Extremal$ω$-plurisubharmonic functions as envelopes of disc functionals

Benedikt Steinar Magnússon Science Institute, University of Iceland

Complex Variables and Complex Analysis mathscidoc:1701.01014

Arkiv for Matematik, 49, (2), 383-399, 2009.6
For each closed, positive (1,1)-current$ω$on a complex manifold$X$and each$ω$-upper semicontinuous function$φ$on$X$we associate a disc functional and prove that its envelope is equal to the supremum of all$ω$-plurisubharmonic functions dominated by$φ$. This is done by reducing to the case where$ω$has a global potential. Then the result follows from Poletsky’s theorem, which is the special case$ω$=0. Applications of this result include a formula for the relative extremal function of an open set in$X$and, in some cases, a description of the$ω$-polynomial hull of a set.
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  title={Extremal$ω$-plurisubharmonic functions as envelopes of disc functionals},
  author={Benedikt Steinar Magnússon},
  booktitle={Arkiv for Matematik},
Benedikt Steinar Magnússon. Extremal$ω$-plurisubharmonic functions as envelopes of disc functionals. 2009. Vol. 49. In Arkiv for Matematik. pp.383-399. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203630597623004.
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