Theorem: Authorized-extension theorem

Suppose:

1. Assumption ass:expansion is verified;

2. the raw family has proved reference convergence $t_{\delta }{|}_{V_{\mathrm{ref}}}\to t^{\circ }$;

3. $S_{\delta }\in \mathcal{B}$, with the authorized subtraction sign $t_{\delta }-S_{\delta }$;

4. $T$ satisfies the required tensor, growth, support, microlocal, and overlap conditions;

5. additions from $\mathcal{A}$ preserve those conditions.

Then $T$ is an authorized subtracted extension of $t^{\circ }$. Its normalized extensions in this ambiguity class are exactly

They exist if and only if

If a solution exists, it is unique if and only if

Proof. The complete-expansion theorem supplies the subtracted limit. Proposition prop:support gives its reference agreement. The authorization and admissibility hypotheses supply the additional extension requirements.

The normalization equation is $\mathcal{N}(a)=n_{\ast }-\mathcal{N}(T)$. Its solvability is the image condition. Differences of solutions form the kernel of the restricted map.