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A simple result of microlocal analysis would be, if P is a differential operator then $WF(Pu) \subseteq WF(u)$, where WF denotes the wave front set of the distribution. Can anyone give me an example where the inclusion is strict.

Thanks in advance.

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    $\begingroup$ Take $P = 0$ and $u$ anything with a non-empty wavefront set? $\endgroup$ May 28, 2020 at 15:25
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    $\begingroup$ If you don't like $P =0$: work on $\mathbb{R}^2$ and let $P = \partial_y$ and $u = H(x)$ where $H$ denotes the Heaviside function. $\endgroup$ May 28, 2020 at 15:27
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    $\begingroup$ @hsm: If you have equality, then $P$ is hypoelliptic, by definition. In order to have strict inclusion, then, it is enough to choose $P$ not hypoelliptic; the example that immediately comes to mind is the wave operator $P = \partial_t^2 - \partial_x^2$. $\endgroup$
    – Alex M.
    May 28, 2020 at 16:51

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