then the following general Poincar´e inequality holds: (1.1) Z Ω hA(x)∇φ,∇φi|∇u|2≥ Z ΩT {∇u6=0} W(x)φ2, where W(x

CARNOT GROUPS

YING-XIONG XIAO

Abstract. We prove some weighted Poincar´e inequalities on Carnot groups and obtain the sharp constants. These improve the corresponding results of Fausto Ferrari and Enrico Valdinoci.

1. Introduction

Let Ω be a domain inRⁿ and Abe a smooth symmetric matrix function on Ω.

Assume that there exist Λ > 0 such that for every ξ ∈ Rⁿ and all x∈ Ω, there hiolds

0≤ hA(x)ξ, ξi ≤Λ|ξ|².

F. Ferrari and E. Valdinocihad [4] have showed that ifuis aC²stable week solution of the following PDE

div(A(x)∇u(x)) =f(u(x)), then the following general Poincar´e inequality holds:

(1.1)

Z

Ω

hA(x)∇φ,∇φi|∇u|²≥ Z

ΩT {∇u6=0}

W(x)φ², where

W(x) = Xn

i=1

hA(x)∇ui,∇uii − hA(x)(D²u·N),(D²u·N)i − Xn

i=1

div(Ai(x)∇u)ui

and N = _|∇u(x)|^∇u(x). By the choice of suitable A(x) and stable solutions, F. Fer- rari and E. Valdinocihad [4] obtained some weighted Poincar´e inequalities for the Laplace operator in the Euclidean space Rⁿ, Kohn’s sublaplace operator in the Heisenberg group Hⁿ, the sublaplace operator in the Engel group, the Franchi- Grushin-Lanconelli operators and thep-laplacian in the Euclidean spaceRⁿ.

The aim of this note is to prove analogous Poincar´e inequalities with weights on Carnot groups. We refer to [7, 11] for Hardy inequalities on this groups. Our proof shows if one considers the weighted Poincar´e inequalities on Carnot groups (not the general Poincar´e inequality (1.1)), one needs not use the stable solutions. In fact, we offer here an alternative elementary proof, in which some calculus is enough.

To state our results, we need some notations. Recall that a Carnot groupGis a stratified, simply connected nilpotent Lie group with the Lie algebra g=L_r

i=1V_i satisfying [V1, Vj] = Vj+1 for all 1 ≤ j ≤ r−1. The integer r is called the step of the group G. Set nj = dimVj (1 ≤ j ≤ r). Let {X1,· · · , Xn1} be a basis

2000Mathematics Subject Classification. Primary 26D10; 22E25.

Key words and phrases. weighted Poincar´e inequalities; Carnot group; best constant.

1

of V1 and denote by ∇G = (X1,· · · , Xn1). Set ξ⁽¹⁾ = ξ1X1+· · ·+ξn1Xn1 and

|ξ⁽¹⁾|= q

ξ₁²+· · ·+ξ_n²₁. To this end, we have

Theorem 1.1. Let p >1 andα≥0. IfGis a Carnot group of step 2, then for all φ∈C₀^∞(G), there holds

(1.2)

µn1+α p

¶_pZ

G

|φ|^p|ξ⁽¹⁾|^α≤ Z

G

|∇Gφ|^p|ξ⁽¹⁾|^α+p and the constant

³n1+α p

´_p

in (1.2) is sharp.

LetHⁿ be the Heisenberg group whose group structure is given by (x, t)◦(x⁰, t⁰) = (x+x⁰, t+t⁰+ 2

Xn

j=1

(x2jx⁰_2j−1−x2j−1x⁰_2j)).

The vector fieldsX2j−1= _∂x^∂

2j−1 + 2x2j ∂

∂t, X2j = _∂x^∂

2j −2x2j−1∂

∂t, (j= 1,· · · , n) are left invariant and generate the Lie algebra of Hⁿ. It is easy to check thatHⁿ is a Carnot group of step two. By Theorem 1.1, we have the following corollary which improves the Theorem 1.3 in [4].

Corollary 1.2. Let p >1 andα≥0. There holds, for allφ∈C₀^∞(Hⁿ), (1.3)

µ2n+α p

¶_pZ

Hⁿ

|φ|^p|x|^α≤ Z

Hⁿ

|∇Hφ|^p|x|^α+p, where∇H= (X1,· · ·, X2n), and the constant ³

2n+α p

´_p

in (1.3) is sharp.

WhenGis a Carnot group of steprwithr >2, we have the following theorem.

Theorem 1.3. Let p >1and α≥0. IfGis a Carnot group of step rwith r >2, then forα≥0andφ∈C₀^∞(G), there holds,

(1.4)

µn1+α p

¶_pZ

G

|φ|^p|ξ⁽¹⁾|^α≤ Z

G

|∇Gφ|^p|ξ⁽¹⁾|^α+p. Furthermore, ifp= 2, the constant ¡_n

1+α 2

¢₂

in (1.4) is sharp.

Denote byEthe Engel group. Let Z₁= ∂

∂x1

+x₂ ∂

∂x3

+x²₂ ∂

∂x4

and Z₂= ∂

∂x2

.

It is known that Z1 and Z2 generate the Lie algebra of Engel group E and E is a Carnot group of step 3. By Theorem 4, we have the following corollary which generalizes the inequality (1.18) of [4].

Corollary 1.4. Let α≥0. There holds, for allφ∈C₀^∞(E), (1.5) (1 +α)²

Z

R⁴

|φ|²(x²₁+x²₂)^α≤ Z

R⁴

|∇Eφ|²(x²₁+x²₂)^α+1, where∇E= (Z1, Z2), and the constant (1 +α)² in (1.5) is sharp.

We fail to show the constant

³n1+α p

´_p

in (1.4) is sharp when p 6= 2. For the reasons, see Remark 3.1.

2. Proof of Theorem 1.1

Recall that ifGis a Carnot group of step two, then there existsm×mlinearly independent skew-symmetric matricesU⁽¹⁾,· · · , U⁽ⁿ⁾, such thatGis isomorphic to (R^m+n,◦) with the following Lie group law (x∈R^m,t∈Rⁿ) (see A.4, [2])

(x, t)◦(x⁰, t⁰) =

µ xi+x⁰_i, i= 1,2,· · ·, m

tj+t⁰_j+¹₂ < x, U^(j)x⁰ >, j= 1,2,· · · , n

¶ .

The vector fields in the Lie algebra gof G = (R^m+n,◦) that agree at the origin with _∂x^∂

j(j= 1,· · ·, m) are given by Xj= ∂

∂xj +1 2

Xn

k=1

Ã_m X

i=1

U_i,j^(k)xi

!

∂

∂tk, and thatgis spanned by the left-invariant vector fields

X1,· · ·, Xm, T1= ∂

∂t1,· · ·, Tn= ∂

∂tn. The Kohn’s sub-Laplacian on the groupGis given by

∆G= Xm

j=1

X_j²= Xm

j=1

Ã

∂

∂xj +1 2

Xn

k=1

Ã_m X

i=1

U_j,i^(k)xi

!

∂

∂tk

!₂

and the corresponding horizontal gradient is them-dimensional vector given by

∇G= (X1,· · · , Xm) =∇x−1

2U⁽¹⁾x ∂

∂t1

− · · · − −1

2U⁽ⁿ⁾x ∂

∂tn

. (2.1)

To finish the proof of Theorem 1.1, it is enough to show that the following inequality is held for allφ∈C₀^∞(R^m+n)

(2.2)

µm+α p

¶_pZ

R^m+n

|φ|^p|x|^α≤ Z

R^m+n

|∇Gφ|^p|x|^α+p and the corresponding constant is sharp.

For a functionφ∈C₀^∞(G), we denote by φ²:= [(|φ|²+²²)^p/2−²^p]^1p with² >0.

Then 0 ≤φ² ∈ C₀^∞(Rⁿ). In fact, φ² has the same support as φ. Notice that for

|x| 6= 0,

∆G|x|^α+2= Xm

j=1

µ ∂

∂xj

¶₂

|x|^α+2= (α+ 2)(m+α)|x|^α. Integrating by parts yields

(α+ 2)(m+α) Z

R^m+n

|φ²|^p|x|^α= Z

R^m+n

|φ²|^p∆G|x|^α+2

=− Z

R^m+n

h∇G|φ²|^p,∇G|x|^α+2i=−p Z

R^m+n

(|φ|²+²²)^p−22 φh∇Gφ,∇G|x|^α+2i

≤p Z

R^m+n

(|φ|²+²²)^p−22 |φ| · |∇Gφ| · |∇G|x|^α+2|

≤p(α+ 2) Z

R^m+n

(|φ|²+²²)^p−12 |∇_Gφ||x|^α+1.

Letting²→0, we obtain (m+α)

Z

R^m+n

|φ|^p|x|^α≤p Z

R^m+n

|φ|^p−1|∇Gφ||x|^α+1. By H¨older’s inequality:

(m+α) Z

R^m+n

|φ|^p|x|^α≤p µZ

R^m+n

|φ|^p|x|^α

¶^p−1

p µZ

R^m+n

|∇_Gφ|^p|x|^α+2

¶¹

p

. Canceling and raising both sides to the powerp, we get (2.2).

To see the constant

³m+α p

´_p

in (2.2) is sharp, we follow [9], Lemma 2.5. Choosing f(x, t) =u(|x|)g(t), whereu(|x|)∈C₀^∞(R^m) andg(t) =Q_n

j=1wj(tj) withwj(tj)∈ C₀^∞(R) for all 1≤j≤n. We have, by (2.1),

|∇Gf(x, t)|²=

¯¯

¯¯

¯¯g(t)∇xu(|x|)−1 2u(|x|)

Xn

j=1

U^(j)x∂g

∂tj

¯¯

¯¯

¯¯

2

=g²|∇xu(|x|)|²+1 4u²(|x|)

¯¯

¯¯

¯¯ Xn

j=1

U^(j)x∂g

∂tj

¯¯

¯¯

¯¯

2

−u(|x|)g· Xn

j=1

h∇_xu(|x|), U^(j)xi∂g

∂tj

.

SinceU^(j)((1≤j ≤n)) is a skew-symmetric matric, h∇xu(|x|), U^(j)xi=u⁰(|x|)

¿ x

|x|, U^(j)x À

= 0.

Therefore,

(2.3) |∇Gf(x, t)|²=|∇xu|²g²+1 4u²

¯¯

¯¯

¯¯ Xn

j=1

U^(j)x∂g

∂tj

¯¯

¯¯

¯¯

2

.

Using the inequality (cf. [9], inequality (4))

(2.4) (s²₁+s²₂)^p/2≤(1−λ)^1−ps^p₁+λ^1−ps^p₂, s1, s2>0, 0< λ <1, we have, for 0< λ <1,

R

R^m+n|∇Gf|^p|x|^α+p R

R^m+n|f|^p|x|^α ≤(1−λ)^1−p R

R^m|∇u|^p|x|^p+αdx R

R^m|u|^p|x|^αdx + λ^1−p2^−p

R

R^m+n|u|^p

¯¯

¯P_n

j=1U^(j)x_∂t^∂g

j

¯¯

¯^p|x|^p+α R

R^m+n|u(|x|)|^p|g|^p|x|^α (2.5)

Notice that there exists a positive constantCn,p, depending only onnandp, such that ¯

¯¯

¯¯

¯ Xn

j=1

U^(j)x∂g

∂tj

¯¯

¯¯

¯¯

p

≤Cn,p

Xn

j=1

¯¯

¯¯U^(j)x∂g

∂tj

¯¯

¯¯

p

=Cn,p

Xn

j=1

¯¯

¯U^(j)x

¯¯

¯^p

¯¯

¯¯∂g

∂tj

¯¯

¯¯

p

.

We have, by using (2.5) andg(t) =Q_n

j=1wj(tj), R

R^m+n|∇Gf|^p|x|^α+p R

R^m+n|f|^p|x|^α ≤(1−λ)^1−p R

R^m|∇u|^p|x|^p+αdx R

R^m|u|^p|x|^αdx + 2^−pλ^1−pCn,p

R

R^m+n|u|^p³P_n

j=1

¯¯U^(j)x¯

¯^p¯

¯¯_∂t^∂g

j

¯¯

¯^p

´

|x|^p+α R

R^m+n|u|^p|g|^p|x|^α

=(1−λ)^1−p R

R^m|∇u|^p|x|^p+αdx R

R^m|u|^p|x|^αdx + 2^−pλ^1−pCn,p

Xn

j=1

R

R^m|u|^p|x|^p+α|U^(j)x|^pdx R

R^m|u(|x|)|^p|x|^α · R

R|w⁰_j(tj)|^pdtj

R

R|wj(tj)|^pdtj

. Since

w(t)∈Cinf₀^∞(R)\{0}

R

R|w⁰(t)|^pdt R

R|w(t)|^pdt = 0, we obtain, for 0< λ <1,

φ∈C₀^∞(Rinf^m+n)\{0}

R

R^m+n|∇Gφ|^p|x|^α+p R

R^m+n|φ|^p|x|^α ≤(1−λ)^1−p inf

u(|x|)∈C₀^∞(R^m)\{0}

R

R^m|∇u|^p|x|^p+αdx R

R^m|u|^p|x|^αdx

= (1−λ)^1−p

µm+α p

¶_p . Here we use the fact (see e.g. [9], Theorem 2.1)

µm+α p

¶p

= inf

u(|x|)∈C₀^∞(R^m)\{0}

R

R^m|∇u(|x|)|^p|x|^p+αdx R

R^m|u(|x|)|^p|x|^αdx . By lettingλ→0+, we can see the constant³

m+α p

´_p

is sharp.

Remark 2.1. A simple calculation shows, if|x| 6= 0, then

|∇G|x|^−α+mp |^p=

µm+α p

¶_p

|x|^{−α−m−p}. It looks like that|x|^−α+mp can realize the sharp constant

³m+α p

´_p

. However, since Z

R^m+n

||x|^−α+mp |^p|x|^αdxdt= +∞,

inequality (1.2) is strict for everyφ6= 0. It seems that one can anticipate improving this inequality by adding some nonnegative correction term to the left-hand side of the inequality (1.2) whenR^m+n is replaced by a bounded domain. We refer to [3] and [10] for more information about this subject. Moreover, in their papers, it also shows that the sharp constant appeared in this type of inequality has some applications in PDE.

We can not directly use the the family of function found in [9], Lemma 2.3. In fact, if we set

u²(x, t) =

½ 1, |x| ≤1;

|x|^{−α+mp −²}, |x| ≥1,

then Z

R^m+n

|u_²(x, t)|^p|x|^αdxdt= Z

Rⁿ

dt· Z

R^m

|u_²(x, t)|^p|x|^αdx= +∞.

3. Proof of Theorem 1.3

We begin by quoting some preliminary facts which will be needed in the sequel and refer to [1, 5, 6, 8] for more precise information about Carnot group. Let G be a Carnot group with the Lie algebrag=L_r

i=1Vi satisfying [V1, Vj] =Vj+1 for all 1 ≤j ≤ r−1. As a simply connected nilpotent group, G is differential with R^N,N =P_r

i=1dimVi=P_r

i=1ni, via the exponential map exp :g→G. The Haar measure on Gis induced by the exponential mapping from the Lebesgue measure ong=R^N and coincides the Lebesgue measure onR^N.

Consider ξ = (ξ⁽¹⁾,· · ·, ξ^(r)) ∈ R^N with ξ⁽ⁱ⁾ = (ξ₁⁽ⁱ⁾,· · ·, ξn⁽ⁱ⁾i) ∈ Rⁿⁱ. Forj = 1,· · · , n1, letXj be the unique vector field ingthat coincides with∂/∂ξ_j⁽¹⁾ at the origin. The second order differential operator

∆G =−

n1

X

j=1

X_j^∗Xj =

n1

X

j=1

X_j²

is called a sub-Laplacian on G. We shall denote by the ∇G = (X1,· · · , Xn1) the related subelliptic gradient. By the Campbell-Hausdorff formula (see e.g. [5], page 2-4),Xj(1≤j≤n1) can be expressed as the following

(3.1) Xj= ∂

∂ξ_j⁽¹⁾ + Xr

k=2 nk

X

s=1

p^j_k,s(ξ⁽¹⁾,· · ·, ξ^(k−1)) ∂

∂ξs^(k)

, wherep^j_k,s(ξ⁽¹⁾,· · ·, ξ^(k−1)) is a polynomial ofξ⁽¹⁾,· · ·, ξ^(k−1).

Now we prove inequality (1.4). The proof is similar to that given in Section 2.

Following the proof of Theorem 1.1, we also denote byφ²:= [(|φ|²+²²)^p/2−²^p]^1p with² >0 ifφ∈C₀^∞(G). Since, for|ξ⁽¹⁾| 6= 0,

∆G|ξ⁽¹⁾|^α+2=

n1

X

j=1

à ∂

∂ξ⁽¹⁾_j

!₂

|ξ⁽¹⁾|^α+2= (α+ 2)(n1+α)|ξ⁽¹⁾|^α, we have, through integrating by parts,

(α+ 2)(n1+α) Z

G

|φ²|^p|ξ⁽¹⁾|^α= Z

G

|φ²|^p∆G|ξ⁽¹⁾|^α+2

=− Z

G

h∇G|φ²|^p,∇G|ξ⁽¹⁾|^α+2i=−p Z

G

(|φ|²+²²)^p−22 φh∇Gφ,∇G|ξ⁽¹⁾|^α+2i

≤p Z

G

(|φ|²+²²)^p−22 |φ| · |∇Gφ| · |∇G|ξ⁽¹⁾|^α+2|

≤p(α+ 2) Z

G

(|φ|²+²²)^p−12 |∇Gφ||ξ⁽¹⁾|^α+1. Letting²→0 yields

(n1+α) Z

G

|φ|^p|ξ⁽¹⁾|^α≤p Z

G

|φ|^p−1|∇Gφ||ξ⁽¹⁾|^α+1. By H¨older’s inequality:

(n1+α) Z

G

|φ|^p|ξ⁽¹⁾|^α≤p µZ

G

|φ|^p|ξ⁽¹⁾|^α

¶^p−1

p µZ

G

|∇Gφ|^p|ξ⁽¹⁾|^α+2

¶¹

p

. Canceling and raising both sides to the powerp, we get (1.4).

Next we shall show the constant¡_n

1+α 2

¢₂

in (1.4) is sharp whenp= 2. Choosing f(ξ) =u(ξ⁽¹⁾)g(ξ⁽²⁾,· · ·, ξ^(r)), whereu(·)∈C₀^∞(Rⁿ¹) andg=Q_r

k=2

Q_nk

s=1w_k,s(ξs^(k)) with wk,s(·)∈C₀^∞(R) for all 2≤k ≤r and 1 ≤s ≤nl. For convenience, we set wk,s≡1 for all 1≤s≤nk ifk= 1. By (3.1),

Z

G

|∇Gf|²|ξ⁽¹⁾|^α+2dξ=

n1

X

j=1



 Z

R^N

Ã

∂u

∂ξ_j⁽¹⁾

!₂

g²|ξ⁽¹⁾|^α+2dξ+

Z

R^N

u²

¯¯

¯¯

¯ Xr

k=2 nk

X

s=1

p^j_k,s ∂g

∂ξ^(k)s

¯¯

¯¯

¯

2

|ξ⁽¹⁾|^α+2dξ+ (∗)



,

where

(∗) =2 Z

R^N

u²g Xr

k=2 nk

X

s=1

p^j_k,s ∂g

∂ξs^(k)

|ξ⁽¹⁾|^α+2dξ

= Xr

k=2 nk

X

s=1

Z

R^N

u²p^j_k,s(ξ⁽¹⁾,· · · , ξ^(k−1)) ∂g²

∂ξ^(k)s

|ξ⁽¹⁾|^α+2dξ.

Sincewk,s(·)∈C₀^∞(R) for all 2≤k≤rand 1≤s≤nk, Z

R

∂w²_k,s(ξ^(k)s )

∂ξs^(k)

dξ_s^(k)= 0 and hence

Z

R

∂g²

∂ξs^(k)

dξ_s^(k)= 0.

Thus (∗) = 0 and Z

G

|∇Gf|²|ξ⁽¹⁾|^α+2dξ=

n1

X

j=1



 Z

R^N

Ã

∂u

∂ξ_j⁽¹⁾

!₂

g²|ξ⁽¹⁾|^α+2dξ+

Z

R^N

u²

¯¯

¯¯

¯ Xr

k=2 n_k

X

s=1

p^j_k,s ∂g

∂ξs^(k)

¯¯

¯¯

¯

2

|ξ⁽¹⁾|^α+2dξ



. (3.2)

Choose a positive constantC such that

¯¯

¯¯

¯ Xr

k=2 nk

X

s=1

p^j_k,s ∂g

∂ξs^(k)

¯¯

¯¯

¯

2

≤C Xr

k=2 nk

X

s=1

¯¯

¯¯

¯p^j_k,s ∂g

∂ξs^(k)

¯¯

¯¯

¯

2

. We have, by (3.2),

R

G|∇Gf|²|ξ⁽¹⁾|^α+2dξ R

Gf²|ξ⁽¹⁾|^αdξ

≤ R

Rⁿ¹

P_n1

j=1

µ

∂u

∂ξ_j⁽¹⁾

¶2

|ξ⁽¹⁾|^α+2dξ⁽¹⁾ R

Rⁿ1u²|ξ⁽¹⁾|^αdξ⁽¹⁾ +C Xr

k=2 n_k

X

s=1

R

Gu²

¯¯

¯p^j_k,s ^∂g

∂ξ^(k)s

¯¯

¯²|ξ⁽¹⁾|^α+2dξ R

Gf²|ξ⁽¹⁾|^αdξ .

Since for all 2≤k≤rand 1≤s≤nl,

wk,s∈Cinf₀^∞(R)\{0}

R

R|w⁰_k,s|²dξ^(k)s

R

R|wk,s|²dξ^(k)s

= 0, we have

wk,s∈Cinf₀^∞(R)\{0}

R

Gu²

¯¯

¯p^j_k,s ^∂g

∂ξ_s^(k)

¯¯

¯²|ξ⁽¹⁾|^α+2dξ R

Gf²|ξ⁽¹⁾|^αdξ

= R

R^{n1+···+nk−1}u²|p^j_k,s|²Q_k−1

l=1

Q_nl

i=1w²_l,i|ξ⁽¹⁾|^α+2 R

R^{n1+···+nk−1}u²Q_k−1

l=1

Q_nl

i=1w²_l,i|ξ⁽¹⁾|^α · inf

wk,s∈C₀^∞(R)\{0}

R

R|w_k,s⁰ |²dξs^(k)

R

R|wk,s|²dξs^(k)

= 0.

Therefore,

φ∈C₀^∞inf(G)\{0}

R

G|∇Gφ|^p|ξ⁽¹⁾|^α+2 R

G|φ|^p|ξ⁽¹⁾|^α ≤ inf

u∈C^∞₀ (Rⁿ1)\{0}

R

Rⁿ¹

P_n1

j=1

µ

∂u

∂ξ⁽¹⁾_j

¶₂

|ξ⁽¹⁾|^α+2dξ⁽¹⁾ R

Rⁿ¹u²|ξ⁽¹⁾|^αdξ⁽¹⁾

=

µn1+α 2

¶₂ .

To get the last inequality above, we use the fact (see e.g. [9], Theorem 2.1) µn1+α

2

¶₂

= inf

u(x)∈C^∞₀ (Rⁿ¹)\{0}

R

Rⁿ¹|∇u(x)|²|x|^2+αdx R

Rⁿ1|u(|x|)|²|x|^αdx . The desired result follows.

Remark 3.1. We note that the equality (2.3) play an important role in the proof of sharpness of constant

³n1+α p

´_p

in (1.2) when G is a Carnot group of step 2.

However, ifG is a Carnot group of stepr with r >2, we can only obtain similar equality forp= 2 (see (3.2)). This is the reason that we can not prove the sharpness of

³n1+α p

´_p

in (1.4) whenp6= 2.

Acknowledgements

The authors thanks the referee for his/her careful reading and very useful com- ments which improved the final version of this paper.

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School of Mathematics and Statistics, Xiaogan University ,Xiaogan, Hubei, 432000, People’s Republic of China

E-mail address:[email protected]