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_\u3b1 : f \u2192 |\ub7|^{ 12\u3b1} L^{ 12\u3b1/2}f, where |\ub7| is a homogeneous norm, 0 < \u3b1 < 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\u2013Pauli\u2013 Weyl inequality, relating the L^p norm of a function f to the L^q norm of |\ub7|^\u3b2 f and the L^r norm of L^{\u3b4/2}f