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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 L^p Hardy type inequality equation*₆V (x) ₆ (x) ^pdx ₆W (x) (x) ^pdxequation* is valid for any φ ∈ C₀^∞ (G) and p>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 L^p Hardy type inequalities with remainder terms on a bounded domain Ω in G via a differential inequality.
Goldstein et al. (Sun,) studied this question.