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Session 5. Complex Analysis
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Geometric characterization of the Shilov boundary for \(q\)-plurisubharmonic functions on bounded convex domains
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T. Pawlaschyk, University of Wuppertal, Germany
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We introduce a notion of the Shilov boundary for some subclasses of
upper semi-continuous 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. We also
show that there is an analytic foliation of parts of the Shilov
boundary for \(q\)-plurisubharmonic functions on smoothly bounded
pseudoconvex domains.
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