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

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

Nonlinear Partial Differential Equations4 citations

TL;DR

This paper establishes weighted Hardy, Hardy-Sobolev, Hardy-Littlewood-Sobolev, and Caffarelli-Kohn-Nirenberg inequalities on general connected Lie groups, including both unimodular and non-unimodular cases. It derives necessary and sufficient conditions for two-weight Hardy inequalities on metric measure spaces with polar decompositions, extending classical inequalities to non-unimodular Lie groups and providing a unified framework with new proofs of Sobolev embeddings and uncertainty principles.

ABSTRACT

In this paper we establish a number of geometrical inequalities such as Hardy, Sobolev, Rellich, Hardy-Littlewood-Sobolev, Caffarelli-Kohn-Nirenberg, Gagliardo-Nirenberg inequalities and their critical versions for an ample class of sub-elliptic differential operators on general connected Lie groups, which include both unimodular and non-unimodular cases in compact and noncompact settings. We also obtain the corresponding uncertainty type principles.

Motivation & Objective

• To extend classical Hardy, Hardy-Sobolev, and related inequalities to general connected Lie groups, including non-unimodular and compact settings.
• To derive necessary and sufficient conditions for two-weight Hardy inequalities on metric measure spaces with polar decompositions.
• To unify and generalize existing results on Sobolev embeddings and uncertainty principles on Lie groups.
• To provide an alternative proof for Sobolev embedding theorems on non-unimodular Lie groups via the new inequalities.

Proposed method

• Derives two-weight Hardy inequalities on metric measure spaces with polar decompositions using analysis of radial and radial-like functions.
• Applies the theory of sub-Laplacians with drift on Lie groups, particularly the operator $\Delta_\chi = -\sum_{j=1}^n (X_j^2 + c_j X_j)$, to define weighted Sobolev spaces $L^p_\alpha(\mu_\chi)$.
• Uses heat kernel estimates and integral representations involving the function $G_{\alpha,\chi}^c(x)$ to control decay and growth of solutions.
• Applies Hölder’s inequality and interpolation techniques to derive embedding results between weighted $L^p$ and $L^q$ spaces.
• Adapts the method to compact Lie groups by leveraging uniform bounds on the heat kernel and character behavior.
• Establishes uncertainty principles as a byproduct of the derived inequalities, particularly through duality and norm comparisons.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient conditions for two-weight Hardy inequalities to hold on general metric measure spaces with polar decompositions?
• RQ2 How can Hardy-Sobolev and related inequalities be extended to non-unimodular Lie groups?
• RQ3 What is the role of the modular function $\delta$ and the character $\chi$ in defining weighted Sobolev spaces and embeddings on Lie groups?
• RQ4 Can the classical Sobolev embedding theorems on Lie groups be re-derived via a unified framework of Hardy-type inequalities?
• RQ5 What uncertainty principles emerge from the derived inequalities on general Lie groups?

Key findings

• The paper establishes a two-weight Hardy inequality on general metric measure spaces with polar decompositions, providing necessary and sufficient conditions for the weights.
• It proves the Hardy-Sobolev inequality on general connected Lie groups: $\left\| \frac{f}{|x|^{\beta/q}} \right\|_{L^q(\mu_{\chi^{q/p}\delta^{1-q/p}})} \lesssim \|f\|_{L^p_\alpha(\mu_\chi)}$ under the condition $1/p - 1/q \leq \alpha/d - \beta/(dq)$.
• For $d/p \leq \alpha < d$, the Hardy-Sobolev inequality holds for all $q \geq p$, extending the range of validity.
• The Caffarelli-Kohn-Nirenberg inequality is derived as a consequence of the main Hardy-Sobolev result, with explicit weight conditions.
• The Gagliardo-Nirenberg and critical versions of the inequalities are established in the same framework.
• An alternative proof for Sobolev embedding theorems on non-unimodular Lie groups is obtained via the new Hardy-type inequalities.