Weighted Hardy type inequalities on the Heisenberg group ℍ^n

Abstract

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.

Adams-Spanne type estimates for the commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups

“…Analysis on the groups is also motivated by their role as the simplest and the most important model in the general theory of vector fields satisfying Hörmander's condition. Due to this reason, many interesting works have been devoted to the theory of harmonic analysis on H n in [6,8,9,19,20,23,26,27].…”

On horizontal Hardy, Rellich, Caffarelli–Kohn–Nirenberg and p-sub-Laplacian inequalities on stratified groups

“…To give an idea for obtaining more general improved weighted Hardy type inequalities let us conclude this paper with the following very short discussion of techniques from [47] (see also [24] and [42]), now in the setting of stratified groups.…”

“…again with p N −p being the best constant (see [11] and [47] for the version on the Heisenberg group). In the abelian case G = (R n , +), n ≥ 3, (3.4) implies the classical Hardy inequality for G ≡ R n :…”

A unified approach to weighted Hardy type inequalities on Carnot groups

“…On the other hand in the case of sub-Riemannian spaces, especially on Carnot groups G, Hardy type inequalities have been also intensively investigated, see [11], [22], [34], [10], [27], [26], [29], [36]. For instance, D'Ambrosio in [11] and Goldstein and Kombe in [22] established, among the other things, the following L p Hardy type inequality on polarizable Carnot groups G,…”