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Separation properties involving harmonic functions. (English) Zbl 0960.31001
The author recalls the following result from the theory of harmonic functions: let \(X\) be a compact, convex set with non-empty interior in the space \(\mathbb{R}^m\). Given a continuous, concave function \(u:X\to \mathbb{R}\) and a continuous convex function \(v:X\to\mathbb{R}\) with the property \(u\leq v\), there exists an affine function \(a\) in \(\mathbb{R}^m\) with the separation property \(u\leq a\leq v\) in \(X\). It is also known that \(u= \inf \{a|_x\); \(a\), affine in \(\mathbb{R}^m\); \(a\geq u\) on \(X\}\); an analogous assertion is valid for \(v\).
The author poses a similar problem in another context. He replaces \(X\) with \(\overline{U}\), \(U\subset \mathbb{R}^m\) a bounded, open set, and affine functions with harmonic ones. He replaces, also, convex and concave functions with subharmonic and superharmonic functions, respectively. Introducing the space of functions \(H(U)= \{h\in C(\overline{U})\); \(h|_U\) harmonic}, he proves that for superharmonic and subharmonic functions, a Hahn-Banach property does not hold with elements from \(H(U)\) if \(m\geq 2\). This result appears as the outcome of a long row of papers elaborated by the author since 1971.
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators
26B25 Convexity of real functions of several variables, generalizations