I need to prove that the wave front set of a distribution (as defined in Hormander's "The analysis of linear partial differential operators I") is equal to the semiclassical frequency set of an h-dependant $L^2$ function minus $R^n\times 0$, and conversely the semiclassical frequency set is the wave front set plus the points of $supp(u)\times 0$, but I don't know how. Some help (or bibliography) on this matter'll be highly appreciated.
1 Answer
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Let $u$ be a distribution on some open subset $\Omega$ of $\mathbb R^n$. A point $(x_0,\xi_0)\in \Omega\times\mathbb S^{n-1}$ does not belong to $WF u$ when there exists a neighborhhod $V$ of $x_0$ and a neighborhood $\Gamma$ of $\xi_0$ (in the sphere $\mathbb S^{n-1}$) such that for all $\chi\in C_c^\infty(V)$ and for all $N\ge 0$, $$ \sup_{\lambda \ge 1,\ \xi\in \Gamma}\lambda^N\vert\widehat{(\chi u)}(\lambda \xi)\vert<+\infty. $$ This is equivalent to: there exists a neighborhood $V$ $\dots$ for all $N\ge 0$, $$ \sum_{\nu\in \mathbb N}2^{2\nu N}\int_{2^\nu\le \vert \xi\vert\le 2^{\nu+1}, \frac{\xi}{\vert\xi\vert}\in \Gamma}\vert\widehat{(\chi u)}(\xi)\vert^2 d\xi<+\infty. $$ This is equivalent to: there exists a neighborhood $V$ $\dots$ for all $N\ge 0$, $$ \Vert(\mathbf 1_\Gamma)(D/\vert D\vert)\phi(h D)(\chi u)\Vert_{L^2}=O(h^N),\quad\text{$h\rightarrow 0_+$}, $$ where and $\phi\in C_c^\infty(\mathbb R^n)$ supported in a ring {$\eta, 1/2\le \vert\eta\vert\le 2$}.
