17 research outputs found
Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group
We prove geometric versions of Hardy's inequality for the sub-elliptic
Laplacian on convex domains in the Heisenberg group ,
where convex is meant in the Euclidean sense. When and is the
half-space given by this generalizes an
inequality previously obtained by Luan and Yang. For such and the
inequality is sharp and takes the form \begin{equation}
\int_\Omega |\nabla_{\mathbb{H}^n}u|^2 \, d\xi \geq \frac{1}{4}\int_{\Omega}
\sum_{i=1}^n\frac{\langle X_i(\xi), \nu\rangle^2+\langle Y_i(\xi),
\nu\rangle^2}{\textrm{dist}(\xi, \partial \Omega)^2}|u|^2\, d\xi,
\end{equation} where denotes the
Euclidean distance from .Comment: 14 page