Lemma: A spacelike sharp comparison region

In the flat half-space $M_{+}=[0,\infty )\times {\mathbb{R}}^n$, suppose $f,g$ are supported at times in $[0,T]$, with spatial supports in balls $\overline{B}(x_0,r)$, $\overline{B}(y_0,s)$, respectively. If

then the upper support sets in (eq:support-bounds) are causally disjoint. Hence both $B_f\gtrsim W_g$ and $W_g\gtrsim B_f$ hold.

Proof. A point of $P_F\cup F$ has time in $[0,T]$ and spatial distance at most $T+r$ from $x_0$. A point of $G$ has spatial distance at most $s$ from $y_0$. Their spatial separation is therefore greater than $T$, whereas their time separation is at most $T$. They are strictly spacelike. The actual functional supports are subsets of these sets.