  We set up a one-parameter family of inequalities that contains both the Hardy
inequalities (when the parameter is 1) and the Caffarelli-Kohn-Nirenberg
inequalities (when the parameter is optimal). Moreover, we study these results
with the exact remainders to provide direct understandings to the sharp
constants, as well as the existence and non-existence of the optimizers of the
Hardy inequalities and Caffarelli-Kohn-Nirenberg inequalities. As an
application of our identities, we establish some sharp versions with optimal
constants and theirs attainability of the stability of the Heisenberg
Uncertainty Principle and several stability results of the
Caffarelli-Kohn-Nirenberg inequalities.
