  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.
