Stationary-direction bound for singular support ($\mathrm{singsupp}{I}_{\Phi }(a)\subset {\pi }_X(C_a\cap \{{\nabla }_{\theta }\Phi =0\})$)

For a homogeneous phase function $\Phi$ and finite-order symbol class amplitude $a$, the singular support of $I_{\Phi }(a)$ lies in the spatial projection of the closed conic amplitude support intersected with ${\nabla }_{\theta }\Phi =0$. Away from that set, the frequency gradient is uniformly nonzero on compact spatial sets and supported directions. Repeated frequency integration by parts lowers the amplitude order until every desired spatial derivative is absolutely integrable. If the ordinary amplitude support is already conic, it gives the same bound.

For $\Phi (s,t,\theta )=s\theta$, take a smooth symbol that turns on increasingly high frequencies in disjoint spatial bumps centered at $t_j\to 0$. Superpolynomially small bump coefficients keep all symbol class estimates valid. The amplitude vanishes near every finite-frequency point over $t=0$, yet its oscillatory integral has delta singularities at $(0,t_j)$. Closedness of singular support forces a singularity at $(0,0)$ as well. A bound using only ordinary, rather than closed conic, amplitude support can consequently fail.