The Bergman-Shilov boundary for subfamilies of $q$-plurisubharmonic functions
Thomas Pawlaschyk
TL;DR
The paper develops a general framework for Shilov (Bergman-Shilov) boundaries for subclasses of upper semi-continuous functions, focusing on $q$-plurisubharmonic and $q$-holomorphic classes on convex and pseudoconvex domains. It introduces closure, peak points, and Bishop-type results to obtain existence and identifications of Shilov boundaries, and then applies these tools to obtain a geometric generalization of Bychkov's theorem: a boundary point lies outside the Shilov boundary for these classes precisely when the boundary contains an open part of a complex plane of dimension at least $q+1$. The work also provides Hausdorff-dimension lower bounds, and shows that for smooth domains the Shilov boundary coincides with the closure of strictly $q$-pseudoconvex points, with parts foliate by complex $q$-dimensional submanifolds, thereby linking boundary geometry to complex-analytic structure across several function classes.
Abstract
We introduce a notion of the Bergman-Shilov (or Shilov) boundary for some subclasses 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 subclasses with simple structure one can show the existence and uniqueness of the Shilov boundary. Then 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. In the case of bounded pesudoconvex domains with smooth boundary we also show that some parts of the Shilov boundary for $q$-plurisubharmonic functions are foliated by $q$-dimensional complex submanifolds.