Cubic endpoint stationary-phase scaling
= Cubic endpoint stationary-phase scaling
If a phase at an endpoint satisfies $\phi(u)=\phi(0)-cu^3+O(u^5)$ with $c>0$, the leading region has width $O(t^{-1/3})$. Rescaling $u=(ct)^{-1/3}v$ reduces the leading term to a cubic oscillatory integral and produces decay $t^{-1/3}$.