Integral representation of harmonic functions defined outside a compact set in a harmonic space
The classical Bôcher theorem in \(\mathbb{R}^n\), after a Kelvin transformation, can be stated as follows: Let \(h\geq 0\) be harmonic outside a compact set in \(\mathbb{R}^n\), \(n\geq 2\). Then \(h\) is of the form \[ h(x): \begin{cases} \alpha\log| x|+ b(x),\quad &\text{if }n=2,\\ \alpha| x|^{2-n}+ b(x),\quad &\text{if }n\geq 3,\end{cases} \] where \(\alpha\) is a constant and \(b(x)\) is bounded harmonic outside a compact set. The authors prove a similar result in the context of the (Brelot) axiomatic potential theory: Let \(\Omega\) be a Brelot harmonic space. Let \(q_y(x)\) be the kernel on \(\Omega\) which behaves like the Newtonian kernel (if \(\Omega\) has potentials \(>0\)) or the logarithmic kernel (if there are no positive potentials on \(\Omega\)). Then, given any harmonic function \(u\) outside a compact set in \(\Omega\), then exists a signed measure \(\mu\) with compact support and a harmonic function \(H\) on \(\Omega\), such that \[ u(x)= \int_\Omega q_y(x)\,d\mu(y)+ H(x) \] outside a compact-set.
scientific article; zbMATH DE number 1901232