# Extension of pseudodifferential operators to Sobolev spaces

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.

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 $$\mathcal S^0$$.