  We develop characterizations for Sobolev spaces of potential type on graded
Lie groups, by means of Littlewood-Paley square functions, and Strichartz
functionals involving second-order differences. A key role is played by some
mean value inequalities that may be of independent interest. Indeed, the
point-wise first-order inequality is generalized in two directions: on
$L^p$-spaces and by considering second-order differences.
