Proposition: Joint seminorm estimate

There are compact sets $K_1,K_2,K_3$, finite orders $r_1,r_2,r_3$, and a constant $C$ such that

Thus ${\mathcal{O}}_{\mathrm{pol}}$ is jointly continuous on the chosen smooth configuration spaces.

The orders are finite but unspecified. One may take $r_1=r_2=r_3=r$ for a finite distributional order bound $r$ of $t_{\mathrm{D}}$. If $N$ is a finite order bound for $B$ on a suitable compact neighborhood, the construction provides an order bound at most $N+2$ for $t_{\mathrm{D}}$. No numerical or optimal value of $N$ or $r$ is asserted.

The constant may depend on the fixed observation data, switch, mass, collar, coefficient functions, and root kernel. No estimate uniform as $\epsilon \downarrow 0$ is claimed.

Proof. A compactly supported distribution has finite order. Choose compact neighborhoods $K_1,K_2$ of $C_0$ and $K_3$ of $O_{f,g}$. There are $r,C$ such that

for every smooth $F$, using the cutoff-independent extension of the pairing.

For $F=a\otimes c\otimes \varphi$, differentiation in the three independent variables gives

Together with (eq:O-t), this proves (eq:target).

Alternatively, an order-$N$ bound for $B$ applied to (eq:t-test) differentiates the evaluated test at most $N$ additional times. The source deltas have at most two derivatives. The chain and product rules therefore require at most $N+2$ derivatives of $F$; derivatives of the fixed coefficients are absorbed into the constant.

The product seminorm estimate proves joint continuity of the trilinear map.