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We give an elementary proof of the classical Hardy inequality on any Carnotgroup, using only integration by parts and a fine analysis of the commutatorstructure, which was not deemed possible until now. We also discuss theconditions under which this technique can be generalized to deal withhypoelliptic families of vector fields, which, in this case, leads to an openproblem regarding the symbol properties of the gauge norm.

A simple proof of the Hardy inequality on Carnot groups and for some hypoelliptic families of vector fields