A unified approach to weighted Hardy type inequalities on Carnot groups

We find a simple sufficient criterion on a pair of nonnegative weight functions V (x) and W (x) on a Carnot group G; so that the general weighted Lp Hardy type inequality (Equation presentted) is valid for any φ ∈ C∞0 (G) and p \u3e 1: It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on G: We also present some new results on two-weight Lp Hardy type inequalities with remainder terms on a bounded domain Ω in G via a differential inequality