Lemma: Domain for the required products

The compactly smeared quadratic Wick vertex preserves ${\mathcal{D}}_{\varphi }$. The time-ordered and ordinary products in (eq:Rdef) are continuous forms in the finite-sector Schwartz topology.

Proof. Creation-pair kernels of a compactly smeared quadratic Wick polynomial contain the Fourier transform of a compact smooth coefficient at energy ${\omega }_{\mathbf{p}}+{\omega }_{\mathbf{q}}$. That energy controls $|\mathbf{p}|+|\mathbf{q}|$, so these kernels, including their derivative factors, are Schwartz. The number-preserving kernel has polynomial growth along $\mathbf{p}=\mathbf{q}$ and rapid decrease in $\mathbf{p}-\mathbf{q}$, with the same property after differentiation. The inequality

shows that it maps Schwartz functions to Schwartz functions. These facts prove invariance of the algebraic finite-sector domain.

Completely normal ordered matrix elements between vectors of this domain are smooth in spacetime, with all derivatives bounded on compact sets by finite Schwartz seminorms. For contraction terms, first smear $D(y-x)$ in $x$ against $f$. The result is smooth in $y$. Multiplication by the compact coefficient $gh$, followed by integration by parts, leaves a linearly smeared scalar field. This proves the asserted form bounds.