This question is motivated by Hille-Yosida semigroup/Markov theory.
Let $P_t:C_0(\mathbb{R}^d) \to C_0(\mathbb{R}^d), t \geq 0$ be a strongly continuous contraction semigroup. Say a function $f \in C_0$ is in $\text{Dom}(L)$ if there exists a $g \in C_0$ such that $\lim \limits_{t \downarrow 0} \lVert \frac{P_t f - f}{t} - g \rVert = 0$ and then $Lf := g$.
Now assume that on the dense subset $C_0^{\infty} \subset \text{Dom}(L) \subset C_0$, $L$ is a second order elliptic differential operator, $Lf(x) = \frac{1}{2} \sum_{i,j} a_{ij}(x) \partial_i \partial_j f(x) + \sum_i b_i(x) \partial_i f(x)$.
A subset $D \subset \text{Dom}(L)$ is called a core if the closure of $L$ restricted to $D$ in the graph norm equals $L$ with its maximal domain.
Question: Is $C_0^{\infty}$ also a core for $L$?
Ideas so far:
- Since the subset is already dense, we could prove that the restriction is closable. For that, we can use the usual lemma and show: If $\text{Dom}(L) \ni f_n \to 0$ and $Lf_n \to g$, then $g=0$. (I only found that for $L^2$ instead of $C_0$, in the appendix.) This probably should work for any linear differential operator $\sum_{|\alpha| \leq N} \lambda_{\alpha} \partial^{\alpha}$.
- A subset $D$ is a core if $D \subset \text{Dom}(L)$ is dense and $P_t : D \to D$ (or even $D_0 \to D$ for $D_0 \subset D$ dense). Here, we could use regularity theory for semigroups. I remember hearing that semigroups always have a smoothing effect. It holds $P_t f(x) = f(x+th)$, so $P_t : C_0^{\infty} \to C_0^{\infty}$ for $L=h \cdot \nabla$, and even something like $P_t : L^1 \to C^{\infty}$ for the heat semigroup for $L= \Delta$, but only due to explicitly knowing its smooth kernel and using the convolution. There must be something similar for general semigroups. Possibly it can even be constructed from the aforementioned cases.
"Appendix": Linear differential operators in $L^2$ are closable. Let $A=\sum_{|\alpha| \leq N} \lambda_{\alpha} \partial^{\alpha}: C_c^{\infty} \subset L^2 \to L^2$. Let $\phi_n \to 0$ and $A \phi_n \to \psi \in L^2$. Let $\xi \in C_c^{\infty}(\mathbb{R}^d)$. Then
$$ \langle \xi, \psi \rangle = \lim \limits_{n \to \infty} \langle \xi, A \phi_n \rangle = \lim \limits_{n \to \infty} \langle A^{*} \xi, \phi_n \rangle = 0$$
where $A^{*} = \sum_{|\alpha| \leq N} (-1)^{\sum_{i=1}^d \alpha_i} \lambda_{\alpha} \partial^{\alpha}$ is the formal adjoint by partial integration. Since $\xi\in C_c^{\infty}$ was arbitrary from a dense set, $A \phi_n \to 0$. Thus, $A$ is closable.