Exercise 23, § 3

Let $n\in \mathbf{N}$ and let U be an open subset of $\mathbf{R}^n$. We denote by $\mu$ the Lebesgue measure on U. For $m\in \mathbf{N}$, we denote by $\boldsymbol{m}= (m, . . . , m)\in \mathbf{N}^n$.

a) Let $f\in \mathscr{D}(U)$. Suppose that there exist an integer $m\geqslant 1$ and a real number $c\geqslant 0$ such that

for every $\varphi \in \mathscr{D}(U)$. Show that there then exists a function $g\in L^{\infty}(U)$ such that $f=\partial^{\boldsymbol{m}}g$. (The linear mapping $\varphi \mapsto \partial^{\boldsymbol{m}}\varphi$ is injective on $\mathscr{D}(U)$; let E be its image; consider the linear form on E into $\mathbf{C}$ which associates with $\partial^{\boldsymbol{m}}\varphi$ the value $\langle f, \varphi \rangle$ and apply the Hahn–Banach theorem.)

b) Let V be a relatively compact open subset contained in U. Show that there exist an integer $m\geqslant 1$ and $c\geqslant 0$ such that

for every $\varphi \in \mathscr{D}(U)$ whose support is contained in V.

c) For every relatively compact open subset $V\subset U$, show that there exist a continuous function $g\in \mathscr{C}(V)$ and $\alpha \in \mathbf{N}^n$ such that the restriction of $f$ to V is equal to $\partial^{\alpha}g$. d) Give an example of a distribution $f\in \mathscr{D}'(\mathbf{R}^n)$ such that $f$ is not of the form $f=\partial^{\alpha}g$ for a continuous function $g\in \mathscr{C}(\mathbf{R}^n)$ and $\alpha \in \mathbf{N}^n$.