Hardy and Rellich inequalities for anisotropic p-sub-Laplacians

HARDY AND RELLICH INEQUALITIES FOR ANISOTROPIC p-SUB-LAPLACIANS

M. RUZHANSKY,¹ B. SABITBEK,²and D. SURAGAN^3∗

Abstract. In this paper we establish the subelliptic Picone type identities.

As consequences, we obtain Hardy and Rellich type inequalities for anisotropic p-sub-Laplacians which are operators of the form

Lpf :=

N

X

i=1

X_i |Xif|^pi−2X_if

, 1< p_i <∞,

whereXi, i= 1, . . . , N, are the generators of the first stratum of a stratified (Lie) group. Moreover, analogues of Hardy type inequalities with multiple sin- gularities and many-particle Hardy type inequalities are obtained on stratified groups.

1. Introduction

1.1. Historical background. Recall the anisotropic Laplacian onR^N forpi >1 where i= 1, . . . , N (see [5]), defined by

N

X

i=1

∂

∂xi

∂u

∂xi

pi−2

∂u

∂xi

!

, (1.1)

which has recently attracted considerable attention. For instance, by takingp_i = 2 or p_i = p = const (see [4]) in (1.1) we get the Laplacian and the pseudo-p- Laplacian, respectively. The anisotropic Laplacian has the theoretical importance not only in mathematics, but also many practical applications in the natural sciences. There are several examples: it reflects anisotropic physical properties of some reinforced materials (Lions [8] and Tang [16]), as well as explains the dynamics of fluids in the anisotropic media when the conductivities of the media are different in each direction [2] and [3]. It has also applications in the image processing [17].

The main purpose of this paper is to obtain the subelliptic Hardy and Rellich type inequalities for the anisotropic p-sub-Laplacian on stratified groups. First, we derive the subelliptic Picone type identities on stratified groups. As conse- quences, Hardy and Rellich type inequalities for anisotropic sub-Laplacians are presented. These results are given in Sections 2 and 3. In Section 4 and 5, we present analogues of Hardy type inequalities with multiple singularities and

Copyright 2018 by the Tusi Mathematical Research Group.

Date: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz.

∗Corresponding author.

2010Mathematics Subject Classification. Primary 35A23; Secondary 35H20.

Key words and phrases. stratified group, anisotropicp-sub-Laplacian, Hardy inequality, Rel- lich inequality, Picone identity.

1

many-particle Hardy type inequalities on stratified groups, respectively. These inequalities are obtained in their horizontal form in the spirit of [13] (see also [14]). In Section 6, Hardy type inequalities with exponential weights are shown.

1.2. Preliminaries. Let G = (R^d,◦, δ_λ) be a stratified Lie group (or a ho- mogeneous Carnot group), with dilation structure δ_λ and Jacobian generators X1, . . . , XN, so that N is the dimension of the first stratum of G. We denote by Q the homogeneous dimension ofG. We refer to [11,12] and to the recent book [15] for extensive discussions of stratified Lie groups and their properties.

The sub-Laplacian on G is given by L=

N

X

k=1

X_k². (1.2)

We also recall that the standard Lebesgue measuredxonRⁿis the Haar measure for G (see, e.g. [15]). The left invariant vector field Xj has an explicit form and satisfies the divergence theorem, see e.g. [15] for the derivation of the exact formula: more precisely, we can formulate

X_k= ∂

∂x⁰_k +

r

X

l=2 Nl

X

m=1

a^(l)_k,m(x⁰, ..., x^(l−1)) ∂

∂x^(l)m

, (1.3)

with x= (x⁰, x⁽²⁾, . . . , x^(r)), where r is the step ofG and x^(l) = (x^(l)₁ , . . . , x^(l)_N

l) are the variables in thel^th stratum anda^(l)_k,mis aδλ-homogeneous polynomial function of degreel−1. The horizontal gradient is given by

∇_G := (X₁, . . . , X_N), and the horizontal divergence is defined by

div_Gv :=∇_G·v.

The horizontal anisotropicp-sub-Laplacian is defined by Lpf :=

N

X

i=1

Xi |Xif|^pi−2Xif

, 1< pi <∞, (1.4) and we use the notation

|x⁰|= q

x⁰²₁ +. . .+x⁰²_N for the Euclidean norm onR^N.

2. Subelliptic anisotropic Hardy type inequality

First, we obtain the subelliptic Picone type identity on a stratified group G. Here we follow the method of Allegretto and Huang [1] for the (Euclidean) p- Laplacian (see also [10]).

Lemma 2.1. Let Ω ⊂ G be an open set, where G is a stratified group with N being the dimension of the first stratum. Letu, v be differentiable a.e. inΩ, v >0 a.e. in Ω and u≥0, and denote

R(u, v) :=

N

X

i=1

|X_iu|^pi−

N

X

i=1

X_i u^pi

v^pi−1

|X_iv|^pi−2X_iv,

L(u, v) :=

N

X

i=1

|X_iu|^pi −

N

X

i=1

p_iu^pi−1

v^pi−1 |X_iv|^pi−2X_ivX_iu +

N

X

i=1

(pi−1)u^pi

v^pi |Xiv|^pi, where p_i >1, i= 1, . . . , N. Then

L(u, v) =R(u, v)≥0. (2.1)

In addition, let Ωbe connected, then we have L(u, v) = 0 a.e. in Ω if and only if u=cv a.e. in Ω with a positive constant c.

Note that the Euclidean case of this lemma was obtained by Feng and Cui [5].

Proof of Lemma 2.1. A direct computation gives R(u, v) =

N

X

i=1

|X_iu|^pi−

N

X

i=1

X_i u^pi

v^pi−1

|X_iv|^pi−2X_iv

=

N

X

i=1

|X_iu|^pi−

N

X

i=1

p_iu^pi−1

v^pi−1|X_iv|^pi−2X_ivX_iu+

N

X

i=1

(p_i−1)u^pi

v^pi|X_iv|^pi

=L(u, v).

This proves the equality in (2.1). Now we rewrite L(u, v) to see L(u, v)≥0, that is,

L(u, v) =

N

X

i=1

|X_iu|^pi−

N

X

i=1

p_iu^pi−1

v^pi−1|X_iv|^pi−1|X_iu|+

N

X

i=1

(p_i−1)u^pi

v^pi|X_iv|^pi +

N

X

i=1

p_iu^pi−1

v^pi−1|X_iv|^pi−2(|X_iv||X_iu| −X_ivX_iu)

=S₁ +S₂, where we denote

S₁ :=

N

X

i=1

p_i

"

1

p_i|X_iu|^pi +pi−1 p_i

u

v|X_iv|pi−1_pi^pi₋₁#

−

N

X

i=1

p_iu^pi−1

v^pi−1|X_iv|^pi−1|X_iu|,

and

S₂ :=

N

X

i=1

p_iu^pi−1

v^pi−1|X_iv|^pi−2(|X_iv||X_iu| −X_ivX_iu).

We can see that S2 ≥0 due to |Xiv||Xiu| ≥XivXiu. To check S1 ≥ 0 we need to use Young’s inequality fora ≥0 and b ≥0

ab≤ a^pi p_i + b^qi

q_i , (2.2)

where p_i > 1, q_i > 1 and _p¹

i + _q¹

i = 1 for i = 1, . . . , N. It holds if and only if a^pi =b^qi, i.e. if a =b^pi1−1. Let us takea =|X_iu| and b= ^u_v|X_iv|pi−1

in (2.2) to get

p_i|X_iu|u

v|X_iv|pi−1

≤p_i

"

1

p_i|X_iu|^pi+ p_i−1 p_i

u

v|X_iv|pi−1_pi^pi₋₁#

. (2.3) From this we see that S₁ ≥ 0 which proves that L(u, v) = S₁ +S₂ ≥ 0. It is easy to see thatu=cv implies R(u, v) = 0 since in the case R(cv, v) both terms cancel each other out. Now let us prove that L(u, v) = 0 implies u=cv. Due to u(x)≥0 and L(u, v)(x₀) = 0, x₀ ∈Ω, we consider the two casesu(x₀)>0 and u(x₀) = 0.

(1) For the caseu(x₀)>0 we conclude from L(u, v)(x₀) = 0 that S₁ = 0 and S₂ = 0. Then S₁ = 0 implies

|Xiu|= u

v|Xiv|, i= 1, . . . , N, (2.4) and S₂ = 0 implies

|X_iv||X_iu| −X_ivX_iu= 0, i= 1, . . . , N. (2.5) The combination of (2.4) and (2.5) gives

X_iu X_iv = u

v =c, with c6= 0, i= 1, . . . , N. (2.6) (2) Let us denote Ω^∗ := {x ∈ Ω|u(x) = 0}. If Ω^∗ 6= Ω, then suppose that x₀ ∈ ∂Ω^∗. Then there exists a sequence x_k ∈/ Ω^∗ such that x_k → x₀. In particular, u(x_k) 6= 0, and hence by the case 1 we have u(x_k) = cv(x_k).

Passing to the limit we get u(x₀) =cv(x₀). Since u(x₀) = 0, v(x₀)6= 0, we get thatc= 0. But then by the case 1 again, sinceu=cvand u6= 0 in Ω\Ω^∗, it is impossible thatc= 0. This contradiction implies that Ω^∗ = Ω.

This completes the proof of Lemma 2.1.

As a consequence of the subelliptic Picone type identity, we present the Hardy type inequality for the anisotropic sub-Laplacian onG. We recall that forx∈G we write x= (x⁰, x⁰⁰), withx⁰ being in the first stratum of G.

Theorem 2.2. Let Ω⊂G\{x⁰ = 0} be an open set, whereG is a stratified group with N being the dimension of the first stratum. Then we have

N

X

i=1

Z

Ω

|X_iu|^pidx≥

N

X

i=1

p_i−1 p_i

piZ

Ω

|u|^pi

|x⁰_i|^pidx, (2.7) for all u∈C¹(Ω) and where 1< p_i < N for i= 1, . . . , N.

Before we start the proof of Theorem2.2, let us establish the following Lemma 2.3.

Lemma 2.3. Let Ω ⊂ G be an open set, where G is a stratified group with N being the dimension of the first stratum. Let constants K_i > 0 and functions H_i(x) with i = 1, . . . , N, be such that for an a.e. differentiable function v, such that v >0 a.e. in Ω, we have

−X_i(|X_iv|^pi−2X_iv)≥K_iH_i(x)v^pi−1, i= 1, . . . , N. (2.8) Then, for all nonnegative functions u∈C¹(Ω) we have

N

X

i=1

Z

Ω

|X_iu|^pidx≥

N

X

i=1

K_i Z

Ω

H_i(x)u^pidx. (2.9) Proof of Lemma 2.3. In view of (2.1) and (2.8) we have

0≤ Z

Ω

L(u, v)dx= Z

Ω

R(u, v)dx

=

N

X

i=1

Z

Ω

|X_iu|^pidx−

N

X

i=1

Z

Ω

X_i u^pi

v^pi−1

|X_iv|^pi−2X_ivdx

=

N

X

i=1

Z

Ω

|X_iu|^pidx+

N

X

i=1

Z

Ω

u^pi

v^pi−1X_i |X_iv|^pi−2X_iv dx

≤

N

X

i=1

Z

Ω

|X_iu|^pidx−

N

X

i=1

K_i Z

Ω

H_i(x)u^pidx.

This completes the proof of Lemma 2.3.

Proof of Theorem 2.2. Before using Lemma2.3, we shall introduce the auxiliary function

v :=

N

Y

j=1

|x⁰_j|^αj =|x⁰_i|^αiV_i, (2.10) where V_i =QN

j=1,j6=i|x⁰_j|^αj and α_j = ^pj_p⁻¹

j . Then we have X_iv =α_iV_i|x⁰_i|^αi−2x⁰_i,

|X_iv|^pi−2 =α^p_iⁱ⁻²V_i^pi−2|x⁰_i|^{αipi−2αi−pi+2},

|X_iv|^pi−2X_iv =α^p_iⁱ⁻¹V_i^pi−1|x⁰_i|^{αipi−αi−pi}x⁰_i.

Consequently, we also have

−X_i(|X_iv|^pi−2X_iv) =

p_i−1 p_i

pi

v^pi−1

|x⁰_i|^pi. (2.11) To complete the proof of Theorem 2.2, we choose K_i =

pi−1 pi

pi

and H_i(x) =

1

|x⁰_i|^pi, and use Lemma2.3.

3. Subelliptic anisotropic Rellich type inequality

We now present the (second order) subelliptic Picone type identity. As a con- sequence, we obtain the Rellich type inequality for the anisotropic sub-Laplacian onG.

Lemma 3.1. Let Ω ⊂ G be an open set, where G is a stratified group with N being the dimension of the first stratum. Let u, v be twice differentiable a.e. in Ω and satisfying the following conditions: u ≥ 0, v > 0, X_i²v < 0 a.e. in Ω for p_i >1, i= 1, . . . , N. Then we have

L₁(u, v) =R₁(u, v)≥0, (3.1) where

R₁(u, v) :=

N

X

i=1

|X_i²u|^pi −

N

X

i=1

X_i² u^pi

v^pi−1

|X_i²v|^pi−2X_i²v,

and

L₁(u, v) :=

N

X

i=1

|X_i²u|^pi −

N

X

i=1

p_iu v

pi−1

X_i²uX_i²v|X_i²v|^pi−2

+

N

X

i=1

(pi −1) u

v pi

|X_i²v|^pi

−

N

X

i=1

p_i(p_i−1)u^pi−2

v^pi−1|X_i²v|^pi−2X_i²v

X_iu− u vX_iv2

.

Proof of Lemma 3.1. A direct computation gives X_i²

u^pi v^pi−1

=Xi

pi

u^pi−1

v^pi−1Xiu−(pi−1)u^pi v^piXiv

=pi(pi−1)u^pi−2 v^pi−2

(X_iu)v−u(X_iv) v²

Xiu+pi

u^pi−1 v^pi−1X_i²u

−p_i(p_i−1)u^pi−1 v^pi−1

(Xiu)v−u(Xiv) v²

X_iv−(p_i−1)u^pi v^piX_i²v

=p_i(p_i−1)

u^pi−2

v^pi−1|X_iu|²−2u^pi−1

v^pi X_ivX_iu+ u^pi

v^pi+1|X_iv|²

+pi

u^pi−1

v^pi−1X_i²u−(pi−1)u^pi v^piX_i²v

=p_i(p_i−1)u^pi−2 v^pi−1

X_iu− u vX_iv2

+p_iu^pi−1

v^pi−1X_i²u−(p_i−1)u^pi v^piX_i²v, which gives (3.1). By Young’s inequality we have

u^pi−1

v^pi−1X_i²uX_i²v|X_i²v|^pi−2 ≤ |X_i²u|^pi p_i + 1

q_i u^pi

v^pi|X_i²v|^pi, i= 1, . . . , N, where p_i >1, q_i >1 _p¹

i + _q¹

i = 1. Since X_i²v <0 we arrive at L₁(u, v)≥

N

X

i=1

|X_i²u|^pi+

N

X

i=1

(p_i−1)u^pi

v^pi|X_i²v|^pi−

N

X

i=1

p_i

|X_i²u|^pi pi

+ 1 qi

u^pi

v^pi|X_i²v|^pi

−

N

X

i=1

p_i(p_i−1)u^pi−2

v^pi−1|X_i²v|^pi−2X_i²v

X_iu− u vX_iv

2

=

N

X

i=1

pi−1−p_i q_i

u^pi

v^pi|X_i²v|^pi

−

N

X

i=1

p_i(p_i−1)u^pi−2

v^pi−1|X_i²v|^pi−2X_i²v

X_iu− u vX_iv

2

≥0.

This completes the proof of Lemma 3.1.

Now we are ready to prove the subelliptic Rellich type inequality onG. Theorem 3.2. Let Ω⊂G\{x⁰ = 0} be an open set, whereG is a stratified group with N being the dimension of the first stratum. Then for a function u ≥ 0, u∈C²(Ω), and 2< α_i < N −2 we have the following inequality

N

X

i=1

Z

Ω

|X_i²u|^pidx≥

N

X

i=1

C_i(α_i, p_i) Z

Ω

|u|^pi

|x⁰_i|^2pidx, (3.2) where 1< p_i < N for i= 1, . . . , N, and

C_i(α_i, p_i) = (α_i(α_i−1))^pi−1(α_ip_i−2p_i−α_i+ 2)(α_ip_i−2p_i−α_i+ 1).

Proof of Theorem 3.2. We introduce the auxiliary function v :=

N

Y

j=1

|x⁰_j|^αj =|x⁰_i|^αiV_i, we chooseα_j later, and letV_i =QN

j=1,j6=i|x⁰_j|^αj. Then we have X_i²v =X_i(α_iV_i|x⁰_i|^αi−2x⁰_i) =α_i(α_i−1)V_i|x⁰_i|^αi−2,

|X_i²v|^pi−2 = (α_i(α_i−1))^pi−2V_i^pi−2|x⁰_i|^{αipi−2pi−2αi+4},

|X_i²v|^pi−2X_i²v = (α_i(α_i−1))^pi−1V_i^pi−1|x⁰_i|^{αipi−2pi−αi+2}.

Consequently, we obtain

X_i²(|X_i²v|^pi−2X_i²v) =(α_i(α_i−1))^pi−1V_i^pi−1X_i²(|x⁰_i|^{αipi−2pi−αi+2})

=(α_i(α_i−1))^pi−1(α_ip_i−2p_i−α_i+ 2)V_i^pi−1X_i |x⁰_i|^{αipi−2pi−αi}x⁰_i

=(α_i(α_i−1))^pi−1(α_ip_i−2p_i−α_i+ 2)(α_ip_i−2p_i−α_i+ 1)

×V_i^pi−1|x⁰_i|^{αi(pi−1)−2pi}.

Thus, for twice differentiable functionv >0 a.e. in Ω with X_i²v <0 we have X_i²(|X_i²|^pi−2X_i²v) = Ci(αi, pi)v^pi−1

|x⁰_i|^2pi (3.3) a.e. in Ω. Using (3.3) we compute

0≤ Z

Ω

L₁(u, v)dx= Z

Ω

R₁(u, v)dx

=

N

X

i=1

Z

Ω

|X_i²u|^pidx−

N

X

i=1

Z

Ω

X_i² u^pi

v^pi−1

|X_i²v|^pi−2X_i²vdx

=

N

X

i=1

Z

Ω

|X_i²u|^pidx−

N

X

i=1

Z

Ω

u^pi

v^pi−1X_i² |X_i²v|^pi−2X_i²v dx

=

N

X

i=1

Z

Ω

|X_i²u|^pidx−

N

X

i=1

C_i(α_i, p_i) Z

Ω

|u|^pi

|x⁰_i|^2pidx.

The proof of Theorem 3.2 is complete.

4. Hardy type inequalities with multiple singularities on G

In this section we obtain the analogue of the Hardy inequality with multiple singularities on a stratified group. The singularities are represented by a family {ak}^m_k=1 ∈G, where we write ak = (a⁰_k, a⁰⁰_k), with a⁰_k being in the first stratum of G. We can also write a⁰_k = (a⁰_k1, . . . , a⁰_kN), so (xa⁻¹_k )⁰ =x⁰−a⁰_k.

Theorem 4.1. Let Ω ⊂ G be an open set, where G is a stratified group with N being the dimension of the first stratum. Let N ≥ 3, x = (x⁰, x⁰⁰) ∈ G with x⁰ = (x⁰₁, . . . , x⁰_N) being in the first stratum of G, and let a_k ∈G, k= 1, . . . , m,be the singularities. Then we have

Z

Ω

|∇_Gu|²dx≥

N −2 2

2Z

Ω

PN j=1

Pm k=1

(xa⁻¹_k )⁰_j

|(xa⁻¹_k )⁰|^N

2

Pm k=1

1

|(xa⁻¹_k )⁰|^N−2

2 |u|²dx, (4.1) for all u∈C₀^∞(Ω).

The Euclidean case of this inequality was obtained by Kapitanski and Laptev [7]. In (4.1), (xa⁻¹_k )⁰_j =x⁰_j−a⁰_kj denotes the j^th component of xa⁻¹_k .

Proof of Theorem 4.1. Let us introduce a vector-fieldA(x) = (A₁(x), . . . ,A_N(x)) to be specified later. Also let λ be a real parameter for optimisation. We start with the inequality

0≤ Z

Ω N

X

j=1

(|X_ju−λA_ju|²)dx

= Z

Ω

|∇_Gu|²−2λRe

N

X

j=1

AjuXju+λ²

N

X

j=1

|Aj|²|u|²

! dx.

By using the integration by parts we get

− Z

Ω

λ²

N

X

j=1

|Aj|²+λdiv_GA

!

|u|²dx≤ Z

Ω

|∇_Gu|²dx. (4.2) We differentiate the integral on the left-hand side with respect to λ to optimise it, yielding

2λ|A|² + div_GA = 0,

for all x∈Ω. This is a restriction onA(x) giving ^div_|A(x)|^GA(x)2 =const.For λ= ¹₂ we get

div_GA(x) =−|A(x)|². (4.3) Then putting (4.3) in (4.2) we have the following Hardy inequality

1 4

Z

Ω N

X

j=1

|A_j(x)|²|u|²dx≤ Z

Ω

|∇_Gu|²dx.

Now if we assume thatA =∇_Gφ for some functionφ, then (4.3) becomes Lφ+|∇_Gφ|² = 0.

It follows that the function is harmonic (with respect to the sub-LaplacianL).

w=e^φ≥0

Then w is a constant >0 or has a singularity. Let us consider w:=

m

X

k=1

1

|(xa⁻¹_k )⁰|^N−2, and then take

φ(x) = ln(w).

Therefore

A(x) = ∇_G(lnw) =1 w∇_G

m

X

k=1

|(xa⁻¹_k )⁰|^2−N

!

=1 w

m

X

k=1

∇_G

N

X

j=1

((xa⁻¹_k )⁰_j)²

!^2−N₂

=− N−2 w

m

X

k=1

(xa⁻¹_k )⁰

|(xa⁻¹_k )⁰|^N

! ,

and

|A(x)|² =

N

X

j=1

|A_j(x)|² =

N −2 w

2 N

X

j=1

m

X

k=1

(xa⁻¹_k )⁰_j

|(xa⁻¹_k )⁰|^N

2

.

This completes the proof of Theorem4.1.

We then also obtain the corresponding uncertainty principle.

Corollary 4.2. Let Ω ⊂ G be an open set, where G is a stratified group with N being the dimension of the first stratum. Let N ≥ 3, x = (x⁰, x⁰⁰) ∈ G with x⁰ = (x⁰₁, . . . , x⁰_N) being in the first stratum of G. Let a_k ∈ G, k = 1, . . . , m, be the singularities. Then we have

N −2 2

Z

Ω

|u|²dx≤ Z

Ω

|∇_Gu|²dx ¹₂









 Z

Ω

Pm k=1

1

|(xa⁻¹_k )⁰|^N−2

2

PN j=1

Pm k=1

(xa⁻¹_k )⁰_j

|(xa⁻¹_k )⁰|^N

2|u|²dx











1 2

,

(4.4) for all u∈C₀^∞(Ω) and 1< p_i < N for i= 1, . . . , N.

Proof of Corollary 4.2. By (4.1) and the Cauchy-Schwarz inequality we get Z

Ω

|∇_Gu|²dx Z

Ω

Pm k=1

1

|(xa⁻¹_k )⁰|^N−2

2

PN j=1

Pm k=1

(xa⁻¹_k )⁰_j

|(xa⁻¹_k )⁰|^N

2|u|²dx

≥

N −2 2

2Z

Ω

PN j=1

Pm k=1

(xa⁻¹_k )⁰_j

|(xa⁻¹_k )⁰|^N

2

Pm k=1

1

|(xa⁻¹_k )⁰|^N−2

2 |u|²dx Z

Ω

Pm k=1

1

|(xa⁻¹_k )⁰|^N−2

2

PN j=1

Pm k=1

(xa⁻¹_k )⁰_j

|(xa⁻¹_k )⁰|^N

2|u|²dx

≥

N −2 2

2Z

Ω

|u|²dx 2

.

The proof is complete.

5. Many-particle Hardy type inequality on G

Suppose now that we have n particles, where n is a positive integer. Let Gⁿ be the product Gⁿ :=

n

z }| {

G×. . .×G. Now let us consider x = (x₁, . . . , x_n) ∈ Gⁿ, x_j ∈ G. Let x ∈ Gⁿ with x⁰ = (x⁰₁, . . . , x⁰_n) and x⁰_i = (x⁰_i1, . . . , x⁰_iN) being the coordinates on the first stratum of G for i = 1, . . . , n. The distance between particles x_i, x_j ∈Gcan be defined by

r_ij :=|(x_ix⁻¹_j )⁰|=|x⁰_i−x⁰_j|= v u u t

N

X

k=1

(x⁰_ik−x⁰_jk)². We also use the following notation

∇_Gi = (X_i1, . . . , X_iN)

for the gradient associated to thei-th particle. We denote∇_Gⁿ := (∇_G1, . . . ,∇_Gn), and

Li =

N

X

k=1

X_ik²

is the sub-Laplacian associated to the i-th particle. We note that L = PN i=1L_i. We are new ready to prove the following crucial inequality inR^m.

Lemma 5.1. Let m ≥1, and let

A= (A₁(x), . . . ,A_m(x))

be a mapping in A :R^m → R^m whose components and their first derivatives are uniformly bounded in R^m. Then for u∈C₀¹(R^m) we have

Z

R^m

|∇u|²dx≥ 1 4

R

R^mdivA|u|²dx2

R

R^m|A|²|u|²dx . (5.1) Proof of Lemma 5.1. We have

Z

R^m

divA|u|²dx

= 2

Re Z

R^m

hA,∇uiudx

≤2 Z

R^m

|A|²|u|²dx

1/2Z

R^m

|∇u|²dx 1/2

.

We have used the Cauchy-Schwarz inequality in the last line. The proof is finished

by squaring this inequality.

Theorem 5.2. Let Ω ⊂ Gⁿ be an open set, where G is a stratified group with N being the dimension of the first stratum. Let N ≥ 2 and n ≥ 3. Let rij =

|(x_ix⁻¹_j )⁰|=|x⁰_i−x⁰_j|. Then we have Z

Ω

|∇_Gⁿu|²dx≥ (N −2)² n

Z

Ω

X

1≤i<j≤n

|u|²

r_ij² dx, (5.2) for all u∈C¹(Ω).

The Euclidean case of inequality (5.2) is obtained by M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, A. Laptev, and J. Tidblom [6].

Proof of Theorem 5.2. Let us choose a mappingB₁ in the following form B1(x⁰_i, x⁰_j) := (xix⁻¹_j )⁰

r_ij² , 1≤i < j≤n.

And putting the mappingB₁ in (5.1) we have Z

Ω

|(∇_Gi− ∇_Gj)u|²dx≥ 1 4

R

Ω (div_Gi−div_Gj)B₁

|u|²dx2

R

Ω|B₁|²|u|²dx

= 1 4

R

Ω

2(N−2)

|(x_ix⁻¹_j )⁰|²|u|²dx 2

R

Ω

|u|²

|(x_ix⁻¹_j )⁰|²dx

= (N −2)² Z

Ω

|u|²

r_ij² dx. (5.3)

Also, we introduce another mapping B₂ B₂(x) :=

Pn j=1x⁰_j

Pn j=1x⁰_j

2,

and

div_GiB₂ =∇_Gi· B₂ =

N

X

k=1

X_ik

Pn j=1x⁰_jk

|Pn j=1x⁰_j|²

!

=

N n|Pn

j=1x⁰_j|²−2n (Pn

j=1x⁰_j1)²+. . .+ (Pn

j=1x⁰_jN)²

|Pn j=1x⁰_j|⁴

= N n−2n

|Pn

j=1x⁰_j|².

As before we put the mapping B₂ in (5.1) and using above computation yielding Z

Ω

n

X

i=1

∇_Giu

2

dx≥ 1 4

R

Ω(Pn

i=1div_GiB₂)|u|²dx2

R

Ω|B₂|²|u|²dx

= 1 4

R

Ω

Pn i=1

N n−2n

|Pn

j=1x⁰_j|²|u|²dx2

R

Ω

|u|²

|^Pnj=1x⁰_j|²dx

= (N −2)²n⁴ 4

Z

Ω

|u|²

Pn j=1x⁰_j

2dx. (5.4)

Adding inequalities (5.3) and (5.4) and using the identity n

n

X

i=1

|∇_Giu|² = X

1≤i<j≤n

∇_Giu− ∇_Gju

2+

n

X

i=1

∇_Giu

2

,

we arrive at

n

X

i=1

Z

Ω

|∇_Giu|²dx≥ (N −2)² n

Z

Ω

X

i<j

|u|²

r²_ij dx+(N −2)²n³ 4

Z

Ω

|u|²

Pn j=1x⁰_j

2dx.

(5.5) Because the last term on right-hand side is positive, we get

n

X

i=1

Z

Ω

|∇_Giu|²dx≥ (N −2)² n

Z

Ω

X

i<j

|u|² r_ij² dx.

Also we have

n

X

i=1

|∇_Giu|² = (∇_G1u)²+. . .+ (∇_Gnu)²

=|(∇_G1u, . . . ,∇_Gnu)|²

=|∇_Gⁿu|².

The proof of Theorem 5.2 is complete.

The following theorem deals with the total separation ofn ≥2 particles.

Theorem 5.3. Let Ω⊂Gⁿ be an open set, where G is a stratified group withN being the dimension of the first stratum. Let ρ² :=P

i<j|(x_ix⁻¹_j )⁰|² =P

i<j|x⁰_i− x⁰_j|² with x⁰_i 6=x⁰_j. Then we have

Z

Ω

|∇_Gu|²dx=n

(n−1) 2 N−1

2Z

Ω

|u|² ρ² dx+

Z

Ω

|∇_Gρ^−2αu|²ρ^4αdx (5.6) for all u∈C₀^∞(Ω) with α= ^2−(n−1)N₄ .

The Euclidean case of inequality (5.6) was obtained by Douglas Lundholm [9].

Proposition 5.4. Let Ω ⊂ Gⁿ be an open set, where G is a stratified group with N being the dimension of the first stratum. Let f : Ω → (0,∞) be twice differentiable. Then for any function u∈C₀^∞(Ω) and α∈R, we have

Z

Ω

|∇_Gu|²dx= Z

Ω

α(1−α)|∇_Gf|²

f² −αLf f

|u|²dx+ Z

Ω

|∇_Gv|²f^2αdx, (5.7) where v :=f^−αu.

Proof of Proposition 5.4. Let us compute for u=f^αv, that

∇_Gu=αf^α−1(∇_Gf)v+f^α∇_Gv.