Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function

Magnus Carlehed; Urban Cegrell; Frank Wikström

Annales Polonici Mathematici (1999)

  • Volume: 71, Issue: 1, page 87-103
  • ISSN: 0066-2216

Abstract

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We characterise hyperconvexity in terms of Jensen measures with barycentre at a boundary point. We also give an explicit formula for the pluricomplex Green function in the Hartogs triangle. Finally, we study the behaviour of the pluricomplex Green function g(z,w) as the pole w tends to a boundary point.

How to cite

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Magnus Carlehed, Urban Cegrell, and Frank Wikström. "Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function." Annales Polonici Mathematici 71.1 (1999): 87-103. <http://eudml.org/doc/262654>.

@article{MagnusCarlehed1999,
abstract = {We characterise hyperconvexity in terms of Jensen measures with barycentre at a boundary point. We also give an explicit formula for the pluricomplex Green function in the Hartogs triangle. Finally, we study the behaviour of the pluricomplex Green function g(z,w) as the pole w tends to a boundary point.},
author = {Magnus Carlehed, Urban Cegrell, Frank Wikström},
journal = {Annales Polonici Mathematici},
keywords = {hyperconvexity; Jensen measures; Reinhardt domains; Hartogs' triangle; pluricomplex Green function; hyperconvex domain; Jensen measure; Lempert function; Hartogs triangle},
language = {eng},
number = {1},
pages = {87-103},
title = {Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function},
url = {http://eudml.org/doc/262654},
volume = {71},
year = {1999},
}

TY - JOUR
AU - Magnus Carlehed
AU - Urban Cegrell
AU - Frank Wikström
TI - Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function
JO - Annales Polonici Mathematici
PY - 1999
VL - 71
IS - 1
SP - 87
EP - 103
AB - We characterise hyperconvexity in terms of Jensen measures with barycentre at a boundary point. We also give an explicit formula for the pluricomplex Green function in the Hartogs triangle. Finally, we study the behaviour of the pluricomplex Green function g(z,w) as the pole w tends to a boundary point.
LA - eng
KW - hyperconvexity; Jensen measures; Reinhardt domains; Hartogs' triangle; pluricomplex Green function; hyperconvex domain; Jensen measure; Lempert function; Hartogs triangle
UR - http://eudml.org/doc/262654
ER -

References

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  1. [1] A. Aytuna, On Stein manifolds M for which (M) is isomorphic to ( Δ n ) as Fréchet spaces, Manuscripta Math. 62 (1988), 297-316. Zbl0662.32014
  2. [2] E. Bedford and J. E. Fornæss, A construction of peak functions on weakly pseudoconvex domains, Ann. of Math. 107 (1978), 555-568. Zbl0392.32004
  3. [3] Z. Błocki, The complex Monge-Ampère operator in hyperconvex domains, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), 721-747. Zbl0878.31003
  4. [4] M. Carlehed, Comparison of the pluricomplex and the classical Green functions, Michigan Math. J. 45 (1998), 399-407. Zbl0960.32021
  5. [5] U. Cegrell, Capacities in Complex Analysis, Aspects Math. E14, Vieweg, 1988. 
  6. [6] D. Coman, Remarks on the pluricomplex Green function, preprint, 1996. 
  7. [7] K. Diederich and G. Herbort, Pseudoconvex domains of semiregular type, in: Contributions to Complex Analysis and Analytic Geometry, H. Skoda and J.-M. Trépreau (eds.), Aspects Math. E26, Vieweg, 1994, 127-161. Zbl0845.32019
  8. [8] A. Edigarian, On definitions of the pluricomplex Green function, Ann. Polon. Math. 67 (1997), 233-246. Zbl0909.31007
  9. [9] J. E. Fornæss and J. Wiegerinck, Approximation of plurisubharmonic functions, Ark. Mat. 27 (1989), 257-272. Zbl0693.32009
  10. [10] T. Gamelin, Uniform Algebras and Jensen Measures, London Math. Soc. Lecture Note Ser. 32, Cambridge Univ. Press, 1978. 
  11. [11] M. Hervé, Lindelöf's principle in infinite dimensions, in: Proc. on Infinite Dimensional Holomorphy (Berlin), T. L. Hayden and T. J. Suffridge (eds.), Lecture Notes in Math. 364, Springer, 1974, 41-57. 
  12. [12] M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter, 1993. Zbl0789.32001
  13. [13] N. Kerzman et J.-P. Rosay, Fonctions plurisousharmoniques d'exhaustion bornées et domaines taut, Math. Ann. 257 (1981), 171-184. Zbl0451.32012
  14. [14] M. Klimek, Pluripotential Theory, London Math. Soc. Monographs (N.S.) 6, Oxford Univ. Press, 1991. 
  15. [15] S. G. Krantz, Function Theory of Several Complex Variables, 2nd ed., Wadsworth & Brooks/Cole, 1992. Zbl0776.32001
  16. [16] F. Lárusson and R. Sigurdsson, Plurisubharmonic functions and analytic discs on manifolds, J. Reine Angew. Math. 501 (1998), 1-39. Zbl0901.31004
  17. [17] P. Lelong, Fonction de Green pluricomplexe et lemme de Schwarz dans les espaces de Banach, J. Math. Pures Appl. 68 (1989), 319-347. Zbl0633.32019
  18. [18] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France 109 (1981), 427-474. 
  19. [19] L. Lempert, Intrinsic distances and holomorphic retracts, in: Complex Analysis and Applications '81, Bulgar. Acad. Sci., Sophia, 1984, 341-364. 
  20. [20] E. A. Poletsky, Holomorphic currents, Indiana Univ. Math. J. 42 (1993), 85-144. Zbl0811.32010
  21. [21] E. A. Poletsky and B. V. Shabat, Invariant metrics, in: Several Complex Variables III, G. M. Khenkin (ed.), Encyclopaedia Math. Sci., 9, Springer, 1989, 63-111. 
  22. [22] H. Wu, Normal families of holomorphic mappings, Acta Math. 119 (1967), 193-233. Zbl0158.33301
  23. [23] J. Yu, Peak functions on weakly pseudoconvex domains, Indiana Univ. Math. J. 43 (1994), 1271-1295. Zbl0828.32003
  24. [24] W. Zwonek, On Carathéodory completeness of pseudoconvex Reinhardt domains, preprint, 1998. Zbl0939.32025

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