Let $X=(X_{1},X_{2},\cdots,X_{m})$ be $C^{\infty}$ real vector fields defined in $\mathbb{R}^{n} (n>2)$ satisfying the Hormander's condition $$ \text{rankLie}[X_{1},\cdots,X_{m}]=n $$ at every point $x\in\mathbb{R}^{n}$. For any open set $W\subset \mathbb{R}^{n}$, Define $$ H_{X}^{1}(W)=\{u\in L^{2}(W)|X_{j}u\in L^{2}(W), j=1,\cdots,m\}.$$ $H_{X}^{1}(W)$ is a Hilbert space with norm $\|u\|^2_{H^{1}_{X}(W)}=\|u\|_{L^2(W)}^2+\|Xu\|_{L^2(W)}^2$ , $\|Xu\|_{L^2(W)}^2=\sum_{j=1}^{m}\|X_{j}u\|_{L^2}^2$. Next, for any open connected set $\Omega\subset W$, we define $H_{X,0}^{1}(\Omega)$ to be the closure of $C_{0}^{\infty}(\Omega)$ in $H_{X}^{1}(W)$. If $\partial\Omega$ is $C^{\infty}$ and non characteristic for $X$, then $H_{X,0}^{1}(\Omega)$ is well-defined, and also a Hilbert space. If $u\in C^{\infty}(\Omega), u|_{\partial\Omega}=0$, I wonder will $u\in H_{X,0}^{1}(\Omega)$ or not? I know $u\in H_{X}^{1}(\Omega)$.
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