Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups
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,...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
World Scientific Publishing
2020-12-01
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| Series: | Bulletin of Mathematical Sciences |
| Subjects: | |
| Online Access: | http://www.worldscientific.com/doi/epdf/10.1142/S1664360720500162 |
| Summary: | 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. |
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| ISSN: | 1664-3607 1664-3615 |