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.


31D05 Axiomatic potential theory
Full Text: DOI Numdam EuDML


[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.