## 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

### Keywords:

Bocher theorem; harmonic dimension at infinity
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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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