Research Information System
Rocky Mountain Journal of Mathematics, vol.48, no.7, pp.2405-2430, 2018 (SCI-Expanded, Scopus)
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 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 Ω in R 2 n +1 . We also give several applications leading to various weighted Hardy inequalities with remainder terms.