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I'm trying to prove 2.74 Theorem of Folland's book, Introduction to PDE. It says:

If $u$ is harmonic on the complement of a bounded set in $\mathbb{R}^n$, the following are equivalent:

a) $u$ is harmonic at infinity.

b) $u(x)\to0$ as $x\to\infty$ if n>2, or $|u(x)|=o(log|x|)$ as $x\to\infty$ if $n=2$.

c)$|u(x)|=O(|x|^{2-n })$ as $x\to\infty$.

I've proved a) implies b) and c) implies a) but I'm stuck proving b) implies c).

I started with n=2, it's required to prove $|u(x)|=O(1)$. The assumption means that for every $\epsilon >0$ exists $M>0$ such that $ |u(x)|<\epsilon\, log|x|$ for every $|x|\geq M.$ From this, ¿How can I prove that $|u(x)|<c$ for every $|x|\geq |x_0|$ for some $C>0$ and some $x_0$?.

For $n>2$, I have the same problem. I think I'm missing some theorem for harmonic functions. I was thinking about maximum principle or maybe the fact that Kelvin transform is harmonic in a punctured bounded set, but I'm not sure how to use them. Please, give me some hint for this implication or tell me if I'm using wrong something.

ifthenelse
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  • I don't have access to the book right now, what is the definition of "harmonic at infinity"? – Jose27 Feb 23 '22 at 04:15
  • @Jose27 It means that Kelvin transform has a removable singularity at zero. – ifthenelse Feb 23 '22 at 04:18
  • Do you know that if a harmonic function in $B(0,r)\setminus{ 0}$ is $o(|x|^{2-n})$ if $n\geq 3$, or $o(\ln |x|)$ if $n=2$ (as $|x|\to 0$ of course); then it extends to a harmonic function on $B(0,r)$? – Jose27 Feb 23 '22 at 04:21
  • @Jose27 Yes, I use it for proving b) implies a) but I need to prove a) implies c) and again I'm in a problem more about limits and Big O notation, or ¿what scheme do you suggest for this problem? – ifthenelse Feb 23 '22 at 04:51
  • It's Okay , the theorem you mentioned is enough. Thanks. – ifthenelse Feb 23 '22 at 05:07
  • See here (the first paragraph in the answer) for a way to prove b) implies c) in the case $n\geq 3$. For the $n=2$ case, I think the last part of the answer could be useful as well. – Jose27 Feb 23 '22 at 06:22

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