Skip to main content
SpringerLink
Log in

On convex hulls and pseudoconvex domains generated by q-plurisubharmonic functions, part II

  • Original Article
  • Published:
Boletín de la Sociedad Matemática Mexicana Aims and scope Submit manuscript

Abstract

We study the properties of hulls of compact sets in \(\mathbb {C}^n\) that are generated by certain subfamilies of q-plurisubharmonic functions. We consider in particular those functions that are plurisubharmonic on the level sets of holomorphic mappings defined from \(\mathbb {C}^n\) into \(\mathbb {C}^q\). We also compare the above-mentioned hulls against the generalised polynomially and rationally convex hulls already defined in the literature. Our main result yields that the hulls defined by q-plurisubharmonic functions or q-pseudoconvex sets are all q-maximum, so that their complement is relatively \((n{\text {-}}q{\text {-}}2)\)-pseudoconvex.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Basener, R.F.: Nonlinear Cauchy–Riemann equations and \(q\)-pseudoconvexity. Duke Math. J. 43(1), 203–213 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  2. Basener, R.F.: Several dimensional properties of the spectrum of a uniform algebra. Pac. J. Math. 74(2), 297–306 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  3. Colţoiu, M.: Traces of Runge domains on analytic subsets. Math. Ann. 290(3), 545–548 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Diederich, K., Fornæss, J.E.: Smoothing \(q\)-convex functions and vanishing theorems. Invent. Math. 82(2), 291–305 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Dieu, N.Q.: \(q\)-plurisubharmonicity and \(q\)-pseudoconvexity in \({\mathbb{C}}^{n}\). Publ. Mat. 50, 349–369 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Fujita, O.: Domaines pseudoconvexes d’ordre général et fonctions pseudoconvexes d’ordre général. J. Math. Kyoto Univ. 30(4), 637–649 (1990)

    MathSciNet  MATH  Google Scholar 

  7. Fujita, O.: On the equivalence of the q-plurisubharmonic functions and the pseudoconvex functions of general order. Ann. Reports of Graduate School of Human Culture. Nara Women’s Univ. 7, 77–81 (1992)

    Google Scholar 

  8. Henkin, G.M., Leiterer, J.: Andreotti–Grauert Theory by Integral Formulas, Progress in Mathematics, vol. 74. Birkhäuser Boston Inc, Boston (1988)

    Book  MATH  Google Scholar 

  9. Hunt, L.R., Murray, J.J.: \(q\)-plurisubharmonic functions and a generalized Dirichlet problem. Mich. Math. J. 25(3), 299–316 (1978)

  10. Lupacciolu, G., Stout, E.L.: Holomorphic hulls in terms of curves of nonnegative divisors and generalized polynomial hulls. Manuscr. Math. 98(3), 321–331 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Pawlaschyk, T.: On some classes of \(q\)-pseudoconcave sets. Dissertation (2015) (to be published)

  12. Pawlaschyk, T., Zeron, E.S.: On convex hulls and pseudoconvex domains generated by \(q\)-plurisubharmonic functions, part I. J. Math. Anal. Appl. 408, 394–408 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Słodkowski, Z.: The Bremermann–Dirichlet problem for \(q\)-plurisubharmonic functions. Ann. Scuola Norm. Super. Pisa-Cl. Sci. (4) 11(2), 303–326 (1984)

  14. Słodkowski, Z.: Local maximum property and \(q\)-plurisubharmonic functions in uniform algebras. J. Math. Anal. Appl. 115(1), 105–130 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  15. Stout, E.L.: Polynomial Convexity, Progress in Mathematics, vol. 261. Birkhäuser Boston Inc, Boston (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Wuppertal, Gaußstr. 20, 42119, Wuppertal, Germany

    Thomas Pawlaschyk

  2. Departamento de Matemáticas, CINVESTAV del IPN, Apartado Postal 14-740, Ciudad de México, 07000, Mexico

    Eduardo S. Zeron

Authors
  1. Thomas Pawlaschyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eduardo S. Zeron
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eduardo S. Zeron.

Additional information

To Professor Sergei Grudsky on the occasion of his 60th anniversary.

Research supported by the Deutscher Akademischer Austauschdienst (DAAD) and Conacyt México under the PPP Proalmex Project No. 51240052. T. Pawlaschyk was supported by the Deutsche Forschungsgemeinschaft (DFG) under the Grant SH 456/1-1, Pluripotential Theory, Hulls and Foliations; and E. S. Zeron was supported by Cinvestav del IPN in México.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pawlaschyk, T., Zeron, E.S. On convex hulls and pseudoconvex domains generated by q-plurisubharmonic functions, part II. Bol. Soc. Mat. Mex. 22, 367–388 (2016). https://doi.org/10.1007/s40590-016-0123-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40590-016-0123-9

Keywords

Mathematics Subject Classification

Search

Navigation