Fixed-width switching is not a smoothing theorem

If the chosen switch is smooth at fixed $\epsilon$, then smooth multiplication is defined and does not enlarge wavefront sets:

The wavefront set of a sum is contained in the union of the wavefront sets of its summands:

These general facts are discussed in BDHintro.

They do not identify $\mathrm{WF}(D_{\sigma })$ or the wavefront sets of the observable kernels derived from $J_D$. In particular, (eq:smoothmult) is not a statement that ${\sigma }_{\epsilon }{J}_3$ is smooth.

deduction. On an open set where a smooth multiplier $a$ is nowhere zero,

after restriction to that open set.

. One inclusion follows from smooth multiplication. Locally, $1/a$ is smooth and $v=(1/a)(av)$, which gives the reverse inclusion.

Thus switching does not remove a singularity at a point where the switch is nonzero merely by being smooth. Possible cancellation in the total source must be proved for the actual distributions; the upper bound (eq:sumbound) neither proves nor excludes such cancellation.

All statements here are at fixed $\epsilon$. No uniform estimate as $\epsilon \downarrow 0$, and no sharp-width limit, is asserted.