Cylindrical extensions of critical Sobolev type inequalities and identities

Ruzhansky M. Shaimerdenov Y. Yessirkegenov N.
1 February 2026 Academic Press Inc.

Journal of Mathematical Analysis and Applications
2026 #554 Issue 1

In this paper, we investigate cylindrical extensions of critical Sobolev type (improved Hardy) inequalities and identities in the style of Badiale-Tarantello [6], 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∈(1,∞) 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 Steins 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.

Critical Caffarelli-Kohn-Nirenberg inequality , Critical Sobolev type inequality , Heisenberg group , Homogeneous Lie group , Stratified Lie group

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Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Belgium
School of Mathematical Sciences, Queen Mary University of London, United Kingdom
SDU University, Kaskelen, Kazakhstan
Institute of Mathematics and Mathematical Modeling, Kazakhstan
KIMEP University, Almaty, Kazakhstan

Department of Mathematics: Analysis
School of Mathematical Sciences
SDU University
Institute of Mathematics and Mathematical Modeling
KIMEP University

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