Cylindrical extensions of critical Sobolev type inequalities and identities

TL;DR

This paper extends critical Sobolev inequalities to cylindrical and Lie group settings, deriving new identities and inequalities with sharp constants, including applications to uncertainty principles and subelliptic PDEs.

Contribution

It introduces novel cylindrical extensions of Sobolev inequalities, higher-order identities involving special numbers, and broadens the scope to Lie groups and subelliptic PDEs.

Findings

Derived new cylindrical Sobolev inequalities with sharp constants.

Established higher-order identities involving special combinatorial numbers.

Applied results to uncertainty principles and subelliptic PDEs in Lie groups.

Abstract

In this paper, we investigate cylindrical extensions of critical Sobolev type (improved Hardy) inequalities and identities in the style of Badiale-Tarantello [BT02], which in a special case give a critical Hardy inequality and its stability results. We also obtain higher-order identities, which interestingly include well-known numbers like double factorial, Oblong numbers, and Stirling numbers of the second kind. All functional identities are obtained in $L^{p}$ for $p\in (1,\infty)$ without the real-valued function assumption, which gives a simple and direct understanding of the corresponding inequalities as well as the nonexistence of nontrivial extremizers. As applications, we obtain Caffarelli-Kohn-Nirenberg type inequalities with logarithmic weights, which in a particular case give the critical case of the Heisenberg-Pauli-Weyl type uncertainty principle. We also discuss these results in the setting of Folland and Stein's homogeneous Lie groups. A special focus is devoted to stratified Lie groups, where Sobolev type inequalities become intricately intertwined with the properties of sub-Laplacians and more general subelliptic partial differential equations. The obtained results are already new even in the classical Euclidean setting with respect to the range of parameters and the arbitrariness of the choice of any homogeneous quasi-norm. Most inequalities are obtained with sharp constants.