  In this note we extend several integral inequalities to the context of
noncommutative Vilenkin groups. We prove some sharp weak and strong type
estimates for the Hardy operator and the Hardy-Littlewood-P{\'o}lya operator on
constant-order noncommutative Vilenkin groups. In particular for graded
$\K$-Lie groups, where $\K$ is a non-archimedean local field, we additionally
provide some functional inequalities, like the Hardy-Littlewood-Sobolev
unequality and the Stein-Weiss inequality, linking some classes of homogeneous
pseudo-differential operators, like the Vladimirov-Taibleson operator and the
Vladimirov Laplacian, with Hardy inequalities.
