Lemma: Fourier existence

The locked formula defines a tempered distribution in $(t,s,\mathbf{z})$, smooth in $t,s$ with values in ${\mathcal{S}}^{\prime }({\mathbb{R}}^3)$. The common-Abel family converges to it distributionally.

Proof. After spatial Fourier transformation of a joint test, the multiplier is bounded on compact time sets by $C(|p{|}^{-1}+|p{|}^{-2})$. Both powers are locally integrable with three-dimensional momentum measure. At infinity the transformed test is rapidly decreasing. Dominated convergence proves the pairing and the Abel limit. Temporal derivatives introduce only polynomial factors in momenta and times, handled by the same estimates. Temporal Schwartz weights absorb the explicit factor $t$, giving temperedness.