= Liouville lemma for globally Hölder harmonic functions
{c}
{title2=$[w]_{\mu;\mathbb R^n}<\infty\Longrightarrow w\text{ constant}$}
A <harmonic function> on all of $\mathbb R^n$ with a finite global <Hölder seminorm> of exponent $0<\mu<1$ is constant. Indeed $|w(x)-w(0)|\leq C|x|^\mu$, and the <interior derivative estimate for a harmonic function> on balls of radius $R$ bounds $|\nabla w(0)|$ by $CR^{\mu-1}$. Letting $R\to\infty$ makes this <gradient> zero, and translating the centre proves the assertion everywhere.
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