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Choquet simplices and Harnack inequalities. (English) Zbl 0639.46012
Abstract analysis, Proc. 14th Winter Sch., Srnî/Czech. 1986, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 14, 339-343 (1987).
[For the entire collection see Zbl 0627.00012.]
Let S be a metrizable Choquet simplex and E the set of all extreme points of S. Given $$x\in S$$, let $$\mu_ x$$ be the unique maximal measure representing the point x and let face(x) be the smallest face of S containing x. By the solution of the Dirichlet problem for a boundary function f on E we understand the affine function $$u_ f$$ defined on $$D_ f=\{x\in S:f\in L$$ $$1(\mu_ x)\}$$ (which is always a face of S) by $$u_ f(x)=\mu_ x(f).$$
The aim of this paper is to describe the points $$x\in S$$ for which an analogue of the Harnack inequality is satisfied on face(x), i.e., for any compact set $$K\subset face(x)$$ there is a number $$\alpha_ K$$ such that for every continuous affine function f:face(x)$$\to [0,\infty)$$ we have $$\sup_{y\in K}f(y)\leq \alpha_ Kf(x)$$. The author proves that this is the case iff the restriction to face(x) of the solution of the Dirichlet problem is continuous for every boundary function from L $$1(\mu_ x)$$.
Reviewer: I.Raşa
##### MSC:
 46A55 Convex sets in topological linear spaces; Choquet theory
Zbl 0627.00012
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