Let $M$ be a compact manifold without boudary and $X_{1},\ldots,X_{m}$ be the real smooth vector fields on $M$. Consider the following weighted Sobolev space: $$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(M)， 1\leq j\leq m\}.$$ We can prove that $W_{X}^{1}(M)$ is a Hilbert space. My question is: Can we claim that $C^{\infty}(M)$ dense in $W_{X}^{1}(M)$?

I found some results about above question. For a bounded domain $\Omega$ in $\mathbb{R}^n$, the Meyers-Serrin theorems for function spaces associated with a family of vector fields were studided by N. Garofalo and D.M. Nhieu in [1], which shows that the space $$\overline{C^{\infty}(\Omega)\cap W_{X}^{1}(\Omega)}^{\|\cdot\|_{W_{X}^{1}}}=W_{X}^{1}(\Omega).$$ Does this result also hold for compact manifold without boudary? Thank you very much!

[1] Garofalo, Nicola; Nhieu, Duy-Minh, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74, 67-97 (1998). ZBL0906.46026.

Let $M$ be a compact manifold without boudary and let $X_{1},\ldots,X_{m}$ be smooth vector fields on $M$. Consider the following weighted Sobolev space: $$ W_{X}^{1}(M)=\{f\in L^{2}(M)|X_{j}f\in L^2(M)， 1\leq j\leq m\}.$$ We can prove that $W_{X}^{1}(M)$ is a Hilbert space. My question is: Can we claim that $C^{\infty}(M)$ dense in $W_{X}^{1}(M)$?

I found some results about the above question. For a bounded domain $\Omega$ in $\mathbb{R}^n$, the Meyers-Serrin theorems for function spaces associated with a family of vector fields were studided by N. Garofalo and D.M. Nhieu in [1], which shows that the space $$\overline{C^{\infty}(\Omega)\cap W_{X}^{1}(\Omega)}^{\|\cdot\|_{W_{X}^{1}}}=W_{X}^{1}(\Omega).$$ Does this result also hold for compact manifolds without boudary? Thank you very much!

[1] Garofalo, Nicola; Nhieu, Duy-Minh, Lipschitz continuity, global smooth approximations and extension theorems for Sobolev functions in Carnot-Carathéodory spaces, J. Anal. Math. 74, 67-97 (1998). ZBL0906.46026.