Questions tagged [microlocal-analysis]

I'm trying to show that the wavefront set of a function $f \in C^{\infty}(\mathbb{R}^{n})$ is empty, $WF(f) = \emptyset$. Can anyone help me prove this? This is my first exposition into wavefront sets so I have little progress with regards to…

I read a statment on a book saying that the FBI transform $$\mathcal{F}_u(x,\xi)=\int_{\mathbb{R}^m} e^{i\xi.(x-y)-|\xi||x-y|^2}u(y)\,dy ,\; (x,\xi)\in \mathbb{R}^m \times \mathbb{R}^m$$ is nonlinear. I can not see how this is nonlinear.