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Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups

Michael Ruzhansky, Nurgissa Yessirkegenov

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.

Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups

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.

Paper Structure

This paper contains 8 sections, 18 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Main results
  4. Proof of Theorem \ref{['main1_thm']}
  5. Proof of Theorem \ref{['main3_thm']}
  6. Proof of Theorems \ref{['CKN_thm']}, \ref{['CKN_thm_crit']}, \ref{['HLS_thm']} and Corollary \ref{['uncer_cor']}
  7. Appendix: The case of compact Lie groups
  8. Acknowledgments

Key Result

Theorem 1.1

Let ${\mathbb G}$ be a connected Lie group. Let $e$ be the identity element of ${\mathbb G}$, and let $\chi$ be a positive character of ${\mathbb G}$. Let $|x|:=d_{C}(e,x)$ denote the Carnot-Carathéodory distance from $e$ to $x$. Let $d$ be the local dimension of ${\mathbb G}$ as recalled in (V_less for all $q\geq p$ such that $1/p-1/q\leq \alpha/d-\beta/(dq)$.

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • ...and 22 more