in this paper http://arxiv.org/pdf/1412.1626.pdf it says that Lemma 3.1/(3.1) follows from Theorem 1.3 in http://arxiv.org/pdf/math/0402192.pdf without extra details. Can somebody please explain that?

I can see, that these two estimates are every similar as $q=2$, $r=\infty$ , $n=3$ are admissable in Theorem 1.3. Furthermore the $\gamma=1$ in Theorem 1.3 fits perfectly with the $\lambda$ in Lemma 3.1.

The only real difference seems to be that Theorem 1.3 concerns the wave equation and Lemma 3.1 the Klein-Gordon-Equation, I know, that the wave equation is just the Klein-Gordon-Equation with $m=0$ but I still don't see why this local Klein-Gordon-Estimate follows directly from the corresponding local Wave-Estimate.

Thanks a lot for your help!