The characterization of Sobolev space

Question

If $\Omega$ is a bounded open set in $\mathbb R^n$ and $u$ is a distribution with supp$u \subset \subset \Omega$. For any $s \in \mathbb R$, if ${(I - \Delta )^{\frac{s}{2}}}u \in L_{loc}^2(\Omega )$, can we conclude that $u \in H^s(\Omega)$? (We know that since ${(I - \Delta )^{\frac{s}{2}}}$ is a pseudodifferential operator of order $s$, the converse holds.)

Answer 1

I think you should be able to go the other way too. You can conclude, I think, that this definition does not depend on the specific elliptic operator you use, as long you are using an elliptic pseudo of order $s$, maybe properly supported. If I am correct, Mikhail Shubin's book on Pseudodifferential operators and Spectral Theory proves that in this case you will be able to make the conclusion that $u \in H^s_{loc}(\Omega)$. You only have to prove that $H^s_{loc}(\Omega) = H^s(\Omega)$. Please look up Shubin's book (Chapter $1$, I presume).