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Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and define \begin{eqnarray*} \mathcal{S}^m(\Omega ) &=&\left\{ \begin{array}{c} a\in C^\infty(\Omega \times \mathbb{R}^{d}): \hbox{for all compact set }K\subset \Omega \hbox{ and }\alpha, \beta \in \mathbb{N}^{d}, \\ \hbox{there exists }C_{K,\alpha,\beta} > 0 \hbox{ such that }\underset{x \in K}{\sup}| D_{x}^\beta D_\xi^{\alpha} a(x, \xi)| \leq C(1+|\xi|)^{m-|\alpha|} \end{array}% \right\} \end{eqnarray*}

Definition 1. An operator $P:C_0^\infty(\Omega) \rightarrow C^{\infty}(\Omega)$ is said properly supported, if its distributional kernel $K_P$ has proper support, that is, $\pi_x^{-1}(A)\cap K_P$ and $\pi_y^{-1}(A)\cap K_P$ are compact subsets of $\Omega \times \Omega$ for all compact $A \subset \Omega$.

Classic results in the literature show that if $P \in \Psi^m(\Omega)$ is properly supported then $P$ maps continuously $C^\infty(\Omega)$ into $C^\infty(\Omega)$, $\mathcal{E}'(\Omega)$ into $\mathcal{E}'(\Omega)$, $\mathcal{D}'(\Omega)$ into $\mathcal{D}'(\Omega)$ and $H^{s}_{loc}(\Omega)$ into $H^{s-m}_{loc}(\Omega)$.

I would like to know if there is an extension of $P$ from $H^{s}(\Omega)$ to $H^{s-m}(\Omega)$.

PS: Note that I am not asking for uniform limitation on the symbol.

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No. Take $\Omega=\mathbb R^d$; with your definition of $\mathcal S^m$, you have no constraint at infinity in the $x$ variable. For instance in one dimension, the multiplication by $x$ is not bounded on $L^2(\mathbb R)$ although $x$ is a symbol in your class $\mathcal S^0$.

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