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 Ruzhansky1,2 · Bolys Sabitbek2,3 · Durvudkhan Suragan4
Received: 6 November 2019 / Accepted: 26 February 2020 / Published online: 2 April 2020
© The Author(s) 2020
Abstract
In this paper we present L2 and Lp 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+∶= {(x1, x2, x3) ∈ℍ,|x3>0} for u∈C∞0 (ℍ+)
Moreover, the geometric Lp-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 L2 and Lp versions of the subelliptic geometric Hardy type inequalities on the half-space. In Sect. 3, we show subelliptic L2 and Lp versions of the geometric Hardy type inequalities on the convex domains.
1.1 Preliminaries
Let 𝔾= (ℝn,◦,𝛿𝜆) be a stratified Lie group (or a homogeneous Carnot group), with dilation structure 𝛿𝜆 and Jacobian generators X1,…, XN , 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 ℝn is the Haar measure for 𝔾 (see, e.g. [8, Proposition 1.6.6]). Each left invariant vector field Xk 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 lth stratum, see also [8, Section 3.1.5] for a general presentation. The horizontal gradient is given by
(1.1)
�Ω|∇u|2dx≥ 1
4�Ω |u|2 dist(x,𝜕Ω)2dx,
(1.2)
�ℍ+
|∇ℍu|2dx≥ �ℍ+
|x1|2+|x2|2 x23 |u|2dx.
(1.3) L=
∑N k=1
X2k.
(1.4) Xk= 𝜕
𝜕x�k +
∑r l=2
Nl
∑
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 fk∈C1(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 L2‑Hardy inequality on the half‑space of 𝔾
In this section we present the geometric L2-Hardy inequality on the half-space of 𝔾 . We define the half-space as follows
where 𝜈∶= (𝜈1,…,𝜈r) with 𝜈j∈ℝNj, 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 ∶= (X1,…, XN),
div𝔾v∶= ∇𝔾⋅v.
(1.5)
∫ΩXkfkdz=
∫𝜕Ωfk⟨Xk, dz⟩.
(1.6)
∫Ω
�N k=1
Xkfkdz=
∫𝜕Ω
�N k=1
fk⟨Xk, dz⟩.
𝔾+∶= {x∈𝔾∶⟨x,𝜈⟩>d},
(2.1) dist(x,𝜕𝔾+) =⟨x,𝜈⟩−d.
(2.2) W(x) =
��
���N
i=1
⟨Xi(x),𝜈⟩2.
(2.3)
�𝔾+
�∇𝔾u�2dx≥C1(𝛽)�𝔾+
W(x)2
dist(x,𝜕𝔾+)2�u�2dx +𝛽�𝔾+
�N i=1
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)�u�2dx,
for all u∈C∞0(𝔾+) and where C1(𝛽) ∶= −(𝛽2+𝛽).
Remark 2.2 If 𝔾 has step r=2 , then for i=1,…, N we have the following left- invariant vector fields
where asm,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�= (𝜈1�,…,𝜈N�) and 𝜈��= (𝜈1��,…,𝜈N��
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∈C0∞(𝔾+) we have
where C1(𝛽) ∶= −(𝛽2+𝛽) and K(a,𝜈,𝛽) ∶=𝛽∑N2
s=1
∑N i=1asi,i𝜈s��.
Proof of Theorem 2.1 To prove inequality (2.3) we use the method of factorization.
Thus, for any W ∶= (W1,…, WN), Wi∈C1(𝔾+) real-valued, which will be chosen later, by a simple computation we have
(2.4) Xi= 𝜕
𝜕x�i +
N2
∑
s=1
∑N m=1
asm,ix�m 𝜕
𝜕x��s ,
(2.5)
�𝔾+
|∇𝔾u|2dx≥C1(𝛽)
�𝔾+
W(x)2
dist(x,𝜕𝔾+)2|u|2dx +K(a,𝜈,𝛽)�𝔾+
|u|2 dist(x,𝜕𝔾+)dx,
0≤ �𝔾+
|∇𝔾u+𝛽Wu|2dx=
�𝔾+
|(X1u,…, XNu) +𝛽(W1,…, WN)u|2dx
=�𝔾+
|(X1u+𝛽W1u,…, XNu+𝛽WNu)|2dx
=�𝔾+
∑N i=1
|Xiu+𝛽Wiu|2dx
=�𝔾+
∑N i=1
[|Xiu|2+2Re𝛽WiuXiu+𝛽2Wi2|u|2] dx
=�𝔾+
∑N i=1
[|Xiu|2+𝛽WiXi|u|2+𝛽2Wi2|u|2] dx
=�𝔾+
∑N i=1
[|Xiu|2−𝛽(XiWi)|u|2+𝛽2Wi2|u|2] dx.
From the above expression we get the inequality
Let us now take Wi in the form
where
and
Now Wi(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+
|∇𝔾u|2dx≥ �𝔾+
∑N i=1
[(𝛽(XiWi) −𝛽2Wi2)
|u|2] dx.
(2.7) Wi(x) = ⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+) = ⟨Xi(x),𝜈⟩
⟨x,𝜈⟩−d,
Xi(x) = (
i
⏞⏞⏞
0,…, 1,…, 0, a(2)i,1(x�),…, a(r)i,N
r(x�, x(2),…, x(r−1))),
𝜈= (𝜈1,𝜈2,…,𝜈r), 𝜈j∈ℝNj.
Wi(x) = 𝜈1,i+∑r l=2
∑Nl m=1a(l)i,m�
x�,…, x(l−1)� 𝜈l,m
∑r
l=1x(l)⋅𝜈l−d .
(2.8) XiWi(x) = Xi⟨Xi(x),𝜈⟩dist(x,𝜕𝔾+) −⟨Xi(x),𝜈⟩Xi�
dist(x,𝜕𝔾+)� dist(x,𝜕𝔾+)2
= Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)− ⟨Xi(x),𝜈⟩2 dist(x,𝜕𝔾+)2,
Xi(dist(x,𝜕𝔾+)) =Xi
� N
�
k=1
x�k𝜈1,k+
�r l=2
Nl
�
m=1
x(l)m𝜈l,m−d
�
=𝜈1,i+
�r l=2
Nl
�
m=1
a(l)i,m�
x�,…, x(l−1)� 𝜈l,m
=⟨Xi(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)∈ℝNj, j=2,…, r . By taking 𝜈∶= (𝜈�, 0,…, 0) with 𝜈�= (𝜈1�,…,𝜈N�), 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+
�∇𝔾u�2dx≥− (𝛽2+𝛽)
�𝔾+
�N i=1
⟨Xi(x),𝜈⟩2 dist(x,𝜕𝔾+)2�u�2dx +𝛽�𝔾+
�N i=1
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)�u�2dx.
(2.9)
�𝔾+
|∇𝔾u|2dx≥ 1 4�𝔾+
|u|2 dist(x,𝜕𝔾+)2dx,
Xi(x) = (
i
⏞⏞⏞
0,…, 1,…, 0, a(2)i,1(x�),…, a(r)i,N
r(x�, x(2),…, x(r−1)))
�N i=1
⟨Xi(x),𝜈⟩2=
�N i=1
(𝜈i�)2=�𝜈��2=1,
Xi⟨Xi(x),𝜈⟩=Xi𝜈i�=0.
�𝔾+
|∇𝔾u|2dx≥−(𝛽2+𝛽)�𝔾+
|u|2 dist(x,𝜕𝔾+)2dx.
−2𝛽−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+= {(x1, x2, x3) ∈ℍ|x3>0} be a half-space of the Heisenberg group ℍ. Then for any u∈C∞0 (ℍ+) we have
where ∇ℍ= {X1, X2}.
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+
|∇𝔾u|2dx )12(
�𝔾+
dist(x,𝜕𝔾+)2|u|2dx )12
≥ 1 2�𝔾+
|u|2dx
�𝔾+
|∇𝔾u|2dx
�𝔾+
dist(x,𝜕𝔾+)2|u|2dx
≥ 1 4�𝔾+
1
dist(x,𝜕𝔾+)2|u|2dx�𝔾+
dist(x,𝜕𝔾+)2|u|2dx
≥ 1 4 (
�𝔾+
|u|2dx )2
.
(2.11)
�ℍ+
|∇ℍu|2dx≥ �ℍ+
|x1|2+|x2|2 x23 |u|2dx,
X1= 𝜕
𝜕x1 +2x2 𝜕
𝜕x3, X2= 𝜕
𝜕x2 −2x1 𝜕
𝜕x3,
For x= (x1, x2, x3) , choosing 𝜈= (0, 0, 1) as the unit vector in the direction of x3 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 ℝ4 with the group law of 𝔼 , which is given by
where
The left-invariant vector fields of 𝔼 are generated by the basis [X1, X2] = −4 𝜕
𝜕x3.
X1(x) = (1, 0, 2x2) and X2(x) = (0, 1,−2x1),
⟨X1(x),𝜈⟩=2x2, and ⟨X2(x),𝜈⟩= −2x1, X1⟨X1(x),𝜈⟩=0, and X2⟨X2(x),𝜈⟩=0.
W(x)2
dist(x,𝜕𝔾+)2 =4|x1|2+|x2|2 x23 .
�ℍ+
|∇ℍu|2dx≥ �ℍ+
|x1|2+|x2|2 x23 |u|2dx,
x◦y=(
x1+y1, x2+y2, x3+y3+P1, x4+y4+P2
),
P1=1 2
(x1y2−x2y1) ,
P2=1 2
(x1y3−x3y1) + 1
12
(x21y2−x1y1(x2+y2) +x2y21) .
X1= 𝜕
𝜕x1 − x2 2
𝜕
𝜕x3 − (x3
2 −x1x2 12
) 𝜕
𝜕x4, X2= 𝜕
𝜕x2 + x1
2
𝜕
𝜕x3 + x21 12
𝜕
𝜕x4, X3= 𝜕
𝜕x3 + x1 2
𝜕
𝜕x4, X4= 𝜕
𝜕x4.
Corollary 2.7 Let 𝔼+= {x∶= (x1, x2, x3, x4) ∈𝔼� ⟨x,𝜈⟩>0} be a half-space of the Engel group 𝔼. Then for all 𝛽 ∈ℝ and u∈C0∞(𝔼+) we have
where ∇𝔼= {X1, X2}, 𝜈∶= (𝜈1,𝜈2,𝜈3,𝜈4), and C1(𝛽) = −(𝛽2+𝛽).
Remark 2.8 If we take 𝜈4=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+
�∇𝔼u�2dx≥C1(𝛽)�𝔼+
⟨X1(x),𝜈⟩2+⟨X2(x),𝜈⟩2 dist(x,𝜕𝔼+)2 �u�2dx +𝛽
3�𝔼+
x2𝜈4
dist(x,𝜕𝔼+)�u�2dx,
�𝔼+
�∇𝔼u�2dx≥ 1 4�𝔼+
⟨X1(x),𝜈⟩2+⟨X2(x),𝜈⟩2 dist(x,𝜕𝔼+)2 �u�2dx.
X1= 𝜕
𝜕x1 − x2 2
𝜕
𝜕x3 − (x3
2 −x1x2 12
) 𝜕
𝜕x4, X2= 𝜕
𝜕x2 + x1
2
𝜕
𝜕x3 + x21 12
𝜕
𝜕x4,
X3 = [X1, X2] = 𝜕
𝜕x3 +x1 2
𝜕
𝜕x4, X4 = [X1, X3] = 𝜕
𝜕x4.
X1(x) = (
1, 0,−x2 2,−
(x3
2 −x1x2 12
)) ,
X2(x) = (
0, 1,x1 2,x21
12 )
.
⟨X1(x),𝜈⟩=𝜈1− x2 2𝜈3−
�x3 2 − x1x2
12
� 𝜈4,
⟨X2(x),𝜈⟩=𝜈2+ x1 2𝜈3+ x21
12𝜈4, X1⟨X1(x),𝜈⟩= x2
12𝜈4+x2
4𝜈4= x2𝜈4
3 , X2⟨X2(x),𝜈⟩=0.
Now substituting these into inequality (2.3) we obtain the desired result. ◻
2.2 Lp‑Hardy inequality on 𝔾+
Here we construct an Lp 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 Wp , 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 C2(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 ∈C1(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) Wp(x) =
� N
�
i=1
�⟨Xi(x),𝜈⟩�p
�1p .
(2.14)
�𝔾+
�N i=1
�Xiu�pdx≥C2(𝛽, p)
�𝔾+
Wp(x)p
dist(x,𝜕𝔾+)p�u�pdx +𝛽(p−1)
�𝔾+
�N i=1
��⟨Xi(x),𝜈⟩�
dist(x,𝜕𝔾+)
�p−2
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)�u�pdx
(2.15)
�𝔾+
div𝔾(fW)�u�pdx= −
�𝔾+
fW⋅∇𝔾�u�pdx
= −p�𝔾+
f⟨W,∇𝔾u⟩�u�p−1dx
≤p
�
�𝔾+
�⟨W,∇𝔾u⟩�pdx
�1p�
�𝔾+
�f�
p p−1�u�pdx
�p−1p .
ab≤ ap p + bq
q, for a≥0, b≥0.
By using Young’s inequality in (2.15) and rearranging the terms, we arrive at
We choose W ∶=Ii , which has the following form Ii= (
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,∇𝔾u⟩�pdx
�1
p
and b∶=
�
∫𝔾+
�f�
p p−1�u�pdx
�p−1
p
.
(2.16)
�𝔾+
�⟨W,∇𝔾u⟩�pdx≥ �𝔾+
�
div𝔾(fW) − (p−1)�f�
p p−1
�
�u�pdx.
f =𝛽 �⟨Xi(x),𝜈⟩�p−1 dist(x,𝜕𝔾+)p−1.
div𝔾(Wf) = (∇𝔾⋅Ii)f =Xif =𝛽Xi
��⟨Xi(x),𝜈⟩�
dist(x,𝜕𝔾+)
�p−1
=𝛽(p−1)
��⟨Xi(x),𝜈⟩�
dist(x,𝜕𝔾+)
�p−2
Xi
� ⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)
�
=𝛽(p−1)
��⟨Xi(x),𝜈⟩�
dist(x,𝜕𝔾+)
�p−2�
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)− �⟨Xi(x),𝜈⟩�2 dist(x,𝜕𝔾+)2
�
=𝛽(p−1)
���⟨Xi(x),𝜈⟩�
dist(x,𝜕𝔾+)
�p−2�Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)
�
− �⟨Xi(x),𝜈⟩�p dist(x,𝜕𝔾+)p
� ,
�f�
p p−1 =�𝛽�
p
p−1 �⟨Xi(x),𝜈⟩�p dist(x,𝜕𝔾+)p.
⟨W,∇𝔾u⟩= (
i
⏞⏞⏞
0,…, 1,…, 0)⋅�
X1u,…, Xiu,…, XNu�T
=Xiu.
(2.17)
�𝔾+
�N i=1
�Xiu�pdx≥−(p−1)
�
�𝛽�
p p−1 +𝛽
�
�𝔾+
�N i=1
�⟨Xi(x),𝜈⟩�p dist(x,𝜕𝔾+)p�u�pdx +𝛽(p−1)
�𝔾+
�N i=1
��⟨Xi(x),𝜈⟩�
dist(x,𝜕𝔾+)
�p−2
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)�u�pdx.
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) ∶=W2(x).
3.1 Geometric L2‑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 C1(𝛽) ∶= −(𝛽2+𝛽).
(2.18)
|∇𝔾u|p= ( N
∑
i=1
|Xiu|2 )p2
≥
∑N i=1
(|Xiu|2)p
2,
(2.19)
�𝔾+
�∇𝔾u�pdx≥C2(𝛽, p)
�𝔾+
Wp(x)p
dist(x,𝜕𝔾+)p�u�pdx +𝛽(p−1)
�𝔾+
�N i=1
��⟨Xi(x),𝜈⟩�
dist(x,𝜕𝔾+)
�p−2
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕𝔾+)�u�pdx.
Wp(x) =
� N
�
i=1
�⟨Xi(x),𝜈⟩�p
�1p ,
(3.1)
�Ω�∇𝔾u�2dx≥C1(𝛽)
�Ω
W(x)2
dist(x,𝜕Ω)2�u�2dx+𝛽�Ω
�N i=1
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕Ω)�u�2dx
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 {Fj}j and unit normals of these facets by {𝜈j}j , which are directed inward. Then Ω can be con- structed by the union of the disjoint sets Ωj∶= {x∈ Ω ∶dist(x,𝜕Ω) =dist(x,Fj)} . Now we apply the same method as in the case of the half-space 𝔾+ for each element Ωj 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 nj is the unit normal of 𝜕Ωj which is directed outward. Since Fj⊂ 𝜕Ωj we have nj= −𝜈j.
The boundary terms on 𝜕Ω vanish since u is compactly supported in Ω . So we only deal with the parts of 𝜕Ωj in Ω . Note that for every facet of 𝜕Ωj there exists some 𝜕Ωl which shares this facet. We denote by Γjl the common facet of 𝜕Ωj and
𝜕Ωl , with nk|Γjl = −nl|Γjl . From the above expression we get the following inequality
Now we choose Wi in the form
and a direct computation shows that
Inserting the expression (3.3) into inequality (3.2) we get 0≤ �Ω
j
�∇𝔾u+𝛽Wu�2dx=
�Ω
j
�N i=1
�Xiu+𝛽Wiu�2dx
=�Ω
j
�N i=1
��Xiu�2+2Re𝛽WiuXiu+𝛽2Wi2�u�2� dx
=�Ω
j
�N i=1
��Xiu�2+𝛽WiXi�u�2+𝛽2Wi2�u�2� dx
=�Ω
j
�N i=1
��Xiu�2−𝛽(XiWi)�u�2+𝛽2Wi2�u�2� dx
+𝛽�𝜕Ω
j
�N i=1
Wi⟨Xi(x), nj(x)⟩�u�2dΓ𝜕Ω
j(x),
(3.2)
�Ω
j
�∇𝔾u�2dx≥ �Ω
j
�N i=1
��𝛽(XiWi) −𝛽2Wi2�
�u�2� dx
−𝛽�𝜕Ω
j
�N i=1
Wi⟨Xi(x), nj(x)⟩�u�2dΓ𝜕Ω
j(x).
Wi(x) = ⟨Xi(x),𝜈j⟩
dist(x,𝜕Ωj) = ⟨Xi(x),𝜈j⟩
⟨x,𝜈j⟩−d,
(3.3) XiWi(x) = Xi⟨Xi(x),𝜈j⟩
dist(x,𝜕Ωj) − ⟨Xi(x),𝜈j⟩2 dist(x,𝜕Ωj)2.
Now we sum over all partition elements Ωj and let njl=nk|Γjl , i.e. the unit normal of Γjl pointing from Ωj into Ωl . Then we get
Here we used the fact that (by the definition) Γjl is a set with dist(x,Fj) =dist(x,Fl) . From
rearranging x⋅(𝜈j−𝜈l) −dj+dl=0 we see that Γjl is a hyperplane with a nor- mal 𝜈j−𝜈l . Thus, 𝜈j−𝜈l is parallel to njl and one only needs to check that (𝜈j−𝜈l)⋅njl>0 . Observe that njl points out and 𝜈j points into jth partition element, so 𝜈j⋅njl is non-negative. Similarly, we see that 𝜈l⋅njl is non-positive. This means we have (𝜈j−𝜈l)⋅njl>0 . In addition, it is easy to see that
which implies that
where 𝛼jl is the angle between 𝜈j and 𝜈l . So we obtain
(3.4)
�Ω
j
�∇𝔾u�2dx≥−(𝛽2+𝛽)
�Ω
j
�N i=1
⟨Xi(x),𝜈j⟩2 dist(x,𝜕Ωj)2�u�2dx
+𝛽�Ω
j
�N i=1
Xi⟨Xi(x),𝜈j⟩
dist(x,𝜕Ωj)�u�2dx−𝛽�Γ
jl
�N i=1
⟨Xi(x),𝜈j⟩⟨Xi(x), njl⟩
dist(x,Fj) �u�2dΓjl.
�Ω�∇𝔾u�2dx≥−(𝛽2+𝛽)�Ω
�N i=1
⟨Xi(x),𝜈⟩2 dist(x,𝜕Ω)2�u�2dx +𝛽�Ω
�N i=1
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕Ω)�u�2dx
−𝛽�
j≠l �Γ
jl
�N i=1
⟨Xi(x),𝜈j⟩⟨Xi(x), njl⟩ dist(x,F
j) �u�2dΓjl
= −(𝛽2+𝛽)
�Ω
�N i=1
⟨Xi(x),𝜈⟩2 dist(x,𝜕Ω)2�u�2dx +𝛽�Ω
�N i=1
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕Ω)�u�2dx
−𝛽�
j<l �Γ
jl
�N i=1
⟨Xi(x),𝜈j−𝜈l⟩⟨Xi(x), njl⟩
dist(x,Fj) �u�2dΓjl.
Γjl={
x∶x⋅𝜈j−dj=x⋅𝜈l−dl}
|𝜈j−𝜈l|2= (𝜈j−𝜈l)⋅(𝜈j−𝜈l) =2−2𝜈j⋅𝜈l
=2−2 cos(𝛼jl),
(𝜈j−𝜈l)⋅njl=
√
2−2 cos(𝛼jl),
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 {Ωj}∞j=1 such that u∈C∞0 (Ω1),Ωj⊂Ω and Ωj→Ω as j→∞ . Assume that 𝜈j(x) is the above map 𝜈 (corresponding to Ωj ) we compute
Now we obtain the desired result when j→∞ . ◻
3.2 Lp‑Hardy’s inequality on a convex domain of 𝔾
In this section we give the Lp-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�∇𝔾u�2dx≥−(𝛽2+𝛽)
�Ω
�N i=1
⟨Xi(x),𝜈⟩2 dist(x,𝜕Ω)2�u�2dx +𝛽�Ω
�N i=1
Xi⟨Xi(x),𝜈⟩
dist(x,𝜕Ω)�u�2dx
−𝛽�
j<l
�N i=1�Γ
jl
�
1−cos(𝛼jl)⟨Xi(x), njl⟩2
dist(x,Fj) �u�2dΓjl.
�Ω�∇𝔾u�2dx=
�Ωj�∇𝔾u�2dx
≥−(𝛽2+𝛽)�Ω
j
�N i=1
⟨Xi(x),𝜈j⟩2 dist(x,𝜕Ωj)2�u�2dx
+𝛽�Ω
j
�N i=1
Xi⟨Xi(x),𝜈j⟩ dist(x,𝜕Ωj)�u�2dx
= −(𝛽2+𝛽)
�Ω
�N i=1
⟨Xi(x),𝜈j⟩2 dist(x,𝜕Ωj)2�u�2dx +𝛽�Ω
�N i=1
Xi⟨Xi(x),𝜈j⟩ dist(x,𝜕Ωj)�u�2dx
≥−(𝛽2+𝛽)
�Ω
�N i=1
⟨Xi(x),𝜈j⟩2 dist(x,𝜕Ω)2�u�2dx +𝛽�Ω
�N i=1
Xi⟨Xi(x),𝜈j⟩ dist(x,𝜕Ω) �u�2dx