In this paper some Poincaré type inequalities are obtained for the maps of the Heisenberg group target

http://jipam.vu.edu.au/

Volume 6, Issue 2, Article 33, 2005

POINCARÉ INEQUALITIES FOR THE HEISENBERG GROUP TARGET

GAO JIA COLLEGE OFSCIENCE

UNIVERSITY OFSHANGHAI FORSCIENCE ANDTECHNOLOGY

SHANGHAI200093, CHINA. [email protected]

Received 24 November, 2004; accepted 24 February, 2005 Communicated by J. Sándor

ABSTRACT. In this paper some Poincaré type inequalities are obtained for the maps of the Heisenberg group target.

Key words and phrases: Heisenberg group, Sobolev space, Poincaré type inequality.

2000 Mathematics Subject Classification. 22E20, 26D10, 46E35.

1. INTRODUCTION ANDPRELIMINARIES

LetH^m ([1]) denote a Heisenberg group which is a Lie group that has algebrag = R^2m+1, with a non-abelian group law:

(1.1) (x₁, y₁, t₁)·(x₂, y₂, t₂) = (x₁+x₂, y₁+y₂, t₁+t₂+ 2(y₂x₁−x₂y₁)),

for every u₁ = (x₁, y₁, t₁), u₂ = (x₂, y₂, t₂) ∈ H^m. The Lie algebra is generated by the left invariant vector fields

(1.2) X_i = ∂

∂x_i + 2y_i ∂

∂t, Y_i = ∂

∂y_i −2x_i ∂

∂t, i= 1,2, . . . , m,

andT = _∂t^∂. For every u₁ = (x₁, y₁, t₁), u₂ = (x₂, y₂, t₂) ∈ H^m, the metric d(u₁, u₂)in the Heisenberg groupH^mis defined as ([2])

(1.3) d(u1, u2) = u2u⁻¹₁

=

((x2−x1)²+ (y2−y1)²)²+ (t2−t1+ 2(x2y1−x1y2))²¹₄ . We see that H^m possesses the nonlinear structure of group laws. It is one of the differences betweenH^mand general Riemann manifolds.

LetΩ⊂Rⁿ(n≥2)be a bounded and connected Lipschitz domain . Let2≤p < ∞.

L. Capogna and Fang-Hua Lin [3] have provided the characterizations for the Sobolev space W^1,p(Ω,H^m),proved the existence theorem for the minimizer, and established that all critical

ISSN (electronic): 1443-5756

c 2005 Victoria University. All rights reserved.

Foundation item: the National Natural Science Foundation of China (19771048).

057-05

points for the energy are Lipschitz continuous in the 2-dimensional case. However, the higher dimensional regularity problem is still open.

In this paper, we shall give some Poincaré type inequalities for the maps of the Heisenberg group target. The statements of these results are similar to the ones in the classical case. How- ever, since the metric possesses the nonlinear structure of the group law, we require the use of a few techniques in the proofs for our Poincaré type inequalities.

Definition 1.1. Let2 ≤p <∞.A functionu= (z, t) : Ω→ H^mis inL^p(Ω,H^m)if for some h₀ ∈Ω, one has

(1.4)

Z

Ω

(d(u(h), u(h₀))^pdh <∞.

A functionu= (z, t) : Ω →H^m is in the Sobolev spaceW^1,p(Ω,H^m)ifu∈L^p(Ω,H^m)and

(1.5) E_p,Ω(u) = sup

f∈Cc(Ω),0≤f≤1

lim→0sup Z

Ω

f(h)e_u,(h)dh <∞, where

e_u,(h) = Z

|h−q|=

d(u(h), u(q))

p

dσ(q) ⁿ⁻¹ . E_p,Ω(u)is called thep-energy ofuonΩ.

Lemma 1.1. Ifu= (x, y, t)∈W^1,p(Ω,H^m), then

(1.6) ∇t= 2(y∇x−x∇y) in L^p2(Ω).

The maps satisfying (1.6) are called Legendrian maps.

Lemma 1.2. Ifu= (z, t) = (x, y, t)∈W^1,p(Ω,H^m), then Ep,Ω(u) = ωn−1

Z

Ω

|∇z|^p(q)dq.

Lemma 1.1 and Lemma 1.2 are due to L. Capogna and Fang-Hua Lin [3].

Lemma 1.3 ([4]). (Cp−inequality)Letp >0. Then for anyai ∈R,

n

X

i=1

|a_i|

!p

≤C_p

n

X

i=1

|a_i|^p, whereC_p = 1if0< p < 1andC_p =n^p−1 ifp≥1.

Lemma 1.4 ([5]). (Poincaré Inequality in the classical case) LetΩbe a bounded and connected Lipschitz domain inR^m. Letp >1. Then there exists a constantCdepending only onΩ, mand p, such that for every functionu∈W^1,p(Ω,R), we have

Z

Ω

|u(x)−λ_u|^pdx≤C Z

Ω

|∇u|^pdx, whereλu = _|Ω|¹ R

Ωu(x)dx.

2. THEPOINCARÉTYPE INEQUALITIES FOR THE HEISENBERG GROUP TARGET

Theorem 2.1 (Poincaré type inequality). LetΩbe a bounded and connected Lipschitz domain in Rⁿ. Then there exists a constant C depending only onΩ, n, m andp, such that for every functionu= (x, y, t) = (z, t)∈W^1,p(Ω,H^m),

(2.1)

Z

Ω

(d(u(q), λ_u))^pdq ≤C_ΩE_p,Ω(u) =C_Ω Z

Ω

|∇z|^p(q)dq.

Hereλ_u = (λ_x, λ_y, λ_t)andλ_f = _|Ω|¹ R

Ωf(q)dq.

Proof. Obviously,λ_u ∈W^1,p(Ω,H^m). From (1.3), using theC_p−inequality, we have (d(u(q), λu))^p

=

|z(q)−λ_z|⁴+ (t(q)−λ_t+ 2(λ_xy(q)−λ_yx(q)))^2p₄

≤Cp

h

|x(q)−λx|^p+|y(q)−λy|^p+|t(q)−λt+ 2(λxy(q)−λyx(q))|^p2i , (2.2)

whereC_pdepends onp. By the Poincaré inequality in the classical case, noting that 2(λ_xy(q)−λ_yx(q)) = 2λ_x(y(q)−λ_y)−2λ_y(x(q)−λ_x), we obtain

Z

Ω

(d(u(q), λ_u))^pdq

≤C_p Z

Ω

|x(q)−λ_x|^p+|y(q)−λ_y|^p+|t(q)−λ_t+ 2(λ_xy(q)−λ_yx(q))|^p2 dq

≤C₁ Z

Ω

|∇x|^p(q)dq+C₂ Z

Ω

|∇y|^p(q)dq+C₃ Z

Ω

|∇t+ 2(λ_x∇y−λ_y∇x)|^p2dq.

By virtue of the Legendrian condition∇t= 2(y∇x−x∇y), using the Hölder inequality, noting that|∇x| ≤ |∇z|and|∇y| ≤ |∇z|, we have

Z

Ω

(d(u(q), λ_u))^pdq

≤C₁ Z

Ω

|∇x|^pdq+C₂ Z

Ω

|∇y|^p(q) +C₃2^p2

Z

Ω

|∇y(x−λ_x)− ∇x(y−λ_y)|^p2dq

≤C₁ Z

Ω

|∇x|^pdq+C₂ Z

Ω

|∇y|^pdq +C₄

Z

Ω

|∇x|^p2|y−λ_y|^p2dq+ Z

Ω

|∇y|^p2|x−λ_x|^p2dq

≤C₁ Z

Ω

|∇x|^pdq+C₂ Z

Ω

|∇y|^pdq+C₅ Z

Ω

|∇x|^pdq Z

Ω

|∇y|^pdq ¹₂

≤C1

Z

Ω

|∇x|^pdq+C2

Z

Ω

|∇y|^pdq+C6

Z

Ω

|∇x|^pdq+ Z

Ω

|∇y|^pdq

≤C Z

Ω

|∇z|^p(q)dq,

whereC₁, C₂, C₃, C₄, C₅, C₆ andC are dependent onΩ,n, mandp.

Corollary 2.2. Ifu∈W^1,p(B(h₀, r),H^m), then (2.3)

Z

B(h0,r)

(d(u(q), λ_u))^pdq ≤Cr^pE_p,B(h0,r)(u) =Cr^p Z

B(h0,r)

|∇z|^p(q)dq.

Proof. Observe that (d(u(q), λ_u))^p =

|z(q)−λ_z|⁴+ (t(q)−λ_t+ 2(λ_xy(q)−λ_yx(q)))^2p₄

≤C_ph

|x(q)−λ_x|^p+|y(q)−λ_y|^p+|t(q)−λ_t+ 2(λ_xy(q)−λ_yx(q))|^p2i . HereC_p depends onp. By the Poincaré inequality in the classical case, noting that

2(λ_xy(q)−λ_yx(q)) = 2λ_x(y(q)−λ_y)−2λ_y(x(q)−λ_x), we deduce

Z

Br(h0)

(d(u(q), λ_u))^pdq

≤C_p Z

Br(h0)

(|x−λ_x|^p+|y−λ_y|^p+|t−λ_t+ 2(λ_xy−λ_yx)|^p2)dq

≤C1r^p Z

Br(h0)

|∇x|^p(q)dq+C2r^p Z

Br(h0)

|∇y|^p(q)dq +C₃r^p2

Z

Br(h0)

|∇t+ 2(λ_x∇y−λ_y∇x)|^p2dq.

By virtue of the Legendrian condition∇t = 2(y∇x−x∇y), using Hölder’s inequality, noting that|∇x| ≤ |∇z|and|∇y| ≤ |∇z|, we can obtain

Z

Br(h0)

(d(u(q), λu))^pdq

≤C₁r^p Z

Br(h0)

|∇x|^p(q)dq+C₂r^p Z

Br(h0)

|∇y|^p(q)dq +C₃r^p2

Z

Br(h0)

|∇y(q)(x(q)−λ_x)− ∇x(q)(y(q)−λ_y)|^p2dq

≤C₁r^p Z

Br(h0)

|∇x|^p(q)dq+C₂r^p Z

Br(h0)

|∇y|^p(q)dq

+C4r^p Z

Br(h0)

|∇x|^pdq Z

Br(h0)

|∇y|^pdq ¹₂

≤Cr^p Z

Ω

|∇z|^p(q)dq,

whereC₁, C₂, C₃, C₄ andC depend onΩ,n, mandp.

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