Chapter 8 Geometric Hardy Inequalities on Stratified Groups

Given a domain in the space, the ‘geometric’ version of Hardy inequalities usually refers to the Hardy type inequalities where the weight is given in terms of the distance to the boundary of the domain. In this chapter we discuss L2 and Lp versions of the geometric Hardy inequality on the stratified group \(\mathbb{G}\). For the clarity of the exposition, we first deal with the half-space domains, and then with more general convex domains.

2. In the setting of the Heisenberg group H, the geometric Hardy inequality on the half-space takes the form for all u ∈ C ∞ 0 (H + ). This inequality was obtained in [LY08], and we can also recapture it as a consequence in Corollary 8.1.5. There are further extensions to geometric L p -Hardy inequalities as well as to the convex domains of the Heisenberg group obtained in [Lar16].
The following construction can be traced back to Garofalo [Gar08].
Definition 8.1.2 (Half-space and angle function). Let G be a stratified group. In this section the half-space of G will be defined by where d ∈ R, and ν := (ν 1 , . . . , ν r ) with ν j ∈ R Nj , j = 1, . . . , r, is the Riemannian outer unit normal to ∂G + . The Euclidean distance to the boundary ∂G + will be denoted by dist(x, ∂G + ) and given by the formula dist(x, ∂G + ) = x, ν − d.
The angle function on ∂G + is defined by In what follows we will be working in the setting of Definition 8.1.2.
Theorem 8.1.3 (Geometric L 2 -Hardy inequality on half-space). Let G + be a halfspace of a stratified group G.
(2) We have for all u ∈ C ∞ 0 (G + ). Remark 8.1.4 (Uncertainty principle and step 2 case). In the step 2 case we have the following simplification of Part (1) of Theorem 8.1.3.
1. Note that for the stratified groups of step 2 it follows from Proposition 1.2.19 that one can use the following basis of the left invariant vector fields where i = 1, . . . , N and a s m,i are the constants depending on the group. In addition, we can also write x = (x , x ) with Then the statement of Theorem 8.1.3, Part (1), can be simplified as follows: for all u ∈ C ∞ 0 (G + ) and β ∈ R we have In the standard way Theorem 8.1.3, Part (2), implies the geometric uncertainty principle on the half-space G + for general stratified groups G. Indeed, (8.3) and the Cauchy-Schwarz inequality imply That is, we have Proof of Theorem 8.1.3. Proof of Part (1). For the proof we apply the method of factorisation. So, for any real-valued W := (W 1 , . . . , W N ), W i ∈ C 1 (G + ), which will be chosen later, a direct calculation gives From the above expression we get the inequality Let us now take W i in the form i,1 (x ), . . . , a (r) i,Nr (x , x (2) , . . . , x (r−1) )), and ν = (ν 1 , ν 2 , . . . , ν r ), ν j ∈ R Nj . Now W i (x) can be written as By a direct computation we have Now combining (8.8) with (8.6) we arrive at the inequality which completes the proof of Part (1).

Examples of Heisenberg and Engel groups
Let us give examples of the geometric L 2 -Hardy inequality on half-spaces from Theorem 8.1.3 in the cases of groups of steps 2 and 3. The example of general stratified groups of step 2 was considered in Remark 8.1.4, Part 1, and now we look at the special case of the Heisenberg group. In particular, it yields the estimate that was given in Remark 8.1.1, Part 2.
Corollary 8.1.5 (Geometric L 2 -Hardy inequality on half-space of the Heisenberg group).
Next, let us give an example for a class of stratified groups of step r = 3, namely, the case of the Engel group.
Definition 8.1.6 (Engel group). The Engel group E is the space R 4 with the group law given by The left invariant vector fields of E are generated by (the basis) The group E is stratified of step 3, with the nonzero commutation relations given by

So we have
Corollary 8.1.7 (Geometric L 2 -Hardy inequality on half-space of the Engel group).
In particular, if we take ν 4 = 0 in (8.9), then by taking β = − 1 2 , we get the following inequality on such E + : Proof of Corollary 8.1.7. Using the above basis of the left invariant vector fields, we have It is then straightforward to see that Now plugging these in inequality (8.2) we get the desired inequality (8.9).

L p -Hardy inequality on the half-space
Now we discuss an L p version of the geometric Hardy inequality on the half-space of G as an extension of the previous L 2 arguments. We recall that the p-version of Garofalo's angle function from Definition 8.1.2 can be defined by the formula with W(x) := W 2 (x), and where N denotes the dimension of the first stratum of G. As before let G + be a half-space of a stratified group G. The L p version of the geometric Hardy inequality from Theorem 8.1.3 can be written in the following form.
Theorem 8.2.1 (Geometric L p -Hardy inequality on half-space). Let G + be a halfspace of a stratified group G and let 1 < p < ∞. Then for all u ∈ C ∞ 0 (G + ) and all β ∈ R we have Remark 8.2.2. Note that for p ≥ 2, since |X i u| 2 p/2 , the proof will also yield the inequality Proof of Theorem 8.2.1. For W ∈ C ∞ (G + ) and f ∈ C 1 (G + ), a direct computation with Hölder's inequality gives For p > 1 and q > 1 with 1 p + 1 q = 1, we will use Young's inequality Using this Young inequality in (8.13) and rearranging the terms, we get (8.14) Now choosing W := I i , which has the following form I i = ( i 0, . . . , 1, . . . , 0) and setting Moreover, we have Substituting these in (8.14) and summing over i = 1, . . . , N, we obtain This completes the proof.

L 2 -Hardy inequality on convex domains
In this and the following sections we extend the proceeding arguments from halfspaces to convex domains in the stratified groups. Here, however, the convex domain is understood in the sense of the Euclidean space. Thus, let Ω be a convex domain of a stratified group G and let ∂Ω be its boundary.
For the half-space, we have the distance from the boundary dist(x, ∂Ω) = x, ν −d.
As it was already defined in (8.10), we will use the p-version of the angle function We have the following extension of Theorem 8.1.3.
Theorem 8.3.1 (Geometric L 2 -Hardy inequality on convex domains). Let Ω be a convex domain of a stratified group G. Then for all u ∈ C ∞ 0 (Ω) and all β < 0 we have Proof of Theorem 8.3.1. As elsewhere in this chapter, we follow the proof for general stratified groups of [RSS18b], based on the convex polytope approach used by Larson [Lar16] in the case of the Heisenberg group.
We denote the facets of Ω by {F j } j and unit normals of these facets by {ν j } j , which are directed inward. So, Ω can be viewed as the union of the disjoint sets Now we follow the 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 before we calculate 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 . That is, we have (8.17) The boundary terms on ∂Ω disappears since u is compactly supported in Ω. Thus, we only need to 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. Denote by Γ jl the common facet of ∂Ω j and ∂Ω l , with n k | Γ jl = −n l | Γ jl . Now we choose W i in the form and a direct computation shows that 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 have Here we have used the fact that (by the definition) Γ jl is a set with we see that Γ jl is a hyperplane with a normal ν j − ν l . So, ν j − ν l is parallel to n jl and one only needs to check that (ν j − ν l ) · n jl > 0. Since n jl points out and ν j points into jth partition element, ν j · n jl is non-negative. Similarly, we see that ν l · n jl is non-positive. That is, (ν j − ν l ) · n jl > 0. On the other hand, it is easy to see 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 prove the desired inequality for the polytope convex domains. Now we are ready to 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 } ∞ j=1 such that u ∈ C ∞ 0 (Ω 1 ), Ω j ⊂ Ω and Ω j → Ω as j → ∞. Assume that ν j (x) is the above map ν (corresponding to Ω j ), and then we can calculate Now we obtain the desired result by letting j → ∞.

L p -Hardy inequality on convex domains
The same arguments as in the previous section give the general L p -version of Theorem 8.3.1.
Substituting these into (8.24) and summing over all i = 1, . . . , N, we obtain , n j (x) |u| p dΓ ∂Ωj (x). Now summing up over Ω j , and with the interior boundary terms we get