  In this paper we prove semiclassical resolvent estimates for operators with
normally hyperbolic trapping which are lossless relative to non-trapping
estimates but take place in weaker function spaces. In particular, we obtain
non-trapping estimates in standard $L^2$ spaces for the resolvent sandwiched
between operators which localize away from the trapped set $\Gamma$ in a rather
weak sense, namely whose principal symbols vanish on $\Gamma$.
