= Symbol order reduction by a phase integration operator
{title2=$L^t:S^s_{1,0}\to S^{s-1}_{1,0}$}
For a homogeneous <phase function>, let $q=|\nabla_x\Phi|^2+|\theta|^2|\nabla_\theta\Phi|^2$ and $L=(iq)^{-1}(\nabla_x\Phi\cdot\nabla_x+|\theta|^2\nabla_\theta\Phi\cdot\nabla_\theta)$. Then $Le^{i\Phi}=e^{i\Phi}$. The <formal transpose of a differential operator> $L^t$ lowers the <symbol class> order by one: the $x$ coefficients have order $-1$, while the order-zero frequency coefficients accompany a frequency derivative.
Back to article page
