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The Sobolev space $H^s({\mathbb{R}}^n)$ consists of $u\in {\mathcal{S}}^{\prime }({\mathbb{R}}^n)$ for whichThe Local Sobolev space $H_{\mathrm{loc}}^s(X)$ consists of distributions $u$ such that $\chi u\in H^s({\mathbb{R}}^n)$ for every $\chi \in C_c^{\infty }(X)$.

If $P$ has degree $m$ and principal homogeneous part $P_m$, then $P(D)$ is an elliptic differential operator whenEquivalently, $|P(\xi )|\ge c|\xi {|}^m$ for all sufficiently large real $\xi$.