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Levin-Cochran-Lee inequalities and best constants on homogeneous groups

Michael Ruzhansky, Markos Fisseha Yimer

TL;DR

The paper extends Levin-Cochran-Lee type inequalities to homogeneous Lie groups for $0<p\le q<\infty$ using a direct-method approach. It develops a general exponential-averaged inequality over balls and a weight-characterization criterion via $A(\alpha)$ to determine when the inequality holds and to bound the best constant. The authors provide necessary and sufficient conditions, dual formulations, and sharp constants for power-weight cases when $p=q$, along with broader corollaries and extensions to weighted exponential inequalities on non-Euclidean spaces. This work broadens Hardy-type inequalities to homogeneous groups, offering explicit constants and a versatile framework for weighted exponential inequalities in noncommutative, non-Euclidean settings.

Abstract

In this paper, we apply a direct method instead of a limit approach, for proving the Levin-Cochran-Lee inequalities. First, we state and prove Levin-Cochran-Lee type inequalities on a homogeneous group $\mathbb{G}$ with parameters $0<p\leq q<\infty$. Furthermore, for the case $p=q$, we prove the sharp inequalities with power weights and derive some other new inequalities.

Levin-Cochran-Lee inequalities and best constants on homogeneous groups

TL;DR

The paper extends Levin-Cochran-Lee type inequalities to homogeneous Lie groups for using a direct-method approach. It develops a general exponential-averaged inequality over balls and a weight-characterization criterion via to determine when the inequality holds and to bound the best constant. The authors provide necessary and sufficient conditions, dual formulations, and sharp constants for power-weight cases when , along with broader corollaries and extensions to weighted exponential inequalities on non-Euclidean spaces. This work broadens Hardy-type inequalities to homogeneous groups, offering explicit constants and a versatile framework for weighted exponential inequalities in noncommutative, non-Euclidean settings.

Abstract

In this paper, we apply a direct method instead of a limit approach, for proving the Levin-Cochran-Lee inequalities. First, we state and prove Levin-Cochran-Lee type inequalities on a homogeneous group with parameters . Furthermore, for the case , we prove the sharp inequalities with power weights and derive some other new inequalities.
Paper Structure (5 sections, 11 theorems, 86 equations)

This paper contains 5 sections, 11 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics on homogeneous Lie groups
  4. Main results
  5. Final Remarks

Key Result

Theorem 1.1

Let $\beta, \gamma$ be real numbers with $\beta>0$. If $\int_0^\infty x^\gamma f(x)\mathrm{d}x<\infty$, then holds for all positive functions $f$. Moreover, the constant $\exp{\left(\frac{\gamma+1}{\beta}\right)}$ is sharp.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Maximal integral weighted Hardy inequality
  • Remark 1.4
  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Example 3.3
  • Remark 3.4
  • ...and 15 more