We find a simple sufficient criterion on a pair of nonnegative weight functions V (x) and W (x) on a Carnot group G, so that the general weighted L p Hardy type inequality G V (x) |∇ G φ (x)| p dx ≥ G W (x) |φ (x)| p dx is valid for any φ ∈ C ∞ 0 (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 G. We also present some new results on two-weight L p Hardy type inequalities with remainder terms on a bounded domain Ω in G via a differential inequality.

In this paper we study Hardy and Rellich type inequalities for Baouendi-Grushin vector fields : ∇ γ = (∇ x , |x| 2γ ∇ y ) where γ > 0, ∇ x and ∇ y are usual gradient operators in the variables x ∈ R m and y ∈ R k , respectively. In the first part of the paper, we prove some weighted Hardy type inequalities with remainder terms. In the second part, we prove two versions of weighted Rellich type inequality on the whole space. We find sharp constants for these inequalities. We also obtain their improved versions for bounded domains.

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Abstract. In the present article, we provide a sufficient condition on a pair of nonnegative weight functions V and W on the Heisenberg group H n , so that we establish a general L p Hardy type inequality involving these weights with a remainder term. The method we use here is practical enough to get more weighted Hardy type inequalities. We also obtain new results on two-weight Hardy and Hardy-Poincaré type inequalities with remainder terms on H n . Our findings improve and include many previously known results in special cases.Mathematics subject classification (2010): 26D10, 22E30, 43A80.

In this paper we present new results on two-weight Hardy, Hardy-Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.