This question is connected with my previous: Heisenberg group: function without vertical derivative. Here I am trying to look from another side: what is a difference between Sobolev space and horizontal Sobolev space on Carnot group?

Outline some notations.

A **Carnot group** $\mathbb G$ is a connected simply-connected nilpotent Lie group hose Lie algebra $\mathfrak g$ is graded: $\mathfrak{g} = V_1\oplus\cdots\oplus V_m$, и
$[V_1,V_j]=V_{j+1}$ provided $j=1,\ldots, m-1$, and $[V_1,V_m] = \{0\}$.

$V_1$ is horizontal subbundle, $X_1, \dots X_n$ - basis of $V_1$ and $X_1, \dots X_n, X_{21}, \dots X_{2n_2},\cdots X_{m1}, \dots X_{mn_m}$ - basis of $\mathfrak g$.

**Derivatives.** A locally summable function $v$ is called the generalized
derivative of $f$ along the vector field X, whenever
$$
\int v\psi\, dx = -\int f X\psi\, dx,
$$
for every $\psi\in C^{\infty}_0$. We denote $v = Xf$.

**Sobolev spaces.**

The horizontal Sobolev space $HW^1_p$ consists of all locally summable functions of finite norm $$ \|f \mid HW^1_p\| = \left(\int |f|^p\, dx \right)^{\frac1p} + \left(\int |\nabla_{\mathcal L}f|^p\, dx \right)^{\frac1p}, $$ where $\nabla_{\mathcal L}f = (X_1f, \dots X_nf)$ is the generalized subgradient of $f$ at x ∈ D which uses only the derivatives along the horizontal fields.

The Sobolev space $W^1_p$ consists of all locally summable functions of finite norm $$ \|f \mid W^1_p\| = \left(\int |f|^p\, dx \right)^{\frac1p} + \left(\int |\nabla f|^p\, dx \right)^{\frac1p}, $$ where $\nabla_{\mathcal L}f = (X_1f, \dots X_nf, X_{21}f, \dots X_{2n_2}f,\cdots X_{m1}f, \dots X_{mn_m}f)$.

It is clear that if $f\in W^1_p$ then $f\in HW^1_p$. The question is following: if there is a function $f\in HW^1_p$ such that $f\not\in W^1_p$?