A unified approach to weighted Hardy type inequalities on Carnot groups

Jerome A. Goldstein; Ismail Kombe; Abdullah Yener

doi:10.3934/dcds.2017085

Article Contents

2017, Volume 37, Issue 4: 2009-2021. Doi: 10.3934/dcds.2017085

This issue Previous Article Minimal mass non-scattering solutions of the focusing L²-critical Hartree equations with radial data Next Article Performance bounds for the mean-field limit of constrained dynamics

A unified approach to weighted Hardy type inequalities on Carnot groups

1.
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
2.
Department of Mathematics, Faculty of Art and Science, Istanbul Commerce University, Beyoglu 34445, Istanbul, Turkey

Received: September 2016

Revised: October 2016

Published: May 2017

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Abstract

We find a simple sufficient criterion on a pair of nonnegative weight functions $V(x)$ and $W(x) $ on a Carnot group $\mathbb{G},$ so that the general weighted $L^{p}$ Hardy type inequality

$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$

is valid for any $φ ∈ C_{0}^{∞ }(\mathbb{G})$ and $p>1.$ It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on $\mathbb{G}.$ We also present some new results on two-weight $L^{p}$ Hardy type inequalities with remainder terms on a bounded domain $Ω$ in $\mathbb{G}$ via a differential inequality.

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Mathematics Subject Classification: Primary:26D10, 22E30;Secondary:43A80.

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