  In this note, we prove weighted resolvent estimates for the semiclassical
Schr\"odinger operator $-h^2 \Delta + V(x) : L^2(\mathbb{R}^n) \to
L^2(\mathbb{R}^n)$, $n \neq 2$. The potential $V$ is real-valued, and assumed
to either decay at infinity or to obey a radial $\alpha$-H\"older continuity
condition, $0\leq \alpha \leq 1$, with sufficient decay of the local radial
$C^\alpha$ norm toward infinity. Note, however, that in the H\"older case, the
potential need \emph{not} decay. If the dimension $n \ge 3$, the resolvent
bound is of the form $\exp \left(C h^{-1 - \frac{1 - \alpha}{3 + \alpha}}
[(1-\alpha) \log(h^{-1})+c]\right)$, while for $n = 1$ it is of the form
$\exp(Ch^{-1})$. A new type of weight and phase function construction allows us
to reduce the necessary decay even in the pure $L^\infty$ case.
