In Lemma 4.1 "Endpoint Strichartz Estimates" by Keel-Tao the inequality $$ \int_{1 \leq s-t \leq 2} \langle U^*(s)F(s), U^*(t)G(t) \rangle ds dt =:T_0(F,G) \lesssim \lVert F \rVert_{2,a'} \lVert G \rVert_{2,b'}$$ is proved for $F,G$ supported in a time interval of duration $1$. Then, by using the fact that $T_0$ is localized in time, the result is extended to any $F,G$.
I don't understand what it means for $T_0$ to be localized in time, since the integration region is unbounded.