On a question of Sadullaev concerning boundary relative extremal functions


  • Ibrahim K. Djire
  • Jan Wiegerinck




We study the relation between certain alternative definitions of the boundary relative extremal function. For various domains we give an affirmative answer to the question of Sadullaev whether these extremal functions are equal.


Dieu, N. Q. and Wikström, F., Jensen measures and approximation of plurisubharmonic functions, Michigan Math. J. 53 (2005), no. 3, 529–544. https://doi.org/10.1307/mmj/1133894163

Djire, I. K. and Wiegerinck, J., Characterizations of boundary pluripolar hulls, Complex Var. Elliptic Equ. 61 (2016), no. 8, 1133–1144. https://doi.org/10.1080/17476933.2016.1145214

Edigarian, A., Polynomial hulls and proper analytic disks, Ark. Mat. 50 (2012), no. 1, 59–67. https://doi.org/10.1007/s11512-010-0142-0

Edigarian, A. and Sigurdsson, R., The relative extremal function for Borel sets in complex manifolds, preprint arXiv:math/0607313v1 [math.CV], 2006.

Edigarian, A. and Sigurdsson, R., Relative extremal functions and characterization of pluripolar sets in complex manifolds, Trans. Amer. Math. Soc. 362 (2010), no. 10, 5321–5331. https://doi.org/10.1090/S0002-9947-2010-05080-9

Gogus, N. G., Perkins, T. L., and Poletsky, E. A., Non-compact versions of Edwards' Theorem, Positivity 17 (2013), no. 3, 459–473. https://doi.org/10.1007/s11117-012-0181-9

Klimek, M., Pluripotential theory, London Mathematical Society Monographs. New Series, vol. 6, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.

Poletsky, E. A., Holomorphic currents, Indiana Univ. Math. J. 42 (1993), no. 1, 85–144. https://doi.org/10.1512/iumj.1993.42.42006

Ransford, T. J., Jensen measures, in “Approximation, complex analysis, and potential theory (Montreal, QC, 2000)'', NATO Sci. Ser. II Math. Phys. Chem., vol. 37, Kluwer Acad. Publ., Dordrecht, 2001, pp. 221--237. https://doi.org/10.1007/978-94-010-0979-9_8

Sadullaev, A., Plurisubharmonic measures and capacities on complex manifolds, Uspekhi Mat. Nauk 36 (1981), no. 4(220), 53–105.

Wikström, F., Jensen measures and boundary values of plurisubharmonic functions, Ark. Mat. 39 (2001), no. 1, 181–200. https://doi.org/10.1007/BF02388798

Wikström, F., The Dirichlet problem for maximal plurisubharmonic functions on analytic varieties in $mathbb C^n$, Internat. J. Math. 20 (2009), no. 4, 521–528. https://doi.org/10.1142/S0129167X09005376




How to Cite

Djire, I. K., & Wiegerinck, J. (2019). On a question of Sadullaev concerning boundary relative extremal functions. MATHEMATICA SCANDINAVICA, 124(1), 51–60. https://doi.org/10.7146/math.scand.a-105141