General weighted Hardy-type inequalities related to Greiner operators

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.