Proposition: Convergence on the admissible open sets

On every relatively compact open subset of ${U_{\Gamma }}^{\mathrm{good}}$, the regularization (eq:A-reg) converges distributionally to the prescribed complete line product.

The limit there is independent of the smoothing choice among standard approximate identities satisfying the same local microlocal approximation estimates.

Proof. Localize every line near the compact set under consideration. Approximate-identity regularization converges weakly and in the wavefront-restricted sense outside any slightly enlarged cone containing the localized wavefront set. The latter assertion follows from the rapid Fourier decay in those directions; the separated tails are controlled by Lemma lem:smoothing.

The sum condition permits such enlarged cones to be chosen without introducing a zero covector sum on this compact set. The sequentially continuous product, equivalently the admissible diagonal pullback of the tensor product, therefore takes the regularized factors to their canonical product. This is the microlocal continuity statement recalled in BF (Section 2, properties M1–M3).

Smooth weights and finite differential operators preserve the result. The product is canonical on this domain, so two such regularizations give the same limit there.