Weighted L^{p}-Hardy and L^{p}-Rellich inequalities with boundary terms on stratified Lie groups

Michael Ruzhansky,Bolys Sabitbek,Durvudkhan Suragan

doi:10.1007/s13163-018-0268-3

Abstract

In this paper, generalised weighted L^p-Hardy, L^p-Caffarelli–Kohn–Nirenberg, and L^p-Rellich inequalities with boundary terms are obtained on stratified Lie groups. As consequences, most of the Hardy type inequalities and Heisenberg–Pauli–Weyl type uncertainty principles on stratified groups are recovered. Moreover, a weighted L^2-Rellich type inequality with the boundary term is obtained.

Highlights

Let G be a stratified Lie group, with dilation structure δλ and Jacobian generators X1, . . . , X N, so that N is the dimension of the first stratum
It was shown by Folland [10] that the sub-Laplacian has a unique fundamental solution ε, Lε = δ, where δ denotes the Dirac distribution with singularity at the neutral element 0 of G
It is important to note that the above Green’s formulae hold for the fundamental solution of the sub-Laplacian as in the case of the fundamental solution of the (Euclidean) Laplacian since both have the same behaviour near the singularity z = 0

Summary

Introduction

Let G be a stratified Lie group (or a homogeneous Carnot group), with dilation structure δλ and Jacobian generators X1, . . . , X N , so that N is the dimension of the first stratum. The analogue of Green’s first formula for the sub-Laplacian was given in [19] in the following form: if v ∈ C1( ) ∩ C( ) and u ∈ C2( ) ∩ C1( ), (∇v)u + vLu dz = v ∇u, dz , where. It is important to note that the above Green’s formulae hold for the fundamental solution of the sub-Laplacian as in the case of the fundamental solution of the (Euclidean) Laplacian since both have the same behaviour near the singularity z = 0 (see [1, Proposition 4.3]). Some boundary terms have appeared in [24] For these inequalities in the setting of general homogeneous groups we refer to [22]. A weighted L2-Rellich type inequality with the boundary term is obtained together with its consequences. We derive several versions of the L p weighted Hardy inequalities

Weighted Lp-Cafferelli-Kohn-Nirenberg type inequalities with boundary terms

Horizontal Lp-Caffarelli–Kohn–Nirenberg inequalities with the boundary term

Badiale–Tarantello conjecture

Uncertainty type principles

Weighted Lp-Rellich type inequalities