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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.

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