Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group
Tóm tắt
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
$$\langle \xi , \nu \rangle > d$$
this generalizes an inequality previously obtained by Luan and Yang. For such p and
$$\Omega $$
the inequality is sharp and takes the form
$$\begin{aligned} \int _\Omega |\nabla _{\mathbb {H}^n}u|^2 \, d\xi \ge \frac{1}{4}\int _{\Omega } \sum _{i=1}^n\frac{\langle X_i(\xi ), \nu \rangle ^2+\langle Y_i(\xi ), \nu \rangle ^2}{{{\mathrm{\text {dist}}}}(\xi , \partial \Omega )^2}|u|^2\, d\xi , \end{aligned}$$
where
$${{\mathrm{\text {dist}}}}(\, \cdot \,, \partial \Omega )$$
denotes the Euclidean distance from
$$\partial \Omega $$
.
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