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Jensen measures and boundary values of plurisubharmonic functions. (English) Zbl 1021.32014
For a bounded domain \(\Omega\), let \({\mathcal J}_z^c\) and \({\mathcal J}_z\) denote the collections of all Jensen measures with barycentre \(z\in\overline\Omega\) for continuous (resp., upper bounded) plurisubharmonic functions in \(\Omega\). It is shown that \({\mathcal J}_z\) can be a proper subset of \({\mathcal J}_z^c\), however these two classes coincide for all B-regular domains \(\Omega\). The proof of the latter result is based on an approximation theorem for upper bounded plurisubharmonic functions in a B-regular domain \(\Omega\) by continuous plurisubharmonic functions on \(\overline\Omega\). Conversely, if a bounded hyperconvex domain \(\Omega\) is such that \({\mathcal J}_z ={\mathcal J}_z^c\) for all \(z\), then \(\Omega\) possesses this approximation property. The author gives also an exact characterization, in terms of Jensen measures, of those continuous functions on \(\partial\Omega\) that are boundary values of continuous plurisubharmonic functions.

32U05 Plurisubharmonic functions and generalizations
46E27 Spaces of measures
31C10 Pluriharmonic and plurisubharmonic functions
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