Exercise 7, § 3

Let $n\in \mathbf{N}$. We denote by $\widetilde{\boldsymbol{\gamma }}$ the linear representation of $\mathbf{R}^n$ on the space of functions from $\mathbf{R}^n$ into $\mathbf{C}$ defined by

for every $h\in \mathbf{R}^n$ and every function $f:\mathbf{R}^n\rightarrow \mathbf{C}$.

a) For every $h\in \mathbf{R}^n$, the mapping $\widetilde{\boldsymbol{\gamma }}(h)$ defines by passing to the subspaces a locally convex space automorphism of $\mathscr{D}(\mathbf{R}^n)$, which is again denoted by $\widetilde{\boldsymbol{\gamma }}(h)$.

b) We denote by $\boldsymbol{\gamma }$ the transpose automorphism of $\mathscr{D}'(\mathbf{R}^n)$ of $\widetilde{\boldsymbol{\gamma }}(-h)$. If $f\in \mathscr{D}(\mathbf{R}^n)$, then $\boldsymbol{\gamma }(h)f=\widetilde{\boldsymbol{\gamma }}(h)f$ for every $h\in \mathbf{R}^n$.

c) The mapping $h\mapsto \boldsymbol{\gamma }(h)$ is a linear representation of $\mathbf{R}^n$ in $\mathscr{D}'(\mathbf{R}^n)$.

d) For every $h\in \mathbf{R}^n$ and every $\alpha \in \mathbf{N}^n$, one has $\partial^{\alpha}\circ \boldsymbol{\gamma }(h) =\boldsymbol{\gamma }(h)\circ \partial^{\alpha}$.

e) Let $i$ be an integer such that $1\leqslant i\leqslant n$ and let $e_i$ be the $i^e$ vector of the canonical basis of $\mathbf{R}^n$. For every distribution $f\in \mathscr{D}'(\mathbf{R}^n)$, one has

lim$_{hh\rightarrow\not=00}\frac{1}{h}(\boldsymbol{\gamma }(he_i)f-f) =\partial_if$ in $\mathscr{D}'(\mathbf{R}^n)$ endowed with the weak topology.