Exercise 25, § 3

Let $n\in \mathbf{N}$. Let N denote the euclidean norm on $\mathbf{R}^n$ and $\mu$ Lebesgue measure.

Let $\mathscr{D}_{L^1}(\mathbf{R}^n)$ denote the vector subspace of $\mathscr{C}^{\infty}(\mathbf{R}^n)$ whose elements are the functions $\varphi$ such that $\partial^{\alpha}\varphi \in L^1(\mathbf{R}^n)$ and $\partial^{\alpha}\varphi \in \mathscr{C}_0(\mathbf{R}^n)$ for all $\alpha \in \mathbf{N}^n$. a) The space $\mathscr{D}_{L^1}(\mathbf{R}^n)$, endowed with the topology defined by the seminorms

is a Fréchet space.

b) The space $\mathscr{D}(\mathbf{R}^n)$ is contained in and dense in $\mathscr{D}_{L^1}(\mathbf{R}^n)$. One can identify the dual space $\mathscr{D}'_{L_1}(\mathbf{R}^n)$ of $\mathscr{D}_{L^1}(\mathbf{R}^n)$ with a subspace of $\mathscr{D}'(\mathbf{R}^n)$.

c) Let $f\in \mathscr{D}'(\mathbf{R}^n)$. Then $f\in \mathscr{D}'_{L_1}(\mathbf{R}^n)$ if and only if there exist a finite set I, a family $(\alpha_i)_{i\in I}$ in $\mathbf{N}^n$ and a family $(g_i)_{i\in I}$ in $L^{\infty}(\mathbf{R}^n)$ such that

(Determine I and the family $(\alpha_i)$ so that $|\langle f, \varphi \rangle |$ is bounded when $\varphi$ is such that $q_{\alpha_i}(\varphi )\leqslant 1$ for all $i\in I$; identify $\mathscr{D}_{L^1}(\mathbf{R}^n)$ with a vector subspace of $L^1(\mathbf{R}^n)^m$ by the mapping $\varphi \mapsto (\partial^{\alpha_i}\varphi )_{i\in I}$, and apply theorem of Hahn–Banach.)

d) Let $f\in \mathscr{D}'(\mathbf{R}^n)$. Then $f\in \mathscr{D}'_{L_1}(\mathbf{R}^n)$ if and only if there exist a continuous function $g$ of moderate growth on $\mathbf{R}^n$ and $\alpha \in \mathbf{N}^n$ such that $f=\partial^{\alpha}g$. e) Let $f\in \mathscr{D}'(\mathbf{R}^n)$. There exists an integer $m\in \mathbf{N}$ such that $(1 + N^2)^{-m/2}f$ belong to $\mathscr{D}'_{L_1}(\mathbf{R}^n)$.

f) Let $f\in \mathscr{D}'(\mathbf{R}^n)$. Then $f\in \mathscr{S}'(\mathbf{R}^n)$ if and only if there exist a bounded continuous function $g:\mathbf{R}^n\rightarrow \mathbf{C}$, an integer $m\in \mathbf{N}$ and $\alpha \in \mathbf{N}^n$ such that $f=\partial^{\alpha}((1 + N^2)^{m/2}g)$.