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$\DeclareMathOperator{supp}{supp} \DeclareMathOperator{sing}{sing}$Let $\Omega$ be a domain with compact closure in $\mathbb R^n$. Consider a linear operator $A \colon X \to X$ satisfying one of the following conditions:

  1. $X = C^\infty_c(\mathbb R^n)$, $\forall u \in X \; \supp Au \subseteq\supp u$ (locality),
  2. $X = \mathscr E'(\mathbb R^n)$, $\forall u \in X \; \sing\supp Au \subseteq \sing\supp u$ (pseudo-locality),
  3. $X = \mathscr E'(\mathbb R^n)$, $\forall u \in X \; WF(Au) \subseteq WF(u)$ (micro-locality).

The famous Peetre's theorem, 1959, states that if 1. holds then $A|_{\Omega}$ is a linear differential operator with smooth coefficients. The converse is clearly true.

Now if $A$ is a proper pseudodifferential operator then 2. and 3. hold. But is the converse true? Is it true that if 2. or 3. holds then $A|_{\Omega}$ is a pseudodifferential operator?

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basically no.

3 will hold if the Schwartz kernel of A has WF set contained in the conormal bundle of the diagonal. This is a much weaker condition than being a pseudo-differential operator.

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