Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
add comment

1 Answer

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 {$\eta, 1/2\le \vert\eta\vert\le 2$}.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.