= Slice-support obstruction for oscillatory singularities
{title2=$a(0,\theta)=0\ \not\Rightarrow\ I_\Phi(a)\text{ is smooth near }0$}
In one base and one phase variable, take $\Phi=x\theta$ and $a=x\eta(\theta)/|\theta|$, with an even smooth <cutoff function> equal to zero near zero and one for large $|\theta|$. This is a symbol of order $-1$. Its <oscillatory integral> is $-2x\log|x|$ plus a smooth function near zero. Thus it is singular there even though the restricted function $a(0,\cdot)$ has empty support. The joint closed <conic support of an oscillatory amplitude> retains the limiting base point and its stationary directions. A support-sensitive regularity theorem must use that conic neighborhood information, not merely vanishing of an amplitude at one base point.
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