Definition 5

Let E be a vector subspace of $\mathscr{D}'(U)$ containing $\mathscr{D}(U)$. A differential operator associated with D on E is any partial operator on E which is an extension of the partial operator with domain $\mathscr{D}(U)$ defined by $\varphi \mapsto D(\varphi )$.

Suppose for example that the coefficients $n_{\alpha}$ are bounded functions on U. Let $\mu$ be the Lebesgue measure on U. If $p$ is an element of $[1,+\infty [$, one can then define on $L^p(U)$ a differential operator associated with D whose domain is the Sobolev space $W^{k,p}(U)$ (No. 14 of IV, p. 221), since in this case one has $n_{\alpha}\partial^{\alpha}(f)\in L^p(U)$ for every $f\in W^{k,p}(U)$ and every $|\alpha |\leqslant k$.