  In this paper we discuss cylindrical extensions of improved Hardy, Sobolev
type and Caffarelli-Kohn-Nirenberg type inequalities with sharp constants and
identities in the spirit of Badiale-Tarantello [2]. All identities are obtained
in the setting of $L^p$ for all $p\in (1,\infty)$ without the real-valued
function assumption. The obtained identities provide a simple and direct
understanding of these inequalities as well as the nonexistence of nontrivial
extremizers. As a byproduct, we show extended Caffarelli-Kohn-Nirenberg type
inequalities with remainder terms that imply a cylindrical extension of the
classical Heisenberg-Pauli-Weyl uncertainty principle. Furthermore, we prove
$L^p$-Hardy type identities with logarithmic weights that imply the critical
Hardy inequality in the special case. Lastly, we also discuss extensions of
these results on homogeneous Lie groups. Particular attention is paid to
stratified Lie groups, where the functional inequalities become intricately
intertwined with the properties of sub-Laplacians and related hypoelliptic
partial differential equations. The obtained results already imply new insights
even in the classical Euclidean setting with respect to the range of parameters
and the arbitrariness of the choice of any homogeneous quasi-norm.
