Subelliptic geometric Hardy type inequalities on half-spaces and convex domains

In this paper we present $L^2$ and $L^p$ versions of the geometric Hardy inequalities in half-spaces and convex domains on stratified (Lie) groups. As a consequence, we obtain the geometric uncertainty principles. We give examples of the obtained results for the Heisenberg and the Engel groups.

In the case of the Heisenberg group H, Luan and Yang [13] obtained the following Hardy inequality on the half space H + := {(x 1 , x 2 , x 3 ) ∈ H | x 3 > 0} for u ∈ C ∞ 0 (H + ) Moreover, the geometric L p -Hardy inequalities for the sub-Laplacian on the convex domain in the Heisenberg group was obtained by Larson [12] which also generalises the previous result of [13]. In this note by using the approach in [12] we obtain the geometric Hardy type inequalities on the half-spaces and the convex domains on general stratified groups, so our results extend known results of Abelian (Euclidean) and Heisenberg groups. Thus, the main aim of this paper is to prove the geometric Hardy type inequalities on general stratified groups. As consequences, the geometric uncertainty principles are obtained. We also demonstrate the obtained results for some concrete examples of step 2 and step 3 stratified groups. In Section 2 we present L 2 and L p versions of the subelliptic geometric Hardy type inequalities on the half-space. In Section 3, we show subelliptic L 2 and L p versions of the geometric Hardy type inequalities on the convex domains.
1.1. Preliminaries. 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. We denote by Q the homogeneous dimension of G. We refer to [10], or to the recent books [2] and [8] 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 formulate 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 We now recall the divergence formula in the form of [16,Proposition 3.1]. Let f k ∈ C 1 (Ω) C(Ω), k = 1, . . . , N. Then for each k = 1, . . . , N, we have (1.5) Consequently, we also have Hardy type inequalities on half-space 2.1. L 2 -Hardy inequality on the half-space of G. In this section we present the geometric L 2 -Hardy inequality on the half-space of G. We define the half-space as follows where ν := (ν 1 , . . . , ν r ) with ν j ∈ R N j , j = 1, . . . , r, is the Riemannian outer unit normal to ∂G + (see [11]) and d ∈ R. The Euclidean distance to the boundary ∂G + is denoted by dist(x, ∂G + ) and defined as follows Moreover, there is an angle function on ∂G + which is defined by Garofalo in [11] as for all u ∈ C ∞ 0 (G + ) and where C 1 (β) := −(β 2 + β).
As consequences of Theorem 2.1, we have the geometric Hardy inequalities on the half-space without an angle function, which seems an interesting new result on G.
For optimisation we differentiate the right-hand side of integral with respect to β, then we have −2β − 1 = 0, which implies This completes the proof.
We also have the geometric uncertainty principle on the half-space of G + .
Corollary 2.5. Let G + be a half-space of a stratified group G. Then we have for all u ∈ C ∞ 0 (G + ). Proof of Corollary 2.5. By using (2.9) and the Cauchy-Schwarz inequality we get To demonstrate our general result in a particular case, here we consider the Heisenberg group, which is a well-known example of step r = 2 (stratified) group.
Proof of Corollary 2.6. Recall that the left-invariant vector fields on the Heisenberg group are generated by the basis with the commutator , choosing ν = (0, 0, 1) as the unit vector in the direction of x 3 and taking d = 0 in inequality (2.3), we get

Substituting these into inequality (2.3) we arrive at
taking β = − 1 2 . Let us present an example for the step r = 3 (stratified) groups. A well-known stratified group with step three is the Engel group, which can be denoted by E. Topologically E is R 4 with the group law of E, which is given by . The left-invariant vector fields of E are generated by the basis Corollary 2.7. Let E + = {x := (x 1 , x 2 , x 3 , x 4 ) ∈ E | x, ν > 0} be a half-space of the Engel group E. Then for all β ∈ R and u ∈ C ∞ 0 (E + ) we have where ∇ E = {X 1 , X 2 }, ν := (ν 1 , ν 2 , ν 3 , ν 4 ), and C 1 (β) = −(β 2 + β).
Remark 2.8. If we take ν 4 = 0 in (2.12), then we have the following inequality on E, by taking β = − 1 2 , Proof of Corollary 2.7. As we mentioned, the Engel group has the following basis of the left-invariant vector fields with the following two (non-zero) commutators Thus, we have

.
A direct calculation gives that Now substituting these into inequality (2.3) we obtain the desired result.
2.2. L p -Hardy inequality on G + . Here we construct an L p version of the geometric Hardy inequality on the half-space of G as a generalisation of the previous theorem. We define the p-version of the angle function by W p , which is given by the formula (2.13) Theorem 2.9. Let G + be a half-space of a stratified group G. Then for all β ∈ R we have for all u ∈ C ∞ 0 (G + ), 1 < p < ∞ and C 2 (β, p) := −(p − 1)(|β| p p−1 + β).
Proof of Theorem 2.9. We use the standard method such as the divergence theorem to obtain the inequality (2.14). For W ∈ C ∞ (G + ) and f ∈ C 1 (G + ), a direct calculation shows that Here in the last line Hölder's inequality was applied. For p > 1 and q > 1 with Let us set that By using Young's inequality in (2.15) and rearranging the terms, we arrive at (2.16) We choose W := I i , which has the following form I i = ( i 0, . . . , 1, . . . , 0) and set . . . , 1, . . . , 0) · (X 1 u, . . . , X i u, . . . , X N u) T = X i u.
Inserting the above calculations in (2.16) and summing over i = 1, . . . , N, we arrive at (2.17) We complete the proof of Theorem 2.9.
Remark 2.10. For p ≥ 2, since we have the following inequality

Hardy inequalities on a convex domain of G
In this section, we present the geometric Hardy inequalities on the convex domains in stratified groups. The convex domain is understood in the sense of the Euclidean space. Let Ω be a convex domain of a stratified group G and let ∂Ω be its boundary.
Below for x ∈ Ω we denote by ν(x) the unit normal for ∂Ω at a pointx ∈ ∂Ω such that dist(x, Ω) = dist(x,x). For the half-plane, we have the distance from the boundary dist(x, ∂Ω) = x, ν − d. As it is introduced in the previous section we also have the generalised angle function  Let Ω be a convex domain of a stratified group G. Then for β < 0 we have for all u ∈ C ∞ 0 (Ω), and C 1 (β) := −(β 2 + β).
Proof of Theorem 3.1. We follow the approach of Simon Larson [12] by proving inequality (3.1) in the case when Ω is a convex polytope. We denote its facets by {F j } j and unit normals of these facets by {ν j } j , which are directed inward. Then Ω can be constructed by the union of the disjoint sets Ω j := {x ∈ Ω : dist(x, ∂Ω) = dist(x, F j )}. Now we apply the same method as in the case of the half-space G + for each element Ω j with one exception that not all the boundary values are zero when we use the partial integration. As in the previous computation we have where n j is the unit normal of ∂Ω j which is directed outward. Since F j ⊂ ∂Ω j we have n j = −ν j . The boundary terms on ∂Ω vanish since u is compactly supported in Ω. So we only deal with the parts of ∂Ω j in Ω. Note that for every facet of ∂Ω j there exists some ∂Ω l which shares this facet. We denote by Γ jl the common facet of ∂Ω j and ∂Ω l , with n k | Γ jl = −n l | Γ jl . From the above expression we get the following inequality and a direct computation shows that

Inserting the expression (3.3) into inequality (3.2) we get
Now we sum over all partition elements Ω j and let n jl = n k | Γ jl , i.e. the unit normal of Γ jl pointing from Ω j into Ω l . Then we get Here we used the fact that (by the definition) Γ jl is a set with dist(x, F j ) = dist(x, F l ). From we see that Γ jl is a hyperplane with a normal ν j −ν l . Thus, ν j −ν l is parallel to n jl and one only needs to check that (ν j −ν l )·n jl > 0. Observe that n jl points out and ν j points into j-th partition element, so ν j · n jl is non-negative. Similarly, we see that ν l · n jl is non-positive. This means we have (ν j − ν l ) · n jl > 0. In addition, it is easy to see that which implies that where α jl is the angle between ν j and ν l . So we obtain Here with β < 0 and due to the boundary term signs we verify the inequality for the polytope convex domains.
Let us now consider the general case, that is, when Ω is an arbitrary convex domain. For each u ∈ C ∞ 0 (Ω) one can always choose an increasing sequence of convex polytopes , Ω j ⊂ Ω and Ω j → Ω as j → ∞. Assume that ν j (x) is the above map ν (corresponding to Ω j ) we compute Now we obtain the desired result when j → ∞.
3.2. L p -Hardy's inequality on a convex domain of G. In this section we give the L p -version of the previous results.
Now summing up over Ω j , and with the interior boundary terms we have As in the earlier case if the boundary term is positive we can discard it, so we want to show that Noting the fact that n jl = ν j −ν l √ 2−2 cos(α jl ) and dist(x, F j ) = dist(x, F l ) on Γ jl , we arrive at 1 2 − 2 cos(α jl ) Here we have used the equality (a − b)(a p−1 − b p−1 ) = a p − a p−1 b − b p−1 a + b p−1 with a = | X i (x), ν j | and b = | X i (x), ν l |. From the above expression we note that the boundary term in Ω is positive and β < 0. By discarding the boundary term we complete the proof.