Bergman–Shilov boundary for subfamilies of $q$-plurisubharmonic functions

Thomas Patrick Pawlaschyk¹

¹ Faculty of Mathematics and Natural Sciences University of Wuppertal 42119 Wuppertal, Germany

Annales Polonici Mathematici, Tome 117 (2016) no. 1, pp. 17-39

Résumé

We introduce the notion of the Shilov boundary for some subfamilies of upper semicontinuous functions on a compact Hausdorff space. It is by definition the smallest closed subset of the given space on which all functions of that subclass attain their maximum. For certain subfamilies with simple structure we show the existence and uniqueness of the Shilov boundary. We provide its relation to the set of peak points and establish Bishop-type theorems. As an application we obtain a generalization of Bychkov’s theorem which gives a geometric characterization of the Shilov boundary for $q$-plurisubharmonic functions on convex bounded domains.

Détail

DOI : 10.4064/ap3695-1-2016

Keywords: introduce notion shilov boundary subfamilies upper semicontinuous functions compact hausdorff space definition smallest closed subset given space which functions subclass attain their maximum certain subfamilies simple structure existence uniqueness shilov boundary provide its relation set peak points establish bishop type theorems application obtain generalization bychkov theorem which gives geometric characterization shilov boundary q plurisubharmonic functions convex bounded domains