Hardy-Type Inequalities for the Carnot–Carathéodory Distance in the Heisenberg Group

Abstract

In this paper we study Hardy inequalities in the Heisenberg group ${\mathbb {H}}^n$, with respect to the Carnot–Carathéodory distance $\delta$ from the origin. We firstly show that, letting Q be the homogenous dimension, the optimal constant in the (unweighted) Hardy inequality is strictly smaller than $n^2 = (Q-2)^2/4$. Then, we prove that, independently of n, the Heisenberg group does not support a radial Hardy inequality, i.e., a Hardy inequality where the gradient term is replaced by its projection along $\nabla \!_{\mathbb {H}}\delta$. This is in stark contrast with the Euclidean case, where the radial Hardy inequality is equivalent to the standard one, and has the same constant. Motivated by these results, we consider Hardy inequalities for non-radial directions, i.e., directions tangent to the Carnot–Carathéodory balls. In particular, we show that the associated constant is bounded on homogeneous cones $C_\Sigma$ with base $\Sigma \subset {\mathbb {S}}^{2n}$, even when $\Sigma$ degenerates to a point. This is a genuinely sub-Riemannian behavior, as such constant is well known to explode for homogeneous cones in the Euclidean space.

Appendix A: An Euclidean Non-radial Hardy Inequality

In this section we present an Euclidean version of Theorem 5. Let $x=(x',x_d)\in {{\,\mathrm{{\mathbb {R}}}\,}}^{d-1}\times {{\,\mathrm{{\mathbb {R}}}\,}}$. We consider coordinates $(t,\varpi ,\varphi )\in {\mathbb {R}}_+\times {\mathbb {S}}^{d-2}\times (-\pi /2,\pi /2)$ in ${{\,\mathrm{{\mathbb {R}}}\,}}^d$, defined by

$$\begin{aligned} (x', x_d) = t(\varpi \cos \varphi , \sin \varphi ). \end{aligned}$$

(114)

In this case, the polar vector field is $\Xi =\frac{1}{t}\partial _\varphi$, which is unit thanks to the fact that

$$\begin{aligned} \varphi = \arctan \frac{x_d}{\Vert x'\Vert }. \end{aligned}$$

(115)

Moreover, the volume form becomes $t^{d-1}\cos ^{d-2}\varphi \,\mathrm{{d}}t\,\mathrm{{d}}\varphi \,\mathrm{{d}}\sigma (\varpi )$, where $\mathrm{{d}}\sigma$ is the standard volume on ${\mathbb {S}}^{d-2}$.

Let us consider a spherical cap $\Sigma \subset {\mathbb {S}}_+^{d-1}$, and let $C_\Sigma$ be the associated Euclidean cone. We can always assume it to be centered on the d-th coordinate axis, i.e., $C_\Sigma =\{\varphi >a_\Sigma \}$ for some $a_\Sigma \in (0,\pi /2)$. We have the following.

Theorem 24

Let $d\ge 3$. Then, letting $\psi (x',x_d)=x_d/\Vert x'\Vert$, we have

$$\begin{aligned} \int _{C_\Sigma } \frac{|\langle \nabla u, \Xi \rangle |^2}{\psi }\,\mathrm{{d}}x \ge \left( \frac{d-2}{2}\right) ^2 \int _{C_\Sigma } \frac{|u|^2}{|x|^2}\psi \,\mathrm{{d}}x, \qquad \forall u \in C^\infty _c(C_\Sigma ). \end{aligned}$$

(116)

Moreover, the inequality is sharp.

Proof

Let $v\in C^\infty _c(C_\Sigma )$. An integration by part yields

$$\begin{aligned} \begin{aligned} \int _{C_\Sigma } \langle \nabla v, \Xi \rangle \,\mathrm{{d}}x&= \int _0^{+\infty } t^{d-1}\,\mathrm{{d}}t \int _{{\mathbb {S}}^{d-2}}\mathrm{{d}}\sigma (\varpi )\int _{a_\Sigma }^{\pi /2} \frac{1}{t}\partial _\varphi v \cos ^{d-2}\varphi \,\mathrm{{d}}\varphi \\&= (d-2)\int _0^{+\infty } t^{d-1}\,\mathrm{{d}}t \int _{{\mathbb {S}}^{d-2}}\mathrm{{d}}\sigma (\varpi )\int _{a_\Sigma }^{\pi /2} \frac{v}{t} \cos ^{d-3}\varphi \sin \varphi \,\mathrm{{d}}\varphi \\&= (d-2)\int _{C_\Sigma } \frac{v}{|x|}\psi \,\mathrm{{d}}x. \end{aligned}\nonumber \\ \end{aligned}$$

(117)

Here, we used that $\psi =\tan \varphi$ in polar coordinates, thanks to (115). The desired inequality then follows from the above, choosing $v = u^2/t$ and applying Cauchy–Schwarz on the l.h.s..

To obtain the sharpness, observe that choosing $u(t,\varphi ) = \eta (t)(\cos \varphi )^\gamma$ where $\eta (t)$ is any cut-off with compact support and $\gamma \in {\mathbb {R}}$, we have

$$\begin{aligned} \int _{C_\Sigma } \frac{|\langle \nabla u, \Xi \rangle |^2}{\psi }\,\mathrm{{d}}x = \gamma ^2 \int _{C_\Sigma } \frac{|u|^2}{|x|^2}\psi \,\mathrm{{d}}x. \end{aligned}$$

(118)

Direct computations show that the last integral is finite if and only if $\gamma >\frac{2-d}{2}$. Then, the statement follows via a cut-off argument as the one in the proof of Theorem 5, letting $\gamma \rightarrow \left( \frac{2-d}{2}\right) ^+$. $\square$

Since the function $\psi$ is unbounded on the vertical axis $\{x' =0\}$, the above does not yield any upper bound on $c^ {\perp ,\text {eucl}}_n(\Sigma )$. Indeed, we can only recover the following lower bound, which is not sharp but correctly shows that $c^ {\perp ,\text {eucl}}_n(\Sigma )\rightarrow +\infty$ as $\Sigma$ degenerates to a point (i.e., $a_\Sigma \uparrow \pi /2$).

Corollary 25

Let $\Sigma = \{\varphi >a_\Sigma \}\subset {\mathbb {S}}^{d-1}$ be a spherical cap. Then,

$$\begin{aligned} c^ {\perp ,\text {eucl}}_n(\Sigma )\ge \left( \frac{d-2}{2}\right) ^2 \tan ^2 a_\Sigma . \end{aligned}$$

(119)