I've been working on the following Problem from Friedlander's introduction to the theory of distributions:
Show that if $u$ is a distribution of order $N$ and with compact support on $\mathbb{R}^{n}$, and if $\psi \in C^{N}(\mathbb{R}^{n})$, then $u* \psi$ is of the form $u* \psi = \langle u(y), \psi(x-y) \rangle$ and that $u* \psi$ is a continuous function on $\mathbb{R}^{n}$.
- Approach one:
I have tried to use the fact that $C^{\infty}_{c}(\mathbb{R}^{n})$ is dense in the space of distributions. To do so, I chose $\{u_{k}\} \subset C^{\infty}(\mathbb{R}^{n})$ such that $\text{supp}(u_{k}) \subset \text{supp}(u) + B(0,1)$ and $u_{k} \to u$ in the the space of distributions. Then, I can define $g_{k}(x) = \int_{\mathbb{R}^{n}} u_{k}(y) \psi(x-y) d y = u_{k} * \psi(x)$. I want to say that $\{g_{k}\}_{k \in \mathbb{N}}$ is uniformly equicontinuous, and therefore by Arzela-Ascoli, a subsequence converges locally uniformly and is consequently continuous. Unfortunately, if $u = \delta_{0}$, then my functions $u_{k}$ are not locally bounded so I have no hope of applying Arzela-Ascoli.
- Approach two:
Use a standard mollifiers argument, and define $\langle u* \psi, \phi \rangle = \langle u(x), \langle \psi(y), \phi(x+y) \rangle \rangle \quad \forall \phi \in C^{\infty}_{c}(\mathbb{R}^{n}).$
We note that $\langle \psi(y), \phi(x+y) \rangle \in C^{\infty}$, so this is well-defined. We can then use mollifiers $\varphi_{\epsilon}^{x}$ supported in a ball of radius $\epsilon$ about $x$ that integrate to $1$ and approximate the $\delta$ function as $\epsilon \to 0$, and define $$ g(x) = \langle u* \psi, \varphi_{\epsilon}^{x} \rangle. $$
Again, I don't know how to show that this $g$ is continuous, since using the fact that $u$ is of finite order, I can get a bound on $|g(x) - g(y)|$, but it depends on the supremum over the support of $u$ all derivatives of $\psi$ of order less than or equal to $N$ multiplied by $\varphi_{\epsilon}^{x}$, which grows like $\epsilon^{-n}$ as $\epsilon \to 0$, and hence is a problem.
I'm open to any ideas of how to conclude using either of my approaches as well as any new methods.
Cheers.