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2018 General weighted Hardy-type inequalities related to Greiner operators
Abdullah Yener
Rocky Mountain J. Math. 48(7): 2405-2430 (2018). DOI: 10.1216/RMJ-2018-48-7-2405

Abstract

In this article, we present a general method that can be used to deduce weighted Hardy-type inequalities from a particular non-linear partial differential inequality in a relatively simple and unified way on the sub-Riemannian manifold $\mathbb {R}^{2n+1}=\mathbb {R}^{n}\times \mathbb {R}^{n}\times \mathbb {R}$, defined by the Greiner vector fields \[X_{j}=\frac {\partial }{\partial x_{j}}+2ky_{j} \vert z\vert ^{2k-2}\frac {\partial }{\partial l},\] \[Y_{j}=\frac {\partial }{\partial y_{j}}-2kx_{j}|z|^{2k-2}\frac {\partial }{\partial l},\] $j=1,\ldots ,n$, where $z=x+iy\in \mathbb {C}^{n},$ $l\in \mathbb {R}$, $k\geq 1$. Our method allows us to improve, extend, and unify many previously obtained sharp weighted Hardy-type inequalities as well as to yield new ones. These cases are illustrated by giving many concrete examples, including radial, logarithmic, hyperbolic and non-radial weights. Furthermore, we introduce a new technique for constructing two-weight $L^{p}$ Hardy-type inequalities with remainder terms on smooth bounded domains $\Omega $ in $\mathbb {R}^{2n+1}$. We also give several applications leading to various weighted Hardy inequalities with remainder terms.

Citation

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Abdullah Yener. "General weighted Hardy-type inequalities related to Greiner operators." Rocky Mountain J. Math. 48 (7) 2405 - 2430, 2018. https://doi.org/10.1216/RMJ-2018-48-7-2405

Information

Published: 2018
First available in Project Euclid: 14 December 2018

zbMATH: 06999268
MathSciNet: MR3892138
Digital Object Identifier: 10.1216/RMJ-2018-48-7-2405

Subjects:
Primary: 22E30, 26D10, 43A80

Rights: Copyright © 2018 Rocky Mountain Mathematics Consortium

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Vol.48 • No. 7 • 2018
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