[Paper Review] Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups

Michael Ruzhansky, Nurgissa Yessirkegenov|arXiv (Cornell University)|Oct 20, 2018

Geometric Analysis and Curvature Flows19 references5 citations

TL;DR

This paper establishes weighted Hardy-type inequalities on general Lie groups using polar decompositions in metric measure spaces, leading to new proofs and extensions of classical inequalities such as Hardy-Sobolev, Hardy-Littlewood-Sobolev, and Caffarelli-Kohn-Nirenberg in both unimodular and non-unimodular settings. The key contribution is a unified framework that yields necessary and sufficient conditions for weight functions and derives uncertainty principles as a byproduct.

ABSTRACT

In this paper we obtain two-weight Hardy inequalities on general metric measure spaces possessing polar decompositions. Moreover, we also find necessary and sufficient conditions for the weights for such inequalities to be true. As a consequence, we establish Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. As a byproduct, it also gives, as a special case, an alternative proof for Sobolev embedding theorems on general (non-unimodular) Lie groups. We also obtain the corresponding uncertainty type principles.

Motivation & Objective

• To extend classical inequalities like Hardy, Hardy-Sobolev, and Caffarelli-Kohn-Nirenberg to general connected Lie groups.
• To derive necessary and sufficient conditions for two-weight Hardy inequalities on metric measure spaces with polar decompositions.
• To unify the treatment of both unimodular and non-unimodular Lie groups in compact and noncompact settings.
• To provide an alternative proof for Sobolev embedding theorems on non-unimodular Lie groups.
• To establish uncertainty-type principles as a consequence of the derived inequalities.

Proposed method

• Utilizing polar decompositions in metric measure spaces to analyze radial and radial-like structures on Lie groups.
• Deriving two-weight Hardy inequalities by characterizing the weights through integral conditions involving the radial function.
• Applying the obtained inequalities to deduce Hardy-Sobolev, Hardy-Littlewood-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities on Lie groups.
• Extending the results to critical cases by analyzing limiting behavior of the weights and exponents.
• Using the structure of the Lie group's Haar measure and modular function to handle non-unimodular cases.
• Deriving uncertainty principles from the functional inequalities via duality and duality-based estimates.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions on weights for two-weight Hardy inequalities to hold on general Lie groups with polar decompositions?
• RQ2 How can Hardy-Sobolev and Hardy-Littlewood-Sobolev inequalities be extended to non-unimodular Lie groups?
• RQ3 In what way do the critical versions of these inequalities behave on general connected Lie groups?
• RQ4 Can the derived inequalities provide an alternative proof for Sobolev embedding theorems on non-unimodular Lie groups?
• RQ5 What uncertainty principles emerge from the functional inequalities on Lie groups?

Key findings

• The paper establishes necessary and sufficient conditions on weights for two-weight Hardy inequalities on metric measure spaces with polar decompositions.
• It generalizes Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities to all connected Lie groups, including non-unimodular and noncompact cases.
• The framework yields a new proof of Sobolev embedding theorems on non-unimodular Lie groups through the derived inequalities.
• Critical versions of the inequalities are obtained by analyzing limiting cases of the exponents and weights.
• Uncertainty-type principles are derived as a byproduct of the functional inequalities, linking them to harmonic analysis on Lie groups.
• The results are unified across unimodular and non-unimodular Lie groups, demonstrating the robustness of the polar decomposition approach.