I'm trying to prove exercise 25 of chaper 2 in GTM 137 Harmonic Function Theory:
Let $K$ denote a compact line segment contained in $B_{3}$. Show that every bounded harmonic function on $B_{3} \backslash K$ extends to be harmonic on $B_{3}$.
It is a generalization of this statement(a modification of theorem 2.3):
If $u$ is bounded and harmonic on $\bar{B} \backslash\{0\},$ then $u$ has a harmonic extension to $B .$
In the proof of it, one uses $v_{\varepsilon}(x)=u(x)-P[u|_S](x)+\varepsilon\left(|x|^{2-n}-1\right)$ and shows $u(x)\equiv P[u|_S](x)$, thus harmonic in $B$ where $P[u|_S](x)$ is the Poisson integral.
I think I can find a function defined on $B_{3} \backslash K$ which behaves similarly with $|x|^{2-n}$ to play the role of $|x|^{2-n}$ in this proof to prove the exercise. But I can't find such a function. Any suggestions?