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_α : 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.