Hardy and uncertainty inequalities on stratified Lie groups

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group G. In particular, we show that the operators T-alpha : f -> |center dot|(-alpha) L(-alpha/2)f, where |center dot| 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 |center dot|(beta) f and the L-r norm of L(delta/2)f. (C) 2015 Elsevier Inc. All rights reserved.