wavefront is a coisotropic

Question

I'm thinking about the following example: Let $u_k:=e^{i(kx_1+k^2x_2)}$, where $k\in\mathbb{N}$ and $x_1,x_2\in\mathbb{R}$. Then for the sequence $h_k:=1/(k\sqrt{1+k^2})$ ($h_k\rightarrow0$ as $k\rightarrow\infty$), we have that $(h^{2}_{k}\Delta-1)u_k=0$.

Now, one can show (e.g. by a nonstationary phase argument) that $WF_{h_k}(u_k)=\{x_1,x_2,\xi_1=0,\xi_2=1\}\subset T^*\mathbb{R}^2$. Here $WF_{h_k}$ is semiclassical wavefront set.

My question is: In the above example, $WF_{h_k}(u_k)$ ends up being a Lagrangian submanifold of the phase space. Is there an example of a distribution similar to (or very different from) $u_k$ whose wavefront set is a coisotropic submanifold of $T^*\mathbb{R}^2$ (e.g. $\{x_1,x_2,\xi_2\in\mathbb{R},\xi_1=0\}$)?

Answer 1

Yes: look at Theorem 8.1.4 in the first volume of Hörmander's ALPDO (Springer Grund. 256). For the classical (conic) wave-front-set, given any closed conic set $S$, you can construct a distribution $u$ with $WF u=S$. Adding a variable, will provide the semi-classical version.