Refined general weighted $L^p$-Hardy and Caffarelli-Kohn-Nirenberg type inequalities and identities related to the Baouendi-Grushin operator

In this paper, we present a sufficient condition on a pair of nonnegative weights $v$ and $w$ such that we have a general weighted $L^p$-Hardy type identity. The result, for a certain choice of weights, gives weighted $L^p$-Hardy type inequalities and identities with explicit remainder terms, thereby improving previously known results. Furthermore, we obtain the corresponding general weighted Caffarelli-Kohn-Nirenberg type inequality with remainder terms, which, as a result, imply Heisenberg-Pauli-Weyl type inequalities.