Corollary: Wavefront of the causal kernel

The wavefront of $E_{\beta }$ is the union of both frequency orientations of (eq:main):

Proof. Equation (eq:odd) writes $E_{\beta }$ as the difference of two kernels with opposite, disjoint temporal orientations. Theorem thm:main, applied to time reversal, identifies the two sets. A microlocal singularity in one temporal orientation cannot be cancelled by a kernel smooth in that orientation.