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−1p)p∫H+W(ξ)pdist(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⟨Xi(ξ),ν⟩2+⟨Yi(ξ),ν⟩2)12 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−1p)p∫H+W(ξ)pdist(ξ,∂H+)p|u|pdξ)1p≥C(∫H+|u|p∗dξ)1p∗, 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.