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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 homoge …

Your intuitive characterization does not make sense: the function $u$ is defined on some neighborhood of $x_0$ and $(x_0,\xi_0)$ belongs to the sphere bundle. On the other hand, you may salvage part o …

More a comment than an answer. I am not quite sure to understand the property you are looking for. After all the principal symbol of a pseudodifferential operator $P$ of order $m$ is a positively hom …

Let me first begin with an elementary example, taking $X=[0,1]^2$ in $\mathbb R^2$. It is then easy to see directly that the wave-front-set of $\mathbb 1_X$ is everywhere the conormal bundle except at …

The Propagation-of-Singularities Theorem is telling you that for a real-principal type operator $P$, and a given equation $Pu=f$, $$ WF(u)\backslash WF(f)\quad \text{is invariant by the flow of $H_p$} …

If $S(t)$ is such that $S(0)=Id$ and $$ \dot S=iA S, $$ the operator $S$ is a Fourier integral operator which quantizes the canonical transformation $\chi$ given by the (non-autonomous) flow of $H_a$, …

You may be able to read the Sato-Kawai-Kashiwara lecture notes if your algebraic geometry background is sufficient for this non-trivial task. On the other hand, the book by J. Sjöstrand "Singularité …

Yes: look at Theorem 8.1.4 in the first volume of Hörmander's ALPDO (Springer Grund. 256). For the classical (conic) wave-front-set, given any closed conic set $S$, you can construct a distribution $u …

The problem is in fact a division problem: let $p(\xi)$ be a polynomial in $n$ variables. There exists a tempered distribution $T$ such that $$ p(\xi) T(\xi)=T(\xi) p(\xi)=1. $$ This is a result due t …

$\bf\text{Claim:}$ The wave-front-set of $\displaystyle T(x)=\frac{1}{f(x)+i0}$ is $$\bigl(Z_+\times (0,+\infty)\bigr)\cup \bigl(Z_-\times (-\infty,0)\bigr), \tag{1}$$ where $ Z_\pm=\{x\in X, f(x)=0 …

A classical result due to Peetre (Math. Scand. 8, 1960) says that if for all $u$, $$\text{supp}\ Au\subset \text{supp}u,$$ then $A$ is a differential operator. On the other hand if a Fourier multiplie …

When $A_K$ is a (pseudo)differential operator with analytic coefficients, a good answer to your question is given by Theorem 9.5.1 on page 353 of the first volume of Hörmander's ALPDO (Springer Grund. …

Too long for a comment. For $u$ in $C^s$, $s\in (0,1)$, you can indeed define $u^2$ and then the distribution-derivative of $u^2$, which belongs to $B^{s-1}_{\infty,\infty}$. Now that does not define …

You need some ellipticity condition to start off. Let us assume as you do that $$ c=c(x,\xi, h), \quad \vert\partial_x^\alpha\partial_{\xi}^\beta c\vert\le C_{\alpha\beta} h^{\vert\beta\vert}, \quad\ …

You need to check Condition (ii) in Definition 8.2.2 in the first volume of Hörmander's ALPDO. Let us note $f(x+i0)$ the limit-distribution of your question and let $\Gamma$ be its wave-front-set. Let …