Let $\Omega$ be an open bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial\Omega$. Suppose that $X=(X_{1},X_{2},\ldots,X_{m})$ are smooth vector fields defined on $\mathbb{R}^{n}$ and satisfy the H"ormander's condition: $X_{1},X_{2},\ldots,X_{m}$ together with their commutators up to a certain fixed length span the tangent space at each point of $\mathbb{R}^{n}$.
Consider the following Sobolev spaces $$ H_{X}^{1}(\Omega):=\{u\in L^{2}(\Omega)~|~X_{j}u\in L^{2}(\Omega),~ j=1,\ldots,m\}. $$ We can deduce that $H_{X}^{1}(\Omega)$ is a Hilbert space endowed with the norm $\|u\|^2_{H^{1}_{X}(\Omega)}=\|u\|_{L^2(\Omega)}^2+\sum_{j=1}^{m}\|X_{j}u\|_{L^2(\Omega)}^2$. Denoted by $H_{X,0}^{1}(\Omega)$ the closure of $C_{0}^{\infty}(\Omega)$ in $H_{X}^{1}(\Omega)$, which is also a Hilbert space.
Assume that $u\in H_{X}^{1}(\Omega)\cap L^{\infty}(\Omega)$ with $u\in C(\Omega\cup \Gamma_{0})$, where $\Gamma_{0}\subset \partial\Omega$ denote the non-characteristic set in the boundary $\partial\Omega$. My question is that: Does $\left(u-\sup_{x\in \Omega\cup \Gamma_{0}}u(x) \right)_{+}$ belong to the space $H_{X,0}^{1}(\Omega)$? Can we give more discription of the space $H_{X,0}^{1}(\Omega)$ ?
The question arised in studying the characteristic domain for H"ormander vector fields. If we futher assume that the boundary $\partial\Omega$ is non-charactieristic for vector fields $X_{1},X_{2},\ldots,X_{m}$, by Theorem 2 in Derridj, Makhlouf, Sur un théorème de traces. (On a trace theorem), Ann. Inst. Fourier 22, No. 2, 73-83 (1972). ZBL0226.46041. we know the sobolev space $H_{X}^{1}(\Omega)$ admits a continuous trace operator $T:H_{X}^{1}(\Omega)\to L^2(\Omega)$ with $T(u)=u|_{\partial\Omega}$ for $u\in H_{X}^{1}(\Omega)\cap C(\overline\Omega)$. In this case, it seems we can rewrite $ H_{X,0}^{1}(\Omega)=\{u\in H_{X}^{1}(\Omega)| T(u)=0\} $ and $\left(u-\sup_{x\in \Omega\cup \Gamma_{0}}u(x) \right)_{+}$ belong to the space $H_{X,0}^{1}(\Omega)$ since $$T\left(\left(u-\sup_{x\in \Omega\cup \Gamma_{0}}u(x) \right)_{+}\right)=0.$$
For the general smooth boundary $\partial\Omega$, it may contains some characteristic points. Since the vector fields $X_{1},X_{2},\ldots,X_{m}$ are tangent in these characteristic points, we cannot locally straighten the vector fields $X_{1},X_{2},\ldots,X_{m}$ and the boundary $\partial\Omega$ near the characteristic points, and the function $u$ may lose the regularity at the characteristic points. According to Theorem 1 in Derridj, Makhlouf, Sur un théorème de traces. (On a trace theorem), Ann. Inst. Fourier 22, No. 2, 73-83 (1972). ZBL0226.46041., the characteristic set $\Omega\setminus\Gamma_{0}$ has $n-1$ dimensional zero mearsure.
I am also trying to find some sequence in $C_{0}^{\infty}(\Omega)$ that converge to the function $\left(u-\sup_{x\in \Omega\cup \Gamma_{0}}u(x) \right)_{+}$, but it seems difficult for me. Can some one help me with above question? Can someone give some related references in this topic? Thank you very much!
