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Fractional Hardy type inequalities on homogeneous Lie groups in the case $Q<sp$

Aidyn Kassymov, Michael Ruzhansky, Durvudkhan Suragan

Abstract

In this paper, we obtain a fractional Hardy inequality in the case $Q<sp$ on homogeneous Lie groups, and as an application we show the corresponding uncertainty principle. Also, we show a fractional Hardy-Sobolev type inequality on homogeneous Lie groups. In addition, we prove fractional logarithmic Hardy-Sobolev and fractional Nash type inequalities on homogeneous Lie groups. We note that the case $Q>sp$ was extensively studied in the literature, while here we are dealing with the complementary range $Q<sp$.

Fractional Hardy type inequalities on homogeneous Lie groups in the case $Q<sp$

Abstract

In this paper, we obtain a fractional Hardy inequality in the case on homogeneous Lie groups, and as an application we show the corresponding uncertainty principle. Also, we show a fractional Hardy-Sobolev type inequality on homogeneous Lie groups. In addition, we prove fractional logarithmic Hardy-Sobolev and fractional Nash type inequalities on homogeneous Lie groups. We note that the case was extensively studied in the literature, while here we are dealing with the complementary range .

Paper Structure

This paper contains 6 sections, 10 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Fractional Hardy inequality
  4. Fractional Hardy-Sobolev type inequality
  5. Logarithmic Hardy-Sobolev type inequality
  6. Fractional Nash type inequality

Key Result

Theorem 1.1

Let $1<p\le q <\infty$. Let $\mathbb G$ be a homogeneous group of homogeneous dimension $Q$. Let $g,h$ be measurable functions positive a.e in $\mathbb G$ such that $g\in L^1(\mathbb G\backslash \{0\})$ and $h^{1-p'}\in L^1_{loc}(\mathbb \mathbb{G})$. Denote and where $\frac{1}{p}+\frac{1}{p'}=1.$ Then the inequality hold for all measurable functions $f:\mathbb{G}\to{\mathbb C}$ if and only if

Theorems & Definitions (19)

  • Theorem 1.1: RY18b
  • Theorem 1.2: RS17
  • Proposition 2.1: FR, Proposition 3.1.38 and RS18, Proposition 1.2.4
  • Theorem 2.2
  • Remark 2.3
  • proof
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Corollary 2.6: Uncertainty principle
  • ...and 9 more