The paper "A proof of Liouville's theorem" by E. Nelson, published in 1961 in Proceedings of AMS, contains just one paragraph, giving a (now) standard proof that every bounded harmonic function in $\mathbb{R}^n$ is a constant.
I presume there must be a story behind. First, it is hard to imagine that this proof was unknown before 1961. Second, even if this is the case, it doesn't feel usual, for the author, to submit such a paper and, for the editor, to accept it.
So, can anyone tell that story? Or, to make the question precise:
1) are there any earlier references for this proof?
2) what was/were the standard proof(s) before 1961?
3) by a very similar reasoning, one obtains $̣||\nabla h||_{\infty,\Omega}\leq C(\Omega,\Omega')||h||_{\infty, \Omega'}$ for $\Omega\subset\Omega'$. Was that argument also unknown until 1961?
Best Answer
This doesn't answer any of the three specific questions asked, but addresses an implicit question: "Why did the editor accept it?"
In 1961, the Proceedings of the AMS established a section called "Mathematical Pearls" devoted to, I quote:
In the issue in which Nelson's proof appears, that section starts on page 991 and continues to the end, including 7 papers in all, none of which exceeds 2 pages. In a different issue you can find this paper which also contains the quoted disclaimer above.