Yener, Abdullah2020-11-212020-11-2120180035-7596https://doi.org/10.1216/RMJ-2018-48-7-2405https://hdl.handle.net/11467/3521In 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 R 2 n +1 = R n ×R n ×R, defined by the Greiner vector fields ? X j = ?xj + 2ky j |z| 2 k? 2 ? ?l , ? Y j = ?yj ? 2kx j |z| 2 k? 2 ? ?l , j = 1, . . ., n, where z = x + iy ? C n , l ? R, k ? 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 ? in R 2 n +1 . We also give several applications leading to various weighted Hardy inequalities with remainder terms. Copyright © 2018 Rocky Mountain Mathematics Consortium.eninfo:eu-repo/semantics/closedAccessGeneralized Greiner operatorHeisenberg-Pauli-Weyl inequalityRemainder termsTwo-weight Hardy inequalityWeighted Hardy inequalityArticle48724052430Q4WOS:000453227100013Q22-s2.0-8506165736110.1216/RMJ-2018-48-7-2405