Exercise 3, § 3

Let $n\in \mathbf{N}$. Let $\mu_n$ denote Lebesgue measure on $\mathbf{R}^n$.

a) Let $\mathscr{H}_1$ be the image of the mapping $\partial_1:\mathscr{D}(\mathbf{R}^n)\rightarrow \mathscr{D}(\mathbf{R}^n)$; it is the space of $\varphi \in \mathscr{D}(\mathbf{R}^n)$ such that

for every $y\in \mathbf{R}^{n-1}$.

b) Let $\varphi_0\in \mathscr{D}(\mathbf{R})$ whose integral over $\mathbf{R}$ is equal to 1. For every $\varphi \in \mathscr{D}(\mathbf{R}^n)$, there exists $\varphi_1\in \mathscr{H}_1$ such that

for every $(x, y)\in \mathbf{R}\times \mathbf{R}^{n-1}$.

c) For every distribution $f\in \mathscr{D}'(\mathbf{R}^n)$, there exists $g\in \mathscr{D}'(\mathbf{R}^n)$ such that $\partial_1g=f$. If $g_1$ and $g_2$ in $\mathscr{D}'(\mathbf{R}^n)$ satisfy $\partial_1g_1=\partial_1g_2=f$, then there exists a distribution $h\in \mathscr{D}'(\mathbf{R}^{n-1})$ such that

for every $\varphi \in \mathscr{D}(\mathbf{R}^n)$, where $\psi \in \mathscr{D}(\mathbf{R}^{n-1})$ is defined by

d) Let U be a connected open set of $\mathbf{R}^n$. Let $f\in \mathscr{D}'(U)$ be a distribution such that $\partial_if= 0$ for $1\leqslant i\leqslant n$. The distribution $f$ is the distribution associated with a constant function.