Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups

Abstract

In this paper, we present the geometric Hardy inequality for the sub-Laplacian in the half-spaces on the stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space on the Heisenberg group with a sharp constant \begin{equation*} \int_{\mathbb{H}^+} |\nabla_{H}u|^p dξ\geq \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ, \,\, p>1, \end{equation*} which solves the conjecture in the paper \cite{Larson}. Also, we obtain a version of the Hardy-Sobolev inequality in a half-space on the Heisenberg group \begin{equation*} \left(\int_{\mathbb{H}^+} |\nabla_{H} u|^p dξ- \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(ξ)^p}{dist(ξ,\partial \mathbb{H}^+)^p} |u|^p dξ\right)^{\frac{1}{p}} \geq C \left(\int_{\mathbb{H}^+} |u|^{p^*} dξ\right)^{\frac{1}{p^*}}, \end{equation*} where $dist(\xi ,\partial {\mathbb{H}}^{+})$ is the Euclidean distance to the boundary, $p^{\ast }:=Qp/(Q-p)$, $2\le p<Q$, and $$\mathcal{W}(\xi )={\left(\sum _{i=1}^n⟨X_i(\xi ),\nu {⟩}^2+⟨Y_i(\xi ),\nu {⟩}^2\right)}^{\frac{1}{2}},$$ is the angle function. For $p=2$, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group.