Morawetz estimates for the wave equation at low frequency

We consider Morawetz estimates for weighted energy decay of solutions to the wave equation on scattering manifolds (i.e., those with large conic ends). We show that a Morawetz estimate persists for solutions that are localized at low frequencies, independent of the geometry of the compact part of the manifold. We further prove a new type of Morawetz estimate in this context, with both hypotheses and conclusion localized inside the forward light cone. This result allows us to gain a 1/2 power of $t$ decay relative to what would be dictated by energy estimates, in a small part of spacetime.
Comment: New version of Theorem 1.1 added that includes refined $\ell^1$-$\ell^\infty$ estimate in dyadic shells. Some changes to exposition