I want to understand distributions that are bounded linear functionals on smooth functions whose all derivatives vanish at infinity. Before asking questions, let me first define the terms used in this post.
A continuous function $\phi$ on $\mathbb{R}^n$ is said to vanish at infinity if $\phi(x) \to 0$ as $|x| \to \infty$.
Denote $C_0$ the space of all continuous functions on $\mathbb{R}^n$ that vanish at infinity. This is a Banach space with the uniform norm.
Denote $C_0^\infty$ the space of all smooth functions whose all derivatives vanish at infinity, i.e. $$ C_0^\infty = \{ \phi \in C^\infty : \forall \alpha, \partial^\alpha \phi \in C_0 \}. $$ This is a Frechet space with the semi-norms $$ \|\phi\|_m = \sup\{ |\partial^\alpha \phi(x)| : |\alpha| \leq m, x \in \mathbb{R}^n \}. $$
Clearly, Schwartz functions form a dense subspace in $C_0^\infty$, and the injection $ \mathcal{S} \to C_0^\infty $ is continuous. Thus, we have the continuous injection $ (C_0^\infty)' \to \mathcal{S}' $, i.e. any continuous linear functional on $C_0^\infty$ is a tempered distribution. Moreover, a linear functional $u$ on $C_0^\infty$ is continuous if and only if there exists $m \in \mathbb{N}, C > 0$ such that $$ \forall \phi \in C_0^\infty, |u(\phi)| \leq C \|\phi\|_m. $$
Thus, any distribution that belongs to $(C_0^\infty)'$ has a finite order.
Question 1: Is there any special name for distributions that belong to $(C_0^\infty)'$?
The next question is about the extension of distributions in $(C_0^\infty)'$ to smooth functions with bounded derivatives.
- Denote $C_b$ the space of all bounded continuous functions on $\mathbb{R}^n$.
- Denote $C_b^\infty$ the space of all smooth functions with bounded derivatives, i.e. $$ C_b^\infty = \{ \phi \in C^\infty : \forall \alpha, \partial^\alpha \phi \in C_b \}. $$
Clearly, we have the embeddings $C_0 \subset C_b$ and $C_0^\infty \subset C_b^\infty$. Note that $C_0, C_0^\infty$ are not dense in $C_b, C_b^\infty$ because they are closed subspaces.
Now, by Riesz-Markov representation theorem (and also by the fact that $\mathbb{R}^n$ is a locally compact Polish space), the dual space $(C_0)'$ is isomorphic to the space of all finite Borel measures on $\mathbb{R}^n$. Here is the thing: any measure $\mu \in (C_0)'$ actually can be extended to $C_b$ by
$$ \phi \in C_b \mapsto \int \phi d\mu $$
and the extension is a bounded linear functional on $C_b$. Thus, we have the embedding $(C_0)' \subset (C_b)'$. By analogy, I expect any distribution in $(C_0^\infty)'$ can be canonically extended to a continuous linear functional on $C_b^\infty$.
Question 2: Can any distribution in $(C_0^\infty)'$ be canonically extended to a continuous linear functional on $C_b^\infty$?
Edit: I expect $(C_0^\infty)'$ to be canonically identified with a closed subspace of $(C_b^\infty)'$. Moreover, the extension should respect some distributional identities. For example, the Fourier transform of $u \in (C_0^\infty)'$ should be given by a smooth function $ \hat{u}(\xi) = \langle u, e^{ix \xi} \rangle $.