Exercise 3, § 3

Let M be a paracompact differentiable manifold locally of finite dimension. Let $n$ and $p$ be integers and let D be a morphism of the sheaf $\mathscr{C}^{\infty}(M;\mathbf{R}^n)$ into the sheaf $\mathscr{C}^{\infty}(M;\mathbf{R}^p)$. For every $s\in \mathscr{C}^{\infty}(M;\mathbf{R}^n)$, every integer $r$ and every point $x\in M$, we denote by j$^rs(x)$ the jet of order $r$ of the mapping $s$ with source $x$ and target $s(x)$ (VAR, 12.1.2). We equip the space $\mathbf{R}^p$ with the euclidean norm. a) Let $x\in M$. Exercise prove that there exists a neighbourhood U of $x$ and an integer $r$ such that, for every section $s\in \mathscr{C}^{\infty}(M;\mathbf{R}^n)$ and every $y\in U-\{x\}$ such that j$^rs(y) = 0$, one has $\|Ds(y)\|\leqslant 1$.

b) Let U be a relatively compact open subset of M. Now prove that the restriction of D to U is a differential operator (Peetre’s theorem[^1]).

[^1]: Peetre (J.), « Une caractérisation abstraite des opérateurs différentiels », Math. Scand. 7 (1959), 211-218 and « Rectifications à l’article “Une caractérisation abstraite des opérateurs différentiels” », Math. Scand. 8 (1960), 116-120.