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Bĥer’s theorem in a space of dimension one. (English) Zbl 0966.31005

In the axiomatic potential theory, let \(X\) be a harmonic space in the sense of Brelot; \(X\) has a countable base; the constants are assumed to be harmonic; and \(X\) satisfies the local axiom of proportionality (that is, for any \(x\in X\), if \(p_1\) and \(p_2\) are two potentials in a domain \(\delta\), \(x\in \delta\), with harmonic point support \(x\), then \(p_1\) and \(p_2\) are proportional). Then one can define a kernel \(q_y(x)\) in \(X\), which has most of the properties of the Newtonian kernel \(\frac{1}{|x-y|}\) in \(\mathbb{R}^3\) if \(X\) has potentials \(>0\) and those of the logarithmic kernel \(\log|x-y|\) in \(\mathbb{R}^2\) if \(X\) has no positive potentials.
The authors prove in this note, that if \(h\) is a harmonic function defined outside a compact set in \(X\), then there exists a signed (Radon) measure \(\mu\) with compact support and a unique harmonic function \(u\) on \(X\) such that \(h(x)= \int q_y(x) d\mu(y)+ u(x)\) near infinity; moreover, if the harmonic dimension at infinity of \(X\) is one, then \(u\) is a constant if and only if \(h\) is bounded on one side near infinity. They interpret the last remark when \(X= \mathbb{R}^n\), \(n\geq 2\), after an inversion, as the classical Bôcher theorem in \(\mathbb{R}^n\) concerning the point singularity of a positive harmonic function.

MSC:

31D05 Axiomatic potential theory
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References:

[1] Anandam, V., Espaces Harmoniques Sans Potentiel Positif, Ann. Inst. Fourier, Grenoble, 22, 4 (1972) 97-160. · Zbl 0235.31015
[2] Anandam, V., Potentials in a B.S. Harmonic Space, Bull. Math. Roumanie, 183-4 (1974), 233 - 248. · Zbl 0337.31011
[3] Axler, S., Bourdon, T. and Ramey, W., Harmonic Function Theory, GTM, Vol. 137, Springer-Verlag, Germany, 993. · Zbl 0765.31001
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