. 7.

    Charpentier, P., Dupain, Y.: Extremal bases, geometrically separated domains and applications. Algebra i Analiz 26(1), 196–269 (2014)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Charpentier, P., Dupain, Y., Mounkaila, M.: Estimates for solutions of the \(\bar {\partial }\)-equation and application to the characterization of the zero varieties of the functions of the Nevanlinna class for lineally convex domains of finite type. J. Geom. Anal. 24(4), 1860–1881 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Conrad, M.: Anisotrope optimale Pseudometriken für lineal konvex Gebeite von endlichem Typ (mit Anwendungen). Berg. Universität-GHS Wuppertal, PhD thesis (2002)

    Google Scholar 

  10. 10.

    Cumenge, A.: Zero sets of functions in the Nevanlinna or the Nevanlinna-Djrbachian classes. Pac. J. Math. 199(1), 79–92 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Diederich, K., Mazzilli, E.: Zero varieties for the Nevanlinna class on all convex domains of finite type. Nagoya Math. J. 163, 215–227 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Henkin, G.M.: Solutions with bounds for the equations of H. Lewy and Poincaré-Lelong. Construction of functions of Nevanlinna class with given zeros in a strongly pseudoconvex domain. Dokl. Akad. Nauk SSSR 224(4), 771–774 (1975)

    MathSciNet  Google Scholar 

  13. 13.

    John, F., Nirenberg, L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426 (1961)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Nguyen, N.: Un théorème de la couronne H p et zéros des fonctions de H p dans les convexes de type fini Prépublication n, vol. 224. Laboratoire Emile Picard, Université de Toulouse III (2001)

    Google Scholar 

  15. 15.

    Skoda, H.: Valeurs au bord pour les solutions de l’opérateur \(\bar {\partial }\), et caractérisation des zéros des fonctions de la classe de Nevanlinna. Bull. Soc. Math. France 104(3), 225–299 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Varopoulos, N.: Zeros of H p functions in several complex variables. Pac. J. Math. 88(1), 189–246 (1980)

    Article  MATH  Google Scholar 

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Correspondence to Philippe Charpentier.

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Charpentier, P., Dupain, Y. Zeros Sets of Hp Functions in Lineally Convex Domains of Finite Type in \( \mathbb {C}^{n}\). Acta Math Vietnam 44, 449–467 (2019). https://doi.org/10.1007/s40306-018-0284-x

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Keywords

  • Lineally convex
  • Finite type
  • d-equation
  • \(\overline {\partial }\)-equation
  • Zero sets
  • Hardy classes

Mathematics Subject Classification (2010)

  • 32T25
  • 32T27