Vol:.(1234567890)

Annals of Functional Analysis (2020) 11:1042–1061 https://doi.org/10.1007/s43034-020-00067-9

**ORIGINAL PAPER**

**Subelliptic geometric Hardy type inequalities ** **on half‑spaces and convex domains**

**Michael Ruzhansky**^{1,2}** · Bolys Sabitbek**^{2,3}** · Durvudkhan Suragan**^{4}

Received: 6 November 2019 / Accepted: 26 February 2020 / Published online: 2 April 2020

© The Author(s) 2020

**Abstract**

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.

**Keywords** Stratified groups · Geometric Hardy inequality · Half-space · Convex
domain

**Mathematics Subject Classification** 35A23 · 35H20

**1 Introduction**

In the Euclidean setting, a geometric Hardy inequality in a (Euclidean) convex domain Ω has the following form

Research Group

Communicated by Joachim Toft.

* Bolys Sabitbek [email protected]

Michael Ruzhansky [email protected]

Durvudkhan Suragan

1 Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

2 School of Mathematical Sciences, Queen Mary University of London, London, UK

3 Institute of Mathematics and Mathematical Modeling, Al-Farabi Kazakh National University, Almaty, Kazakhstan

4 Department of Mathematics, School of Sciences and Humanities, Nazarbayev University, 53 Kabanbay Batyr Ave, 010000 Nursultan, Kazakhstan

for *u*∈*C*^{∞}_{0} (Ω) with the sharp constant 1/4. There is a number of studies related to
this subject, see e.g. [1–3, 5, 6, 13].

In the case of the Heisenberg group **ℍ** , Luan and Yang [12] 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 [11] which also generalises
the previous result in [12]. In this note by using the approach in [11] 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. In Sect. 2 we present *L*^{2} and *L*^{p} versions of the subelliptic geometric Hardy
type inequalities on the half-space. In Sect. 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 [9],
or to the recent books [4, 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

with *x*= (*x*^{�}, *x*^{(2)},…, *x*^{(r)}) , where *r* is the step of **𝔾** and *x*^{(l)} = (*x*^{(l)}_{1} ,…, *x*^{(l)}_{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

(1.1)

�_{Ω}|∇*u*|^{2}*dx*≥ 1

4�_{Ω} |*u*|^{2}
*dist*(*x*,*𝜕*Ω)^{2}*dx*,

(1.2)

�**ℍ**^{+}

|∇**ℍ***u*|^{2}*dx*≥ �**ℍ**^{+}

|*x*_{1}|^{2}+|*x*_{2}|^{2}
*x*^{2}_{3} |*u*|^{2}*dx*.

(1.3) L=

∑*N*
*k*=1

*X*^{2}_{k}.

(1.4)
*X*_{k}= *𝜕*

*𝜕x*^{�}_{k} +

∑*r*
*l*=2

*N*_{l}

∑

*m*=1

*a*^{(l)}_{k,m}(

*x*^{�},…, *x*^{(l−}^{1}^{)}) *𝜕*

*𝜕x*^{(l)}_{m}
,

and the horizontal divergence is defined by

We now recall the divergence formula in the form of [14, Proposition 3.1]. Let
*f*_{k}∈*C*^{1}(Ω)⋂

*C*(Ω), *k*=1,…, *N* . Then for each *k*=1,…, *N*, we have

Consequently, we also have

**2 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 nor-
mal to *𝜕***𝔾**^{+} (see [10]) 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 [10] as

**Theorem 2.1*** Let ***𝔾**^{+}* be a half*-*space of a stratified group ***𝔾**.* Then for all 𝛽*∈**ℝ*** we *
*have*

∇**𝔾** ∶= (*X*_{1},…, *X*_{N}),

div_{𝔾}*v*∶= ∇**𝔾**⋅*v*.

(1.5)

∫_{Ω}*X*_{k}*f*_{k}*dz*=

∫_{𝜕Ω}*f*_{k}⟨*X*_{k}, *dz*⟩.

(1.6)

∫_{Ω}

�*N*
*k*=1

*X*_{k}*f*_{k}*dz*=

∫_{𝜕Ω}

�*N*
*k*=1

*f*_{k}⟨*X*_{k}, *dz*⟩.

**𝔾**^{+}∶= {*x*∈**𝔾**∶⟨*x*,*𝜈*⟩*>d*},

(2.1)
*dist*(*x*,*𝜕***𝔾**^{+}) =⟨*x*,*𝜈*⟩−*d*.

(2.2)
W(*x*) =

��

���^{N}

*i*=1

⟨*X*_{i}(*x*),*𝜈*⟩^{2}.

(2.3)

�**𝔾**^{+}

�∇**𝔾***u*�^{2}*dx*≥*C*_{1}(*𝛽*)�**𝔾**^{+}

W(*x*)^{2}

*dist*(*x*,*𝜕***𝔾**^{+})^{2}�*u*�^{2}*dx*
+*𝛽*�**𝔾**^{+}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})�*u*�^{2}*dx*,

*for all u*∈*C*^{∞}_{0}(**𝔾**^{+})* and where C*_{1}(*𝛽*) ∶= −(*𝛽*^{2}+*𝛽*).

* Remark 2.2* If

**𝔾**has step

*r*=2 , then for

*i*=1,…,

*N*we have the following left- invariant vector fields

where *a*^{s}_{m,i} are the group constants (see, e.g. [7, Formula (2.14)] for the definition).

Also we have *x*∶= (*x*^{�}, *x*^{��}) with *x*^{�}= (*x*^{�}_{1},…, *x*^{�}_{N}) , *x*^{��}= (*x*^{��}_{1},…, *x*^{��}_{N}

2) , and also
*𝜈*∶= (*𝜈*^{�},*𝜈*^{��}) with *𝜈*^{�}= (*𝜈*_{1}^{�},…,*𝜈*_{N}^{�}) and *𝜈*^{��}= (*𝜈*_{1}^{��},…,*𝜈*_{N}^{��}

2).

**Corollary 2.3*** Let ***𝔾**^{+}* be a half*-*space of a stratified group ***𝔾*** of step r*=2.* For all *
*𝛽* ∈**ℝ*** and u*∈*C*_{0}^{∞}(**𝔾**^{+})* we have*

*where C*1(*𝛽*) ∶= −(*𝛽*^{2}+*𝛽*)* and K*(*a*,*𝜈*,*𝛽*) ∶=*𝛽*∑*N*2

*s*=1

∑*N*
*i*=1*a*^{s}_{i,i}*𝜈*_{s}^{��}.

* Proof of Theorem 2.1* To prove inequality (2.3) we use the method of factorization.

Thus, for any *W* ∶= (*W*_{1},…, *W*_{N}), *W*_{i}∈*C*^{1}(**𝔾**^{+}) real-valued, which will be chosen
later, by a simple computation we have

(2.4)
*X*_{i}= *𝜕*

*𝜕x*^{�}_{i} +

*N*_{2}

∑

*s*=1

∑*N*
*m*=1

*a*^{s}_{m,i}*x*^{�}_{m} *𝜕*

*𝜕x*^{��}_{s} ,

(2.5)

�**𝔾**^{+}

|∇**𝔾***u*|^{2}*dx*≥*C*_{1}(*𝛽*)

�**𝔾**^{+}

W(*x*)^{2}

*dist*(*x*,*𝜕***𝔾**^{+})^{2}|*u*|^{2}*dx*
+*K*(*a*,*𝜈*,*𝛽*)�**𝔾**^{+}

|*u*|^{2}
*dist*(*x*,*𝜕***𝔾**^{+})*dx*,

0≤ �**𝔾**^{+}

|∇**𝔾***u*+*𝛽Wu*|^{2}*dx*=

�**𝔾**^{+}

|(*X*_{1}*u*,…, *X*_{N}*u*) +*𝛽*(*W*_{1},…, *W*_{N})*u*|^{2}*dx*

=�**𝔾**^{+}

|(*X*_{1}*u*+*𝛽W*_{1}*u*,…, *X*_{N}*u*+*𝛽W*_{N}*u*)|^{2}*dx*

=�**𝔾**^{+}

∑*N*
*i*=1

|*X*_{i}*u*+*𝛽W*_{i}*u*|^{2}*dx*

=�**𝔾**^{+}

∑*N*
*i*=1

[|*X*_{i}*u*|^{2}+2Re*𝛽W*_{i}*uX*_{i}*u*+*𝛽*^{2}*W*_{i}^{2}|*u*|^{2}]
*dx*

=�**𝔾**^{+}

∑*N*
*i*=1

[|*X*_{i}*u*|^{2}+*𝛽W*_{i}*X*_{i}|*u*|^{2}+*𝛽*^{2}*W*_{i}^{2}|*u*|^{2}]
*dx*

=�**𝔾**^{+}

∑*N*
*i*=1

[|*X*_{i}*u*|^{2}−*𝛽*(*X*_{i}*W*_{i})|*u*|^{2}+*𝛽*^{2}*W*_{i}^{2}|*u*|^{2}]
*dx*.

From the above expression we get the inequality

Let us now take *W*_{i} in the form

where

and

Now *W*_{i}(*x*) can be written as

By a direct computation we have

where

Inserting the expression (2.8) in (2.6) we get

(2.6)

�**𝔾**^{+}

|∇**𝔾***u*|^{2}*dx*≥ �**𝔾**^{+}

∑*N*
*i*=1

[(*𝛽*(*X*_{i}*W*_{i}) −*𝛽*^{2}*W*_{i}^{2})

|*u*|^{2}]
*dx*.

(2.7)
*W*_{i}(*x*) = ⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+}) = ⟨*X*_{i}(*x*),*𝜈*⟩

⟨*x*,*𝜈*⟩−*d*,

*X*_{i}(*x*) = (

*i*

*⏞⏞⏞*

0,…, 1,…, 0, *a*^{(2)}_{i,1}(*x*^{�}),…, *a*^{(r)}_{i,N}

*r*(*x*^{�}, *x*^{(2)},…, *x*^{(r−1)})),

*𝜈*= (*𝜈*1,*𝜈*2,…,*𝜈**r*), *𝜈**j*∈**ℝ**^{N}*j*.

*W*_{i}(*x*) = *𝜈*1,*i*+∑*r*
*l*=2

∑*N*_{l}
*m*=1*a*^{(l)}_{i,m}�

*x*^{�},…, *x*^{(l−1)}�
*𝜈**l*,*m*

∑*r*

*l*=1*x*^{(l)}⋅*𝜈**l*−*d* .

(2.8)
*X*_{i}*W*_{i}(*x*) = *X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩*dist*(*x*,*𝜕***𝔾**^{+}) −⟨*X*_{i}(*x*),*𝜈*⟩*X*_{i}�

*dist*(*x*,*𝜕***𝔾**^{+})�
*dist*(*x*,*𝜕***𝔾**^{+})^{2}

= *X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})− ⟨*X*_{i}(*x*),*𝜈*⟩^{2}
*dist*(*x*,*𝜕***𝔾**^{+})^{2},

*X*_{i}(*dist*(*x*,*𝜕***𝔾**^{+})) =*X*_{i}

� _{N}

�

*k*=1

*x*^{�}_{k}*𝜈*1,*k*+

�*r*
*l*=2

*N*_{l}

�

*m*=1

*x*^{(l)}_{m}*𝜈**l*,*m*−*d*

�

=*𝜈*1,*i*+

�*r*
*l*=2

*N*_{l}

�

*m*=1

*a*^{(l)}_{i,m}�

*x*^{�},…, *x*^{(l−1)}�
*𝜈**l*,*m*

=⟨*X*_{i}(*x*),*𝜈*⟩.

The proof of Theorem 2.1 is finished. ◻
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**.

**Corollary 2.4*** Let ***𝔾**^{+}* be a half*-*space of a stratified group ***𝔾**.* Then we have*

*for all u*∈*C*^{∞}_{0}(**𝔾**^{+}).

* Proof of Corollary 2.4* Let

*x*∶= (

*x*

^{�},

*x*

^{(2)},…,

*x*

^{(r)}) ∈

**𝔾**with

*x*

^{�}= (

*x*

^{�}

_{1},…,

*x*

^{�}

_{N}) and

*x*

^{(j)}∈

**ℝ**

^{N}

*j*,

*j*=2,…,

*r*. By taking

*𝜈*∶= (

*𝜈*

^{�}, 0,…, 0) with

*𝜈*

^{�}= (

*𝜈*

_{1}

^{�},…,

*𝜈*

_{N}

^{�}), we have that

we have

and

Inserting the above expressions in inequality (2.3) we arrive at

For optimisation we differentiate the right-hand side of integral with respect to *𝛽* ,
then we have

which implies

�**𝔾**^{+}

�∇**𝔾***u*�^{2}*dx*≥− (*𝛽*^{2}+*𝛽*)

�**𝔾**^{+}

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈*⟩^{2}
*dist*(*x*,*𝜕***𝔾**^{+})^{2}�*u*�^{2}*dx*
+*𝛽*�**𝔾**^{+}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})�*u*�^{2}*dx*.

(2.9)

�**𝔾**^{+}

|∇**𝔾***u*|^{2}*dx*≥ 1
4�**𝔾**^{+}

|*u*|^{2}
*dist*(*x*,*𝜕***𝔾**^{+})^{2}*dx*,

*X*_{i}(*x*) = (

*i*

*⏞⏞⏞*

0,…, 1,…, 0, *a*^{(2)}_{i,1}(*x*^{�}),…, *a*^{(r)}_{i,N}

*r*(*x*^{�}, *x*^{(2)},…, *x*^{(r−1)}))

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈*⟩^{2}=

�*N*
*i*=1

(*𝜈*_{i}^{�})^{2}=�*𝜈*^{�}�^{2}=1,

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩=*X*_{i}*𝜈*_{i}^{�}=0.

�**𝔾**^{+}

|∇**𝔾***u*|^{2}*dx*≥−(*𝛽*^{2}+*𝛽*)�**𝔾**^{+}

|*u*|^{2}
*dist*(*x*,*𝜕***𝔾**^{+})^{2}*dx*.

−2*𝛽*−1=0,

We complete 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.

**Corollary 2.6*** Let ***ℍ**^{+}= {(*x*_{1}, *x*_{2}, *x*_{3}) ∈**ℍ**|*x*_{3}*>*0}* be a half*-*space of the Heisenberg *
*group ***ℍ**.* Then for any u*∈*C*^{∞}_{0} (**ℍ**^{+})* we have*

*where *∇**ℍ**= {*X*1, *X*2}.

* Proof of Corollary 2.6* Recall that the left-invariant vector fields on the Heisenberg
group are generated by the basis

with the commutator

*𝛽*= −1
2.

(2.10) (

�**𝔾**+

|∇**𝔾***u*|^{2}*dx*
)^{1}_{2}(

�**𝔾**+

*dist*(*x*,*𝜕***𝔾**^{+})^{2}|*u*|^{2}*dx*
)^{1}_{2}

≥ 1
2�**𝔾**+

|*u*|^{2}*dx*

�**𝔾**^{+}

|∇**𝔾***u*|^{2}*dx*

�**𝔾**^{+}

*dist*(*x*,*𝜕***𝔾**^{+})^{2}|*u*|^{2}*dx*

≥ 1
4�**𝔾**^{+}

1

*dist*(*x*,*𝜕***𝔾**^{+})^{2}|*u*|^{2}*dx*�**𝔾**^{+}

*dist*(*x*,*𝜕***𝔾**^{+})^{2}|*u*|^{2}*dx*

≥ 1 4 (

�**𝔾**^{+}

|*u*|^{2}*dx*
)2

.

(2.11)

�**ℍ**^{+}

|∇**ℍ***u*|^{2}*dx*≥ �**ℍ**^{+}

|*x*_{1}|^{2}+|*x*_{2}|^{2}
*x*^{2}_{3} |*u*|^{2}*dx*,

*X*_{1}= *𝜕*

*𝜕x*_{1} +2*x*_{2} *𝜕*

*𝜕x*_{3},
*X*_{2}= *𝜕*

*𝜕x*_{2} −2*x*_{1} *𝜕*

*𝜕x*_{3},

For *x*= (*x*_{1}, *x*_{2}, *x*_{3}) , 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

and

Therefore, with W(*x*) as in (2.2), we have

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 strati-
fied 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

where

The left-invariant vector fields of **𝔼** are generated by the basis
[*X*_{1}, *X*_{2}] = −4 *𝜕*

*𝜕x*_{3}.

*X*1(*x*) = (1, 0, 2*x*2) and *X*2(*x*) = (0, 1,−2*x*1),

⟨*X*_{1}(*x*),*𝜈*⟩=2*x*_{2}, and ⟨*X*_{2}(*x*),*𝜈*⟩= −2*x*_{1},
*X*_{1}⟨*X*_{1}(*x*),*𝜈*⟩=0, and *X*_{2}⟨*X*_{2}(*x*),*𝜈*⟩=0.

W(*x*)^{2}

*dist*(*x*,*𝜕***𝔾**^{+})^{2} =4|*x*1|^{2}+|*x*2|^{2}
*x*^{2}_{3} .

�**ℍ**^{+}

|∇**ℍ***u*|^{2}*dx*≥ �**ℍ**^{+}

|*x*_{1}|^{2}+|*x*_{2}|^{2}
*x*^{2}_{3} |*u*|^{2}*dx*,

*x*◦*y*=(

*x*1+*y*1, *x*2+*y*2, *x*3+*y*3+*P*1, *x*4+*y*4+*P*2

),

*P*_{1}=1
2

(*x*_{1}*y*_{2}−*x*_{2}*y*_{1})
,

*P*_{2}=1
2

(*x*_{1}*y*_{3}−*x*_{3}*y*_{1})
+ 1

12

(*x*^{2}_{1}*y*_{2}−*x*_{1}*y*_{1}(*x*_{2}+*y*_{2}) +*x*_{2}*y*^{2}_{1})
.

*X*_{1}= *𝜕*

*𝜕x*_{1} − *x*_{2}
2

*𝜕*

*𝜕x*_{3} −
(*x*3

2 −*x*_{1}*x*_{2}
12

) *𝜕*

*𝜕x*_{4},
*X*2= *𝜕*

*𝜕x*_{2} + *x*1

2

*𝜕*

*𝜕x*_{3} + *x*^{2}_{1}
12

*𝜕*

*𝜕x*_{4},
*X*3= *𝜕*

*𝜕x*_{3} + *x*_{1}
2

*𝜕*

*𝜕x*_{4},
*X*_{4}= *𝜕*

*𝜕x*_{4}.

**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

(2.12)

�**𝔼**^{+}

�∇**𝔼***u*�^{2}*dx*≥*C*1(*𝛽*)�**𝔼**^{+}

⟨*X*1(*x*),*𝜈*⟩^{2}+⟨*X*2(*x*),*𝜈*⟩^{2}
*dist*(*x*,*𝜕***𝔼**^{+})^{2} �*u*�^{2}*dx*
+*𝛽*

3�**𝔼**^{+}

*x*_{2}*𝜈*4

*dist*(*x*,*𝜕***𝔼**^{+})�*u*�^{2}*dx*,

�**𝔼**^{+}

�∇**𝔼***u*�^{2}*dx*≥ 1
4�**𝔼**^{+}

⟨*X*_{1}(*x*),*𝜈*⟩^{2}+⟨*X*_{2}(*x*),*𝜈*⟩^{2}
*dist*(*x*,*𝜕***𝔼**^{+})^{2} �*u*�^{2}*dx*.

*X*_{1}= *𝜕*

*𝜕x*_{1} − *x*_{2}
2

*𝜕*

*𝜕x*_{3} −
(*x*3

2 −*x*_{1}*x*_{2}
12

) *𝜕*

*𝜕x*_{4},
*X*2= *𝜕*

*𝜕x*_{2} + *x*1

2

*𝜕*

*𝜕x*_{3} + *x*^{2}_{1}
12

*𝜕*

*𝜕x*_{4},

*X*_{3} = [*X*_{1}, *X*_{2}] = *𝜕*

*𝜕x*_{3} +*x*_{1}
2

*𝜕*

*𝜕x*_{4},
*X*_{4} = [*X*_{1}, *X*_{3}] = *𝜕*

*𝜕x*_{4}.

*X*_{1}(*x*) =
(

1, 0,−*x*_{2}
2,−

(*x*3

2 −*x*_{1}*x*_{2}
12

)) ,

*X*_{2}(*x*) =
(

0, 1,*x*_{1}
2,*x*^{2}_{1}

12 )

.

⟨*X*_{1}(*x*),*𝜈*⟩=*𝜈*1− *x*_{2}
2*𝜈*3−

�*x*_{3}
2 − *x*_{1}*x*_{2}

12

�
*𝜈*4,

⟨*X*_{2}(*x*),*𝜈*⟩=*𝜈*2+ *x*_{1}
2*𝜈*3+ *x*^{2}_{1}

12*𝜈*4,
*X*_{1}⟨*X*_{1}(*x*),*𝜈*⟩= *x*_{2}

12*𝜈*4+*x*_{2}

4*𝜈*4= *x*_{2}*𝜈*4

3 ,
*X*_{2}⟨*X*_{2}(*x*),*𝜈*⟩=0.

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

**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

1
*p* +^{1}

*q* =1 recall Young’s inequality

Let us set that

(2.13)
W_{p}(*x*) =

� _{N}

�

*i*=1

�⟨*X*_{i}(*x*),*𝜈*⟩�^{p}

�^{1}_{p}
.

(2.14)

�**𝔾**^{+}

�*N*
*i*=1

�*X*_{i}*u*�^{p}*dx*≥*C*_{2}(*𝛽*, *p*)

�**𝔾**^{+}

W_{p}(*x*)^{p}

*dist*(*x*,*𝜕***𝔾**^{+})^{p}�*u*�^{p}*dx*
+*𝛽*(*p*−1)

�**𝔾**^{+}

�*N*
*i*=1

��⟨*X*_{i}(*x*),*𝜈*⟩�

*dist*(*x*,*𝜕***𝔾**^{+})

�*p*−2

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})�*u*�^{p}*dx*

(2.15)

�**𝔾**^{+}

div_{𝔾}(*fW*)�*u*�^{p}*dx*= −

�**𝔾**^{+}

*fW*⋅∇**𝔾**�*u*�^{p}*dx*

= −*p*�**𝔾**^{+}

*f*⟨*W*,∇**𝔾***u*⟩�*u*�^{p−1}*dx*

≤*p*

�

�**𝔾**+

�⟨*W*,∇**𝔾***u*⟩�^{p}*dx*

�^{1}_{p}�

�**𝔾**+

�*f*�

*p*
*p*−1�*u*�^{p}*dx*

�^{p−1}_{p}
.

*ab*≤ *a*^{p}
*p* + *b*^{q}

*q*, for *a*≥0, *b*≥0.

By using Young’s inequality in (2.15) and rearranging the terms, we arrive at

We choose *W* ∶=*I*_{i} , which has the following form *I*_{i}= (

*i*

*⏞⏞⏞*

0,…, 1,…, 0) and set

Now we calculate

and

We also have

Inserting the above calculations in (2.16) and summing over *i*=1,…, *N* , we arrive
at

We complete the proof of Theorem 2.9. ◻

*a*∶=

�

∫**𝔾**^{+}

�⟨*W*,∇**𝔾***u*⟩�^{p}*dx*

�^{1}

*p*

and *b*∶=

�

∫**𝔾**^{+}

�*f*�

*p*
*p*−1�*u*�^{p}*dx*

�^{p−1}

*p*

.

(2.16)

�**𝔾**^{+}

�⟨*W*,∇**𝔾***u*⟩�^{p}*dx*≥ �**𝔾**^{+}

�

div_{𝔾}(*fW*) − (*p*−1)�*f*�

*p*
*p*−1

�

�*u*�^{p}*dx*.

*f* =*𝛽* �⟨*X*_{i}(*x*),*𝜈*⟩�^{p−1}
*dist*(*x*,*𝜕***𝔾**^{+})^{p−1}.

div_{𝔾}(*Wf*) = (∇**𝔾**⋅*I*_{i})*f* =*X*_{i}*f* =*𝛽X*_{i}

��⟨*X*_{i}(*x*),*𝜈*⟩�

*dist*(*x*,*𝜕***𝔾**^{+})

�*p*−1

=*𝛽*(*p*−1)

��⟨*X*_{i}(*x*),*𝜈*⟩�

*dist*(*x*,*𝜕***𝔾**^{+})

�*p*−2

*X*_{i}

� ⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})

�

=*𝛽*(*p*−1)

��⟨*X*_{i}(*x*),*𝜈*⟩�

*dist*(*x*,*𝜕***𝔾**^{+})

�*p*−2�

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})− �⟨*X*_{i}(*x*),*𝜈*⟩�^{2}
*dist*(*x*,*𝜕***𝔾**^{+})^{2}

�

=*𝛽*(*p*−1)

���⟨*X*_{i}(*x*),*𝜈*⟩�

*dist*(*x*,*𝜕***𝔾**^{+})

�*p*−2�*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})

�

− �⟨*X*_{i}(*x*),*𝜈*⟩�^{p}
*dist*(*x*,*𝜕***𝔾**^{+})^{p}

� ,

�*f*�

*p*
*p*−1 =�*𝛽*�

*p*

*p*−1 �⟨*X*_{i}(*x*),*𝜈*⟩�^{p}
*dist*(*x*,*𝜕***𝔾**^{+})^{p}.

⟨*W*,∇**𝔾***u*⟩= (

*i*

*⏞⏞⏞*

0,…, 1,…, 0)⋅�

*X*_{1}*u*,…, *X*_{i}*u*,…, *X*_{N}*u*�*T*

=*X*_{i}*u*.

(2.17)

�**𝔾**^{+}

�*N*
*i*=1

�*X*_{i}*u*�^{p}*dx*≥−(*p*−1)

�

�*𝛽*�

*p*
*p*−1 +*𝛽*

�

�**𝔾**^{+}

�*N*
*i*=1

�⟨*X*_{i}(*x*),*𝜈*⟩�^{p}
*dist*(*x*,*𝜕***𝔾**^{+})^{p}�*u*�^{p}*dx*
+*𝛽*(*p*−1)

�**𝔾**^{+}

�*N*
*i*=1

��⟨*X*_{i}(*x*),*𝜈*⟩�

*dist*(*x*,*𝜕***𝔾**^{+})

�*p*−2

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})�*u*�^{p}*dx*.

* Remark 2.10* For

*p*≥2 , since

we have the following inequality

**3 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 point
*x̂*∈*𝜕*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 sec-
tion we also have the generalised angle function

with W(*x*) ∶=W_{2}(*x*).

**3.1 Geometric L**^{2}**‑Hardy inequality on a convex domain of 𝔾**

**Theorem 3.1*** 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*).

(2.18)

|∇**𝔾***u*|^{p}=
( _{N}

∑

*i*=1

|*X*_{i}*u*|^{2}
)^{p}_{2}

≥

∑*N*
*i*=1

(|*X*_{i}*u*|^{2})^{p}

2,

(2.19)

�**𝔾**^{+}

�∇**𝔾***u*�^{p}*dx*≥*C*_{2}(*𝛽*, *p*)

�**𝔾**^{+}

W_{p}(*x*)^{p}

*dist*(*x*,*𝜕***𝔾**^{+})^{p}�*u*�^{p}*dx*
+*𝛽*(*p*−1)

�**𝔾**^{+}

�*N*
*i*=1

��⟨*X*_{i}(*x*),*𝜈*⟩�

*dist*(*x*,*𝜕***𝔾**^{+})

�*p*−2

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕***𝔾**^{+})�*u*�^{p}*dx*.

W_{p}(*x*) =

� _{N}

�

*i*=1

�⟨*X*_{i}(*x*),*𝜈*⟩�^{p}

�^{1}_{p}
,

(3.1)

�_{Ω}�∇**𝔾***u*�^{2}*dx*≥*C*_{1}(*𝛽*)

�_{Ω}

W(*x*)^{2}

*dist*(*x*,*𝜕*Ω)^{2}�*u*�^{2}*dx*+*𝛽*�_{Ω}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕*Ω)�*u*�^{2}*dx*

* Proof of Theorem 3.1* We follow the approach of Simon Larson [11] by proving ine-
quality (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 con- structed 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 par- tial 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

Now we choose *W*_{i} in the form

and a direct computation shows that

Inserting the expression (3.3) into inequality (3.2) we get
0≤ �_{Ω}

*j*

�∇**𝔾***u*+*𝛽Wu*�^{2}*dx*=

�_{Ω}

*j*

�*N*
*i*=1

�*X*_{i}*u*+*𝛽W*_{i}*u*�^{2}*dx*

=�_{Ω}

*j*

�*N*
*i*=1

��*X*_{i}*u*�^{2}+2Re*𝛽W*_{i}*uX*_{i}*u*+*𝛽*^{2}*W*_{i}^{2}�*u*�^{2}�
*dx*

=�_{Ω}

*j*

�*N*
*i*=1

��*X*_{i}*u*�^{2}+*𝛽W*_{i}*X*_{i}�*u*�^{2}+*𝛽*^{2}*W*_{i}^{2}�*u*�^{2}�
*dx*

=�_{Ω}

*j*

�*N*
*i*=1

��*X*_{i}*u*�^{2}−*𝛽*(*X*_{i}*W*_{i})�*u*�^{2}+*𝛽*^{2}*W*_{i}^{2}�*u*�^{2}�
*dx*

+*𝛽*�_{𝜕Ω}

*j*

�*N*
*i*=1

*W*_{i}⟨*X*_{i}(*x*), *n*_{j}(*x*)⟩�*u*�^{2}*d*Γ_{𝜕Ω}

*j*(*x*),

(3.2)

�_{Ω}

*j*

�∇**𝔾***u*�^{2}*dx*≥ �_{Ω}

*j*

�*N*
*i*=1

��*𝛽*(*X*_{i}*W*_{i}) −*𝛽*^{2}*W*_{i}^{2}�

�*u*�^{2}�
*dx*

−*𝛽*�_{𝜕Ω}

*j*

�*N*
*i*=1

*W*_{i}⟨*X*_{i}(*x*), *n*_{j}(*x*)⟩�*u*�^{2}*d*Γ_{𝜕Ω}

*j*(*x*).

*W*_{i}(*x*) = ⟨*X*_{i}(*x*),*𝜈**j*⟩

*dist*(*x*,*𝜕*Ω_{j}) = ⟨*X*_{i}(*x*),*𝜈**j*⟩

⟨*x*,*𝜈**j*⟩−*d*,

(3.3)
*X*_{i}*W*_{i}(*x*) = *X*_{i}⟨*X*_{i}(*x*),*𝜈**j*⟩

*dist*(*x*,*𝜕*Ω_{j}) − ⟨*X*_{i}(*x*),*𝜈**j*⟩^{2}
*dist*(*x*,*𝜕*Ω_{j})^{2}.

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

rearranging *x*⋅(*𝜈**j*−*𝜈**l*) −*d*_{j}+*d*_{l}=0 we see that Γ_{jl} is a hyperplane with a nor-
mal *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

(3.4)

�_{Ω}

*j*

�∇**𝔾***u*�^{2}*dx*≥−(*𝛽*^{2}+*𝛽*)

�_{Ω}

*j*

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈**j*⟩^{2}
*dist*(*x*,*𝜕*Ω_{j})^{2}�*u*�^{2}*dx*

+*𝛽*�_{Ω}

*j*

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈**j*⟩

*dist*(*x*,*𝜕*Ω_{j})�*u*�^{2}*dx*−*𝛽*�_{Γ}

*jl*

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈**j*⟩⟨*X*_{i}(*x*), *n*_{jl}⟩

*dist*(*x*,F_{j}) �*u*�^{2}*d*Γ_{jl}.

�_{Ω}�∇**𝔾***u*�^{2}*dx*≥−(*𝛽*^{2}+*𝛽*)�_{Ω}

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈*⟩^{2}
*dist*(*x*,*𝜕*Ω)^{2}�*u*�^{2}*dx*
+*𝛽*�_{Ω}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕*Ω)�*u*�^{2}*dx*

−*𝛽*�

*j*≠*l* �_{Γ}

*jl*

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈**j*⟩⟨*X*_{i}(*x*), *n*_{jl}⟩
*dist*(*x*,F

*j*) �*u*�^{2}*d*Γ_{jl}

= −(*𝛽*^{2}+*𝛽*)

�_{Ω}

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈*⟩^{2}
*dist*(*x*,*𝜕*Ω)^{2}�*u*�^{2}*dx*
+*𝛽*�_{Ω}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕*Ω)�*u*�^{2}*dx*

−*𝛽*�

*j<l* �_{Γ}

*jl*

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈**j*−*𝜈**l*⟩⟨*X*_{i}(*x*), *n*_{jl}⟩

*dist*(*x*,F_{j}) �*u*�^{2}*d*Γ_{jl}.

Γ_{jl}={

*x*∶*x*⋅*𝜈**j*−*d*_{j}=*x*⋅*𝜈**l*−*d*_{l}}

|*𝜈**j*−*𝜈**l*|^{2}= (*𝜈**j*−*𝜈**l*)⋅(*𝜈**j*−*𝜈**l*) =2−2*𝜈**j*⋅*𝜈**l*

=2−2 cos(*𝛼**jl*),

(*𝜈**j*−*𝜈**l*)⋅*n*_{jl}=

√

2−2 cos(*𝛼**jl*),

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 con-
vex 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} ) 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.

**Theorem 3.2*** Let *Ω* be a convex domain of a stratified group ***𝔾**.* Then for 𝛽 <*0* we *
*have*

�_{Ω}�∇**𝔾***u*�^{2}*dx*≥−(*𝛽*^{2}+*𝛽*)

�_{Ω}

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈*⟩^{2}
*dist*(*x*,*𝜕*Ω)^{2}�*u*�^{2}*dx*
+*𝛽*�_{Ω}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈*⟩

*dist*(*x*,*𝜕*Ω)�*u*�^{2}*dx*

−*𝛽*�

*j<l*

�*N*
*i*=1�_{Γ}

*jl*

�

1−cos(*𝛼**jl*)⟨*X*_{i}(*x*), *n*_{jl}⟩^{2}

*dist*(*x*,F_{j}) �*u*�^{2}*d*Γ_{jl}.

�_{Ω}�∇**𝔾***u*�^{2}*dx*=

�_{Ω}_{j}�∇**𝔾***u*�^{2}*dx*

≥−(*𝛽*^{2}+*𝛽*)�_{Ω}

*j*

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈**j*⟩^{2}
*dist*(*x*,*𝜕*Ω_{j})^{2}�*u*�^{2}*dx*

+*𝛽*�_{Ω}

*j*

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈**j*⟩
*dist*(*x*,*𝜕*Ω_{j})�*u*�^{2}*dx*

= −(*𝛽*^{2}+*𝛽*)

�_{Ω}

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈**j*⟩^{2}
*dist*(*x*,*𝜕*Ω_{j})^{2}�*u*�^{2}*dx*
+*𝛽*�_{Ω}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈**j*⟩
*dist*(*x*,*𝜕*Ω_{j})�*u*�^{2}*dx*

≥−(*𝛽*^{2}+*𝛽*)

�_{Ω}

�*N*
*i*=1

⟨*X*_{i}(*x*),*𝜈**j*⟩^{2}
*dist*(*x*,*𝜕*Ω)^{2}�*u*�^{2}*dx*
+*𝛽*�_{Ω}

�*N*
*i*=1

*X*_{i}⟨*X*_{i}(*x*),*𝜈**j*⟩
*dist*(*x*,*𝜕*Ω) �*u*�^{2}*dx*