Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups

Abstract

In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: ∫ℍ+|∇Hu|pdξ ≥ p − 1 pp∫ℍ+ 𝒲(ξ)p dist(ξ,∂ℍ+)p|u|pdξ,p > 1, which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, 𝒲(ξ) = ∑i=1n〈X i(ξ),ν〉2 + 〈Y i(ξ),ν〉21 2 is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: ∫ℍ+|∇Hu|pdξ −p − 1 pp∫ℍ+ 𝒲(ξ)p dist(ξ,∂ℍ+)p|u|pdξ1 p ≥ C ∫ℍ+|u|p∗dξ 1 p∗, where dist(ξ,∂ℍ+) is the Euclidean distance to the boundary, p∗ := Qp/(Q − p), and 2 ≤ p < Q. For p = 2, this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.

Keywords

• stratified groups
• heisenberg group
• geometric hardy inequality
• half-space
• sharp constant