Hardy and uncertainty inequalities on stratified Lie groups

Abstract

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_α: f \mapsto |.|^{-α} L^{-α/2} f$, where $|.|$ is a homogeneous norm, $0 < α< Q/p$, 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 $|.|^βf$ and the $L^r$ norm of $L^{δ/2} f$.