Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group

Abstract

We prove geometric $L^p$ versions of Hardy's inequality for the sub-elliptic Laplacian on convex domains $\Omega$ in the Heisenberg group ${\mathbb{H}}^n$, where convex is meant in the Euclidean sense. When $p=2$ and $\Omega$ is the half-space given by $⟨\xi ,\nu ⟩>d$ this generalizes an inequality previously obtained by Luan and Yang. For such $p$ and $\Omega$ the inequality is sharp and takes the form \begin{equation} \int_Ω|\nabla_{\mathbb{H}^n}u|^2 \, dξ\geq \frac{1}{4}\int_Ω \sum_{i=1}^n\frac{\langle X_i(ξ), ν\rangle^2+\langle Y_i(ξ), ν\rangle^2}{\textrm{dist}(ξ, \partial Ω)^2}|u|^2\, dξ, \end{equation} where $\text{dist}(\cdot ,\partial \Omega )$ denotes the Euclidean distance from $\partial \Omega$.

14 pages

References in corpus (1)

• The Hessian of the distance from a surface in the Heisenberg group

Cited by in corpus (2)

• Hardy-type inequalities for the Carnot-Carathéodory distance in the Heisenberg group
• A Bakry-Émery criterion for weighted contractivity and $L^2$-Hardy inequalities