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An extremal plurisubharmonic function associated to a convex pluricomplex Green function with pole at infinity. (English) Zbl 0848.31008

If \(K\) is a compact convex non pluripolar subset of \(\mathbb{C}^N\), for the pluricomplex Green function \(g_K\) of \(K\) with pole at infinity, its boundary behavior at \(\partial K\) can be visualized and investigated by the following plurisubharmonic function \(v_K\) on \(\mathbb{C}^N\): If \(H: \mathbb{C}^N\to \mathbb{R}\) is the supporting function of the convex set \(K\), the function \(v_K\) is defined to be the largest plurisubharmonic function bounded from above by \(H\) and having logarithmic growth at the origin.
It is shown for instance that the “slope in direction \(a\)” \((a\in \mathbb{C}^N\), \(|a|=1)\) of \(g_K\) at \(\partial K\) is infinite if and only if \(v_K (\lambda a)\neq H(\lambda a)\) for all \(\lambda> 0\). As an application to functional analysis it is shown how the function \(v_K\) is related to the problem of finding a solution operator for a given constant coefficient linear partial differential operator on the space \(A(K)\) of all analytic functions on \(K\). An essential ingredient of the proofs is a result of L. Lempert on the convexity of the sublevel sets of \(g_K\). This result is published in an appendix.

MSC:

31C10 Pluriharmonic and plurisubharmonic functions
32U05 Plurisubharmonic functions and generalizations
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