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 Tα:f↦|.|^(−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 |.|^β f and the L^r norm of L^(δ/2)f.