  In this paper, we present the geometric Hardy inequality for the
sub-Laplacian in the half-spaces on the stratified groups. As a consequence, we
obtain the following geometric Hardy inequality in a half-space on the
Heisenberg group with a sharp constant
  \begin{equation*}
  \int_{\mathbb{H}^+} |\nabla_{H}u|^p d\xi \geq \left(\frac{p-1}{p}\right)^p
\int_{\mathbb{H}^+} \frac{\mathcal{W}(\xi)^p}{dist(\xi,\partial
\mathbb{H}^+)^p} |u|^p d\xi, \,\, p>1,
  \end{equation*}
  which solves the conjecture in the paper \cite{Larson}. Also, we obtain a
version of the Hardy-Sobolev inequality in a half-space on the Heisenberg group
  \begin{equation*}
  \left(\int_{\mathbb{H}^+} |\nabla_{H} u|^p d\xi -
\left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+}
\frac{\mathcal{W}(\xi)^p}{dist(\xi,\partial \mathbb{H}^+)^p} |u|^p d\xi
\right)^{\frac{1}{p}} \geq C \left(\int_{\mathbb{H}^+} |u|^{p^*}
d\xi\right)^{\frac{1}{p^*}},
  \end{equation*}
  where $dist(\xi,\partial \mathbb{H}^+)$ is the Euclidean distance to the
boundary, $p^* := Qp/(Q-p)$, $2\leq p<Q$, and
  $$\mathcal{W}(\xi)=\left(\sum_{i=1}^{n}\langle X_i(\xi), \nu
\rangle^2+\langle Y_i(\xi), \nu \rangle^2\right)^{\frac{1}{2}},$$
  is the angle function. For $p=2$, this gives the Hardy-Sobolev-Maz'ya
inequality on the Heisenberg group.
