## 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

### Keywords:

unbounded domains; continuous $$h$$-bounded function
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### References:

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