Semiclassical resolvent bounds for weakly decaying potentials

In this note, we prove weighted resolvent estimates for the semiclassical Schrodinger 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 obey a radial $\alpha$-Holder continuity condition, $0\leq \alpha \leq 1$, with sufficient of local $C^\alpha$ norm toward infinity. Note, however, that in Holder case, need \emph{not} decay. If dimension \ge 3$, bound form $\exp \left(C h^{-1 - \frac{1 \alpha}{3 \alpha}} [(1-\alpha) \log(h^{-1})+c]\right)$, while = 1$ it $\exp(Ch^{-1})$. A new type weight phase function construction allows us reduce necessary even pure $L^\infty$ case.