On Morawetz estimates with time-dependent weights for the Klein-Gordon equation

We obtain some new Morawetz estimates for the Klein-Gordon flow of the form \begin{equation*} \big\||\nabla|^{\sigma} e^{it \sqrt{1-\Delta}}f \big\|_{L^2_{x,t}(|(x,t)|^{-\alpha})} \lesssim \|f\|_{H^s} \end{equation*} where $\sigma,s\geq0$ and $\alpha>0$. The conventional approaches to Morawetz estimates with $|x|^{-\alpha}$ are no longer available in the case of time-dependent weights $|(x,t)|^{-\alpha}$. Here we instead apply the Littlewood-Paley theory with Muckenhoupt $A_2$ weights to frequency localized estimates thereof that are obtained by making use of the bilinear interpolation between their bilinear form estimates which need to carefully analyze some relevant oscillatory integrals according to the different scaling of $\sqrt{1-\Delta}$ for low and high frequencies.