Keyword: Sharp Constant : Search

Anywhere

Advanced Search

SEARCH GUIDE

Results: 1 - 2of2

Follow results:

refine search

Filters

per page:

Sort: Relevance

Context for search term 1Search term 1*

All Dates

LastSelect static range

Custom Range

Select starting monthSelect starting year

Select ending monthSelect ending year

Advanced

Search name	Searched On	Run search
Keyword: Reaction Wheel (2)	11 Apr 2025	Run
Keyword: Sharp Constant (2)	11 Apr 2025	Run
Keyword: Degraded Image (2)	11 Apr 2025	Run
Keyword: Dynamic Output (1)	11 Apr 2025	Run
Keyword: Highd Data Set (1)	11 Apr 2025	Run

articleNo Access
HARDY INEQUALITIES ON RIEMANNIAN MANIFOLDS WITH NEGATIVE CURVATURE
- QIAOHUA YANG,
- DAN SU, and
- YINYING KONG
Communications in Contemporary Mathematics01 Apr 2014
Preview Abstract
Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the L^p Hardy inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.
articleOpen Access
Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups
Bulletin of Mathematical Sciences04 Jul 2020
Preview 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:
$\int_{ℍ^{+}} | \nabla_{H} u |^{p} d ξ \geq {(\frac{p - 1}{p})}^{p} \int_{ℍ^{+}} \frac{𝒲 {(ξ)}^{p}}{dist {(ξ, \partial ℍ^{+})}^{p}} | u |^{p} d ξ, p > 1,$
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,
$𝒲 (ξ) = {(\sum_{i = 1}^{n} {〈 X_{i} (ξ), ν 〉}^{2} + {〈 Y_{i} (ξ), ν 〉}^{2})}^{\frac{1}{2}}$
is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group:
${(\int_{ℍ^{+}} | \nabla_{H} u |^{p} d ξ - {(\frac{p - 1}{p})}^{p} \int_{ℍ^{+}} \frac{𝒲 {(ξ)}^{p}}{dist {(ξ, \partial ℍ^{+})}^{p}} | u |^{p} d ξ)}^{\frac{1}{p}} \geq C {(\int_{ℍ^{+}} | u |^{p^{*}} d ξ)}^{\frac{1}{p^{*}}},$
where $dist (ξ, \partial ℍ^{+})$ is the Euclidean distance to the boundary, $p^{*} : = Q p / (Q - p)$ , and $2 \leq p < Q$ . For $p = 2$ , this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.