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Assume $u\in L^2(\mathbb{R}^n)$ and let $(x_0, \xi _0) \in T^\ast \mathbb{R}^n = \mathbb{R}^n_x \times \mathbb{R}^n_\xi $. Assume I can find $a\in C^\infty (T^\ast \mathbb{R}^n)$ which is also bounded with all derivatives and $a(x,\xi ) = 1$ in a neighborhood of $(x_0, \xi _0)$. Without loss of generality we may even assume $a\in C _0 ^\infty (T^\ast \mathbb{R}^n)$ and still equal to 1 near $(x_0, \xi _0)$. Assume moreover that we know $$\| \operatorname{Op} [a]u \|_{L^2(\mathbb{R}^n)} \le Ch^s$$ where $$\operatorname{Op} [a]u(x) := \frac{1}{(2\pi h)^{n/2}}\iint \limits_{T^\ast \mathbb{R}^n}e^{\tfrac{i}{h}(x-y)\cdot \xi }a(\tfrac{x+y}{2},\xi )u(y)\, dy\, d\xi .$$ Is this enough to conclude that there is some neighborhood $U$ of $x_0$ such that $$\| u \|_{L^2(U)} \le Ch^s \quad ? $$ If not, what kind of condition would be sufficient?

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The integral does not seem to depend on $u$? Are you missing a $u(y)$? –  Thomas Kragh Apr 25 '12 at 20:37
    
Oh, yes of course, I'll edit. Thanks for pointing it out! –  flavio Apr 26 '12 at 7:36

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Of course no. Your inequality on $\Vert Op[a] u\Vert$ is only providing some information microlocally at $(x_0,\xi_0)$ and says nothing about what is happening elsewhere. For instance, with the homogeneous wave-front set setting, you may have in one dimension a (complex-valued) distribution with wave-front-set equal to {$0$}$\times\mathbb R_-^*$. It is the case for instance of $\frac{1}{x-i0}$.

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Ok, thank you, it's what I was afraid of. Actually I also have that $u$ satisfies $Pu=g$ where $\|g\| \le Ch^s$ and $P$ is elliptic in the complement of the set where $u$ is microlocally $O(h^s)$. –  flavio Apr 26 '12 at 8:17
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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. –  Bazin Apr 26 '12 at 11:47

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