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A converse to the mean value theorem for harmonic functions. (English) Zbl 0808.31004
Let $$U$$ be a bounded domain in $$\mathbb{R}^ d$$, $$d\geq 1$$, $$\rho(x)= \text{dist} (x,U^ c)$$, and $$r:U\to \mathbb{R}$$ a function satisfying $$0<r\leq\rho$$. Let $$B^ x$$, $$x\in U$$, denote the open ball centered at $$x$$ with radius $$r(x)$$. A Lebesgue measurable function $$f$$ on $$U$$ satisfying $f(x)= {1\over {\lambda(B^ x)}} \int_{B^ x} f d\lambda$ for every $$x\in U$$ (where $$\lambda$$ denotes the Lebesgue measure), is said to be $$r$$-median. Results of the type under what conditions is an $$r$$-median function $$f$$ actually harmonic, are usually known as a converse to the mean value theorem.
The main result of the present paper states that if $$f$$ is $$r$$-median, continuous (on $$U$$) and $$h$$-bounded (i.e. $$| f|\leq h$$ with $$h$$ harmonic on $$U$$), then $$f$$ is harmonic on $$U$$. As a rather simple consequence of this result, the authors show that an $$r$$-median, $$h$$- bounded, Lebesgue measurable function $$f$$ on $$U$$ is harmonic provided that the function $$r$$ is bounded away from zero on compact subsets of $$U$$, thus improving the result of W. A. Veech [Ann. Math., II. Ser. 97, 189-216 (1973; Zbl 0282.60048)], where $$U$$ was assumed to be a Lipschitz domain.
The proof of the results is analytic, but with the strong probabilistic flavor in the background. It uses the minimal fine topology of the Martin compactification of $$U$$, an appropriate (transfinite) sweeping of measures, and certain properties of the Schrödinger equation $$\Delta u- \delta\rho^{-2} 1_ A=0$$ on $$U$$ ($$\delta>0$$, $$A$$ a suitable subset of $$U$$).

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31C35 Martin boundary theory 60J45 Probabilistic potential theory
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##### References:
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