  We prove various Hardy-type and uncertainty inequalities on a stratified Lie
group $G$. In particular, we show that the operators $T_\alpha: f \mapsto
|.|^{-\alpha} L^{-\alpha/2} f$, where $|.|$ is a homogeneous norm, $0 < \alpha
< Q/p$, and $L$ is the sub-Laplacian, are bounded on the Lebesgue space
$L^p(G)$. As consequences, we estimate the norms of these operators
sufficiently precisely to be able to differentiate and prove a logarithmic
uncertainty inequality. We also deduce a general version of the
Heisenberg-Pauli-Weyl inequality, relating the $L^p$ norm of a function $f$ to
the $L^q$ norm of $|.|^\beta f$ and the $L^r$ norm of $L^{\delta/2} f$.
