Uniformization of degenerate equations and semiclassical asymptotics

For a stratified manifold $X$ that is the quotient of a smooth compact manifold $M$ by a compact Lie group action, we define smooth functions on $X$ as invariant smooth functions on $M$ and $h$-(pseudo)differential operators on $X$ as the restrictions of invariant $h$-(pseudo)differential operators on $M$ to the space of smooth functions on $X$. We discuss semiclassical asymptotics for pseudodifferential equations on $X$; examples include boundary degenerate linearized shallow water equations describing the run-up of long waves on a shallow beach and $h$-pseudodifferential equations on orbifolds.