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Mean values and harmonic functions. (English) Zbl 0794.31002
It is shown that for every domain \(U\) in \(\mathbb{R}^ d\), \(d\geq 1\), such that \(\complement U\neq\emptyset\) (non-polar if \(d=2\)) every continuous function \(f\) on \(U\) which is bounded by some harmonic function \(h\geq 0\) is harmonic provided for every \(x\in U\) there exists a ball \(B_ x\) contained in \(U\) and centered at \(x\) such that \(f(x)= 1/\lambda(B_ x) \int_{B_ x} f d\lambda\). This result holds as well for more general means. Moreover, the underlying general theorem can be applied to Lebesgue measurable functions.

MSC:
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
60J65 Brownian motion
31C35 Martin boundary theory
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