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Jensen measures and approximation of plurisubharmonic functions. (English) Zbl 1102.31009

For a bounded domain \(\Omega\subset\mathbb{C}^n\), let \({\mathcal B}(\overline\Omega)\) denote the set of all Borel probability measures on \(\overline\Omega\), \(\text{PSH}(\Omega)\) denote the cone of plurisubharmonic functions on \(\Omega\), and \(\text{PSM}^c(\Omega)= \text{PSH}(\Omega)\cap C(\overline\Omega)\). The authors consider two classes of Jensen measures, \[ \begin{aligned} J_z(\overline\Omega) &= \Biggl\{\mu\in B(\overline\Omega): u(z)\leq \int_{\overline\Omega} u^* \,d\mu,\;\forall u\in \text{PSH}(\Omega),\;\sup u<\infty\Biggr\},\\ J^c_z(\overline\Omega) &= \Biggl\{\mu\in B(\overline\Omega): u(z)\leq\int_{\overline\Omega} u^*\,d\mu,\;\forall u\in \text{PSH}^c(\Omega)\Biggr\},\end{aligned} \] where \(u^*\) is the upper regularization of \(u\). It is shown that \(J_z= J^c_z\,\forall z\in\Omega\) if and only if \(\Omega\) has the following property: for every upper bounded \(u\in\text{PSH}(\Omega)\) there exists a uniformly upper bounded sequence \(u_j\in \text{PSH}^c(\Omega)\) such that \(u_j\to u\) pointwise on \(\Omega\) and \(\varlimsup_{j\to\infty} u_j\leq u^*\) on \(\partial\Omega\). A useful sufficient condition for the equality to hold is given as well.
Another main result states, roughly speaking, that if the set \(A= \{z: J_z= J^c_z\}\) contains a sufficiently large portion near \(\partial\Omega\), then \(\Omega\setminus A\) is pluripolar.

MSC:

31C10 Pluriharmonic and plurisubharmonic functions
32V05 CR structures, CR operators, and generalizations
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References:

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