Exercise 19, § 3

Let $n\in \mathbf{N}$ and let $U\subset \mathbf{R}^n$ be an open subset.

a) Let $f\in \mathscr{D}'(U)$. The union V of the open sets $W\subset U$ such that the restriction of $f$ to W is zero is open in U.

The complement of V is called the support of $f$; it is a closed subset of U.

b) Let $f\in \mathscr{D}'(U)$ and let $\varphi \in \mathscr{D}(U)$ whose support does not meet the support of $f$. Then $\langle f, \varphi \rangle = 0$.

c) Let $f\in \mathscr{D}'(U)$ and $\alpha \in \mathbf{N}^n$. The support of $\partial^{\alpha}f$ is contained in the support of $f$.

d) If $f\in \mathscr{D}'(U)$ is the distribution associated with a measure $\nu$, then the support of $f$ coincides with the support of $\nu$.

e) Let $f$ be a distribution with compact support in U. The linear form $f$ extends to a continuous linear form on the space $\mathscr{C}^{\infty}(U)$.

f) The space of distributions with compact support in U is identified with the dual of the space $\mathscr{C}^{\infty}(U)$.

g) Let $f\in \mathscr{D}'(\mathbf{R}^n)$ be a distribution with compact support. Then $f\in \mathscr{S}'(\mathbf{R}^n)$. h) Let $f\in \mathscr{D}'(\mathbf{R}^n)$ be a distribution with compact support. The Fourier transform of $f$ belongs to $\mathscr{C}^{\infty}(\mathbf{R}^n)$; it is a function of polynomial growth, as are all its derivatives.