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On regularization of plurisubharmonic functions near boundary points. (English) Zbl 1343.32027

The main result of the paper is the following approximation theorem for plurisubharmonic functions. Let \(D\subset\mathbb C^n\) be a domain with Lipschitz boundary. Then every point \(P\in\partial D\) has a neighborhood \(U\) for which \(G:=D\cap U\) is a domain with Lipschitz boundary and for any function \(u\in\mathcal{PSH}(G)\) there exists a sequence \((u_k)_{k=1}^\infty\subset\mathcal{PSH}(G)\cap\mathcal C^\infty(G)\) such that \(u_k\searrow u\).
The assumption that the boundary of \(D\) is Lipschitz is essential.

MSC:

32U05 Plurisubharmonic functions and generalizations
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References:

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