Is $C^{\infty}(M)$ dense in weighted Sobolev space $W_{X}^{1}(M)$?

Question

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.

Answer 1

The density result is true for any family of vector fields with Lipschitz coefficients.

Theorem. Let $X_1,\ldots,X_k$ be a system of vector fields with Lipschitz coefficient on a compact Riemannian manifold (with or without boundary), If $f\in L^p(M)$ and $X_j\in L^p(M)$, $j=1,2,\ldots,k$, then there is a sequence of smooth functions $f_i\in C^\infty(M)$ such that $$ \Vert f-f_i\Vert_{L^p}+\sum_{j=1}^k \Vert X_j f- X_j f_i\Vert_{L^p}\to 0 $$ as $i\to\infty$.

This result is due to Friedrichs. For a short proof as well as relevant references, see Theorem 11.9 in

P. Hajlasz, P. Koskela, Sobolev met Poincare, Memoirs Amer. Math. Soc. 688 (2000).

In that paper the result is proved on domains in $\mathbb{R}^n$, but as it is indicated at the beginning of the proof, partition of unity allows you to assume that the function has compact support and with that argument you can also prove the result on compact manifolds. The proof is tricky, but short (half page long) and rather elementary.