Well, in that case that's OK since your first estimate does control $u$ near the point $(x_0,\xi_0)$ and by the ellipticity of your $P$, you control as well what may happen elsewhere. You can write down the details for that stuff with a partition of unity. Best, Bazin.

For an elliptic operator $P$ of order $m$, you can construct an approximate inverse in the following sense. There exists $Q$ a pseudo differential operator of order $-m$ such that $$ PQ=Id+R,\quad QP=Id +S $$ where $R,S$ are regularizing operators (that is with smooth kernels). That construction can be microlocalized for a (pseudo)differential operator which is elliptic at a point $(x_0,\xi_0)$ and is a way to prove that $$ WF u\subset\WF(Pu)\cup\char P. $$

I disagree with Paul Garrett answer: the Fourier transform of $\vert x\vert^s$ on $\mathbb R^n$ is a constant multiple of $$ \vert x\vert^{-s-n}, $$ not $\vert x\vert^{s-n}$ as written in his answer. To have both sides locally integrable, we need $$ -n<\Re s<0. $$ Bazin.