Exercise 24, § 3

Let $n\in \mathbf{N}$ and let U be an open subset of $\mathbf{R}^n$. Let $f\in \mathscr{D}'(U)$ be a distribution whose support C is compact in U.

a) There exists an integer $m\in \mathbf{N}$ such that the distribution $f$ extends to a continuous linear form on the space $\mathscr{C}^m(U)$ endowed with the topology of compact convergence of the functions and their partial derivatives of order $\leqslant m$.

The smallest integer $m$ satisfying this property is called the order of $f$.

b) Let $\varphi \in \mathscr{D}(U)$ be such that $\partial^{\alpha}\varphi$ is zero on C for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant m$. Then $\langle f, \varphi \rangle = 0$. (For $\delta >0$ sufficiently small, consider the neighbourhood $U_{\delta}$ of C formed by the $x\in U$ such that $d(x,C)\leqslant \delta$, and show that there exists $\eta \geqslant 0$, tending to 0 as $\delta$ tends to 0, such that, for a certain $C_0\geqslant 0$, one has $|\partial^{\alpha}\varphi (x)|\leqslant C_0\delta^{m-|\alpha|}\eta$ for $x\in U_{\delta}$ and $|\alpha |\leqslant m$; show that there exists a function $\beta_{\delta}\in \mathscr{D}(U)$ equal to 1 on $U_{\delta /4}$ and with support contained in $U_{\delta}$ satisfying $|\partial^{\alpha}\beta_{\delta}|\leqslant C_1\delta^{-|\alpha|}$ for a certain $C_1\geqslant 0$; finally observe that $\langle f, \varphi \rangle =\langle f, \varphi \beta_{\delta}\rangle$ and that $\varphi \beta_{\delta}$ tends to 0 in $\mathscr{C}^m(U).$)

c) Let $x\in U$. The space of distributions $f$ whose support is reduced to $x$ admits as an algebraic base the distributions $\partial^{\alpha}\varepsilon_x$, where $\alpha \in \mathbf{N}^n$.

d) Let $n= 1$. The formula

$\langle f, \varphi \rangle =$ lim$_{m\rightarrow+\infty}(\sum_{1\leqslant k\leqslant m}\varphi (k^{-1})-m\varphi (0)-$ log($m$)$\varphi '(0))$

defines a distribution on $\mathbf{R}$, of order 1, whose support is compact, equal to the set $C =\{0\} \cup \{1/(n+ 1)|n\in \mathbf{N}\}$. There exist sequences $(\varphi_n)$ in $\mathscr{D}(\mathbf{R})$ such that $\partial^{\alpha}\varphi_n$ converges uniformly to 0 on C for every $\alpha \in \mathbf{N}^n$ such that $|\alpha |\leqslant 1$, but $\langle f, \varphi_n\rangle \rightarrow +\infty$.