13 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
Geometric extensions of many-particle Hardy inequalities
Certain many-particle Hardy inequalities are derived in a simple and
systematic way using the so-called ground state representation for the
Laplacian on a subdomain of . This includes geometric extensions
of the standard Hardy inequalities to involve volumes of simplices spanned by a
subset of points. Clifford/multilinear algebra is employed to simplify
geometric computations. These results and the techniques involved are relevant
for classes of exactly solvable quantum systems such as the Calogero-Sutherland
models and their higher-dimensional generalizations, as well as for membrane
matrix models, and models of more complicated particle interactions of
geometric character.Comment: Revised version. 28 page