Semiclassical resolvent estimates for Holder potentials

We first prove semiclassical resolvent estimates for the Schr{"o}dinger operator in R d , d $\ge$ 3, with real-valued potentials which are H{"o}lder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in R d , d $\ge$ 2, and real-valued potentials which are H{"o}lder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be H{"o}lder, Lipschitz or just L $\mathrm{\infty }$ .