Dispersive estimates for Klein-Gordon equations via a physical space approach

Building on the hyperboloidal foliation approach of Lefloch and Ma, we extend Klainerman's physical-space approach to dispersive estimates to recover the frequency-restricted $L^1$--$L^\infty$ dispersive estimates for Klein-Gordon equations. The hyperboloidal foliation approach naturally only provide estimates within a fixed forward light-cone, and is based on an initial data norm that is not translation invariant. Both of these problems can be handled with frequency-dependent physical-space truncations. To handle the lack of scaling symmetry for the Klein-Gordon equation and complete the argument, we also need to keep track of the effectiveness of our estimates in the vanishing mass limit.