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Inverse mean value property of harmonic functions. (English) Zbl 0794.31001
Let \(A\) be a subset of \(\mathbb{R}^ d\), \(d\geq 2\), having finite Lebesgue measure \(\lambda(A)\) and let \(B\) denote the open ball of centre 0 such that \(\lambda(B)= \lambda(A)\). The following result is the basis for similar results on various classes of harmonic functions: The equality \[ h(0)= \lambda(A)^{-1} \int_ A h d\lambda \] holds for every bounded function \(h=G^{1_ C \lambda}- G^{1_ D \lambda}\), \(C\), \(D\) compact subsets of \(\complement A\), if and only if \(\lambda(B \setminus A)=0\).

MSC:
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
26B15 Integration of real functions of several variables: length, area, volume
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