Core of the generator of squared bessel process in $L^2(\mathbb{R}_+)$

Question

Consider the squared bessel process with generator $$Gf(x)=xf''(x)+f'(x), \ \ x\in\mathbb{R}_+.$$ It is known that the Lebesgue measure is an invariant measure for this process and thus, can be extended to a semigroup $(Q_t)_{t\ge 0}$ on $L^2(\mathbb{R}_+)$. Also, the resulting semi-group is self-adjoint with essentially self-adjoint generator $G$. It is known that the domain of $G$ in $L^2(\mathbb{R}_+)$ is given by: $$\mathcal{D}(G)=\{f\in L^2(\mathbb{R})\mid f,f'\in\text{ABS}((0,\infty)), \ Gf\in L^2(\mathbb{R}_+), \ \lim_{x\downarrow 0} f'(x)=0\}.$$ Is it known if $C^\infty_c((0,\infty))$ is a core of $G$ in $L^2(\mathbb{R}_+)$? I think this should be true but cannot find a reference.

Edit: From Proposition 3.2.1 in the book "Analysis and Geometry of Markov diffusion operators" (Bakry, Gentil, Ledoux), it is enough to prove that $G$ is hypo-elliptic.

Answer 1

It seems that the answer is no. If $h=Gf$, then $f'(x)=(1/x) \int_0^x h(t)dt$ and then $\|f'\|_2 \le C\|h\|_2$, by Hardy's inequality. This means that the graph norm is stronger than the $H^1$ norm and then a function in the domain of the operator which does not vanish for $x=0$ cannot be approximated by a sequence of compactly supported functions.