Corollary 3.1.22F898AC76FE42

Proof. After changing the sign and adding a constant, we may assume that the function $h$ is nonnegative. Fix $x\in \mathbb R^ d$ and put $r=|x|$. For $R>r$, (3.1.20) on $B(0,R)$ gives $c_ Rh(0)\leq h(x)\leq C_ Rh(0)$, where the coefficients $c_ R$ and $C_ R$ of (3.1.20) tend to one as $R\to \infty $. Letting $R\to \infty $ gives $h(x)=h(0)$. Hence $h$ is constant. $\square$