Hardy type and Rellich type inequalities on the Heisenberg group

Abstract

Abstract. This paper contains some interesting Hardy type inequalities and Rellich type inequalities for the left invariant vector fields on the Heisenberg group.

Anisotropic Picone identities and anisotropic Hardy inequalities

“…Picone [17] (see also Allegretto [18]) generalized (1.2) to a Laplacian that, for differentiable functions and , Allegretto and Huang [19], Dunninger [20] independently extended (1.3) to a p -Laplacian, for differentiable functions and , and applied (1.4) to derive a Sturmian comparison principle, Liouville’s theorem, the Hardy inequality, and some profound results for p -Laplace equations and systems. For other generalizations of the Picone identities and applications, see Bal [21], Dwivedi [22], Dwivedi and Tyagi [23], Niu, Zhang and Wang [24], Tyagi [25]. These results indicate that Picone identities are seemingly simple in form, but extremely useful in the study of partial differential equations, and they have become an important tool in the analysis.…”

A general approach to weighted $L^{p}$ Rellich type inequalities related to Greiner operator

“…Allegretto and Huang [5] generalized (12) to the p−Laplacian case. With a similar approach as in the [5], Niu, Zhang and Wang [25] obtained a Picone type identity for the p−sub-Laplacian and the p−biharmonic operators in the Heisenberg group. Before proving main result of this section, we first give the following Picone type identity for the p−biharmonic operators in the sub-Riemannian space R 2n+1 defined by the Greiner vector fields (7) that will be used in the sequel.…”

Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups

“…The L p version of the inequality (4.1) has been obtained by Niu, Zhang, and Wang [19], among others, which states that for 1 < p < Q:…”

“…In view of the first equality in Lemma 3.2 (for ǫ = 0), namely |∇ X d| = ( |z| d ) 2k−1 , the above inequality can also be written as For the proof of Theorem 4.1, we need the following lemma; see also [19] for the case w = 1.…”