A unified approach to weighted Hardy type inequalities on Carnot groups

Jerome A. Goldstein; Ismail Kombe; Abdullah Yener

doi:10.3934/dcds.2017085

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

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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: August 31, 2016

Revised: September 30, 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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