Semiclassical resolvent estimates and wave decay in low regularity

We study weighted resolvent bounds for semiclassical Schr\"{o}dinger operators. When the potential function is Lipschitz with long range decay, the resolvent norm grows exponentially in the inverse semiclassical parameter $h$. When the potential belongs to $L^\infty$ and has compact support, the resolvent norm grows exponentially in $h^{-4/3}\log(h^{-1})$. This extends the works of Burq, Cardoso-Vodev, and Datchev. Our main tool is a global Carleman estimate. Applying the resolvent estimates along with the resonance theory for blackbox perturbations, we show local energy decay for the wave equation with wavespeeds that are an $L^\infty$ perturbation of unity.