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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 \(U\)).

31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31C35 Martin boundary theory
60J45 Probabilistic potential theory
Full Text: DOI
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