  We present a unified approach to obtain Hardy-type inequalities in the
context of nilpotent Lie groups with sharp constants. The unified methodology
employed herein allows for exploration of the sharp Hardy inequalities on
various Lie group structures and improves previously known inequalities even in
the classical Euclidean setting. By leveraging this framework, as an
application, we derive new Caffarelli-Kohn-Nirenberg type inequalities both in
the classical Euclidean setting and on general homogeneous Lie groups.
