Lemma: Fourier definition and regulator convergence

The locked formula defines a tempered distribution in $(t,s,\mathbf{z})$. Moreover,

The distribution is smooth in $(t,s)$ with values in ${\mathcal{S}}^{\prime }({\mathbb{R}}^3)$.

Proof. Let $\Xi (t,s,\mathbf{z})$ be a compactly supported smooth test function and take its partial spatial Fourier transform. On its time support the absolute value of the momentum coefficient is bounded by

with $C$ independent of $0<\epsilon \le 1$. The partial Fourier transform decreases faster than any inverse power of $|p|$, uniformly on that support.

Near $p=0$, spherical measure gives

At infinity, the Fourier decay supplies an integrable majorant. Dominated convergence therefore defines the unregulated pairing and proves convergence on compactly supported tests.

For Schwartz tests, the same estimate is combined with temporal Schwartz weights sufficient to absorb the explicit factor $t$. This gives the tempered estimate and convergence in ${\mathcal{S}}^{\prime }$. Temporal differentiation of the momentum coefficients introduces only additional nonnegative powers of $|p|$ and polynomial temporal factors. Spatial Fourier decay again dominates them. Differentiation under the paired integral therefore proves smooth dependence on the time parameters.