Cylindrical Hardy, Sobolev type and Caffarelli-Kohn-Nirenberg type inequalities and identities

Abstract

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.