Proposition: Traceable supported-sector reference

Suppose on an open set $O\subset U$:

1. a supported kernel has normal form ${\sum }_{|\alpha |\le 3}{\mathsf{E}}_{\alpha }{v}_{\alpha }$;

2. the cross-line normal traces $c_{D,\beta }=({{\partial }_q}^{\beta }{c}_D){|}_S$ exist;

3. the products $v_{\alpha }{c}_{D,\beta }$, and the relevant distributional product before restriction, satisfy the stated wavefront conditions.

Then its grouped reference product is

Proof. The identity holds for smooth cross-line coefficients. Regularize in a wavefront-compatible topology on the admitted domain. Continuity of the checked restriction and product operations passes the identity to the distributional limit Hormander,BrunettiFredenhagen.