Geometric Hardy and Hardy-Sobolev inequalities on Heisenberg groups

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 d\xi \geq \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(\xi)^p}{dist(\xi,\partial \mathbb{H}^+)^p} |u|^p d\xi, \,\, 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 d\xi - \left(\frac{p-1}{p}\right)^p \int_{\mathbb{H}^+} \frac{\mathcal{W}(\xi)^p}{dist(\xi,\partial \mathbb{H}^+)^p} |u|^p d\xi \right)^{\frac{1}{p}} \geq C \left(\int_{\mathbb{H}^+} |u|^{p^*} d\xi\right)^{\frac{1}{p^*}}, \end{equation*} where $dist(\xi,\partial \mathbb{H}^+)$ is the Euclidean distance to the boundary, $p^* := Qp/(Q-p)$, $2\leq p<Q$, and $$\mathcal{W}(\xi)=\left(\sum_{i=1}^{n}\langle X_i(\xi), \nu \rangle^2+\langle Y_i(\xi), \nu \rangle^2\right)^{\frac{1}{2}},$$ is the angle function. For $p=2$, this gives the Hardy-Sobolev-Maz'ya inequality on the Heisenberg group.

Filippas, Maz'ya and Tertikas in [7] have established the Hardy-Sobolev inequality in the following form for all function u ∈ C ∞ 0 (R n + ), where p * = np n−p and 2 ≤ p < n. There is a different proof of this inequality by Frank and Loss [6].
The Hardy inequality in the half-space on the Heisenberg group was shown by Luan and Young [21] in the form An alternative proof of this inequality was given by Larson in [11], where the author generalised it to any half-space of the Heisenberg group, where X i and Y i (for i = 1, . . . , n) are left-invariant vector fields on the Heisenberg group, ν is the Riemannian outer unit normal (see [9]) to the boundary. Also, there is the L p -generalisation of the above inequality Note also that the authors in [14] have extended this result to general Carnot groups.
The main aim of this paper is to improve the L p -version of the geometric Hardy inequality for the sub-Laplacian in the half-spaces on the stratified groups (the Carnot groups), where the obtained inequality will be a natural extension of the inequality derived by the authors in [11] and [14] on the Heisenberg and the stratified groups, respectively. Note that it proves the inequality conjectured in [11]. Moreover, we obtain a version of the Hardy-Sobolev inequality in the setting of the Heisenberg group.
The main results of this paper are as follows: • Geometric L p -Hardy inequality on G + : Let G + := {x ∈ G : x, ν > d} be a half-space of a stratified group G. Then for all u ∈ C ∞ 0 (G + ), β ∈ R and p > 1 we have , ν 2 ) 1/2 , and L p is the p-sub-Laplacian on G, see (1.6).

1.1.
Preliminaries on stratified groups. Let G = (R n , •, δ λ ) be a stratified Lie group (or a homogeneous Carnot group), with dilation structure δ λ and Jacobian generators X 1 , . . . , X N , so that N is the dimension of the first stratum of G. Let us denote by Q the homogeneous dimension of G. We refer to the recent books [8] and [17] for extensive discussions of stratified Lie groups and their properties.
The sub-Laplacian on G is given by We also recall that the standard Lebesque measure dx on R n is the Haar measure for G (see, e.g. [8, Proposition 1.6.6]). Each left invariant vector field X k has an explicit form and satisfies the divergence theorem, see e.g. [8] for the derivation of the exact formula: more precisely, we can express N l ) are the variables in the l th stratum, see also [8, Section 3.1.5] for a general presentation. The horizontal gradient is given by and the horizontal divergence is defined by The p-sub-Laplacian has the following form (1.6) Let us define the half-space on the stratified group G as where ν := (ν 1 , . . . , ν r ) with ν j ∈ R N j , j = 1, . . . , r, is the Riemannian outer unit normal to ∂G + (see [9]) and d ∈ R. Let us define the so-called angle function such function was introduced by Garofalo [9] in his investigation of the horizontal Gauss map. The Euclidean distance to the boundary ∂G + is denoted by dist(x, ∂G + ) and defined as

Geometric Hardy inequalities
Theorem 2.1. Let G + be a half-space of a stratified group G. Then for all β ∈ R and p > 1 we have Proof of Theorem 2.1. Let us begin with the divergence theorem, then we apply the Hölder inequality and the Young inequality, respectively. It follows for a vector field By rearranging the above expression, we arrive at Now we choose V in the following form Also, we have Indeed, let us show that X i (x), ν = X i x, ν : i,1 (x ′ ), . . . , a , A direct calculation shows that Putting the above expression into inequality (2.2), we arrive at completing the proof.

Preliminaries on the Heisenberg group.
Let us give a brief introduction of the Heisenberg group. Let H n be the Heisenberg group, that is, the set R 2n+1 equipped with the group law where ξ := (x, y, t) ∈ H n , x := (x 1 , . . . , x n ), y := (y 1 , . . . , y n ), and ξ −1 = −ξ is the inverse element of ξ with respect to the group law. The dilation operation of the Heisenberg group with respect to the group law has the following form δ λ (ξ) := (λx, λy, λ 2 t) for λ > 0.
The Lie algebra h of the left-invariant vector fields on the Heisenberg group H n is spanned by and with their (non-zero) commutator The horizontal gradient of H n is given by so the sub-Laplacian on H n is given by Let us define the half-space of the Heisenberg group by where ν := (ν x , ν y , ν t ) with ν x , ν y ∈ R n and ν t ∈ R is the Riemannian outer unit normal to ∂H + (see [9]) and d ∈ R. Let us define the so-called angle function The Euclidean distance to the boundary ∂H + is denoted by dist(ξ, ∂H + ) and defined by dist(ξ, ∂H + ) := ξ, ν − d.

2.2.
Consequences on the Heisenberg group. As a consequence of Theorem 2.1 we have the following inequality.

Corollary 2.2.
Let H + be a half-space of the Heisenberg group H n . Then for all u ∈ C ∞ 0 (H + ) and p > 1 we have where the constant is sharp.
Proof of Corollary 2.2. Let us rewrite the inequality in Theorem 2.1 in terms of the Heisenberg group as follows In the case of the Heisenberg group, we need to show that the last term vanishes to prove Corollary 2.2. Indeed, we have where ξ := (x, y, t) with x, y ∈ R n and t ∈ R, ν := (ν x , ν y , ν t ) with ν x := (ν x,1 , . . . ν x,n ) and ν y := (ν y,1 , . . . ν y,n ). Then we have

So we have
Now we optimise by differentiating the above inequality with respect to β, so that we have p p − 1 |β| 1 p−1 + 1 = 0, which leads to Using this value of β, we arrive at We have finished the proof of Corollary 2.2.

Geometric Hardy-Sobolev inequalities
In this section, we present the geometric Hardy-Sobolev inequality in the half space on the Heisenberg group.
3.1. A lower estimate for the geometric Hardy type inequalities. We start with an estimate for the remainder.
Lemma 3.1. Let H + be a half-space of the Heisenberg group H n . Then for p ≥ 2, there exists a constant C p > 0 such that for all u ∈ C ∞ 0 (H + ), where dist(ξ, ∂H + ) := ξ, ν − d is the distance from ξ to the boundary, C p = (2 p−1 − 1) −1 , and u(ξ) = dist(ξ, ∂H + ) The Euclidean version of such a lower estimate to the Hardy inequality was established by Barbaris, Filippas and Tertikas [3].