= Solution
The <Sobolev space> $H^s(\mathbb R^n)$ consists of $u\in\mathcal S'(\mathbb R^n)$ for which
$$
\int_{\mathbb R^n}\langle\xi\rangle^{2s}
|\widehat u(\xi)|^2\,d\xi<\infty,
\qquad
\langle\xi\rangle=(1+|\xi|^2)^{1/2}.
$$
The <Local Sobolev space> $H^s_{\rm loc}(X)$ 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> when
$$
P_m(\xi)\ne0\qquad(\xi\in\mathbb R^n\setminus\{0\}).
$$
Equivalently, $|P(\xi)|\geq c|\xi|^m$ for all sufficiently large real $\xi$.
Back to article page
