Journal of Inequalities in Pure and
Applied Mathematics
POINCARÉ INEQUALITIES FOR THE HEISENBERG
GROUP TARGET
volume 6, issue 2, article 33,
2005.
GAO JIA
College of Science
University of Shanghai for Science and Technology
Shanghai 200093, China.
Received 24 November, 2004;
accepted 24 February, 2005.
Communicated by: J. Sándor
EMail: gaojia79@sina.com.cn
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
057-05
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Abstract
In this paper some Poincaré type inequalities are obtained for the maps of the
Heisenberg group target.
2000 Mathematics Subject Classification: 22E20, 26D10, 46E35.
Key words: Heisenberg group, Sobolev space, Poincaré type inequality.
Poincaré Inequalities for the
Heisenberg Group Target
Foundation item: the National Natural Science Foundation of China (19771048).
Gao Jia
Contents
1
2
Introduction and Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . .
The Poincaré Type Inequalities for the Heisenberg Group
Target . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
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1.
Introduction and Preliminaries
Let Hm ([1]) denote a Heisenberg group which is a Lie group that has algebra
g = R2m+1 , with a non-abelian group law:
(1.1) (x1 , y1 , t1 ) · (x2 , y2 , t2 ) = (x1 + x2 , y1 + y2 , t1 + t2 + 2(y2 x1 − x2 y1 )),
for every u1 = (x1 , y1 , t1 ), u2 = (x2 , y2 , t2 ) ∈ Hm . The Lie algebra is generated
by the left invariant vector fields
(1.2)
∂
∂
Xi =
+ 2yi ,
∂xi
∂t
∂
∂
Yi =
− 2xi ,
∂yi
∂t
i = 1, 2, . . . , m,
∂
and T = ∂t
. For every u1 = (x1 , y1 , t1 ), u2 = (x2 , y2 , t2 ) ∈ Hm , the metric
d(u1 , u2 ) in the Heisenberg group Hm is defined as ([2])
(1.3)
d(u1 , u2 ) = u2 u−1
1
= ((x2 − x1 )2 + (y2 − y1 )2 )2
1
+ (t2 − t1 + 2(x2 y1 − x1 y2 ))2 4 .
Poincaré Inequalities for the
Heisenberg Group Target
Gao Jia
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We see that Hm possesses the nonlinear structure of group laws. It is one of the
differences between Hm and general Riemann manifolds.
Let Ω ⊂ Rn (n ≥ 2) be a bounded and connected Lipschitz domain . Let
2 ≤ p < ∞.
L. Capogna and Fang-Hua Lin [3] have provided the characterizations for the
Sobolev space W 1,p (Ω, Hm ), proved the existence theorem for the minimizer,
and established that all critical points for the energy are Lipschitz continuous in
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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. However, 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. Let 2 ≤ p < ∞. A function u = (z, t) : Ω → Hm is in
Lp (Ω, Hm ) if for some h0 ∈ Ω, one has
Z
(1.4)
(d(u(h), u(h0 ))p dh < ∞.
Poincaré Inequalities for the
Heisenberg Group Target
Gao Jia
Ω
A function u = (z, t) : Ω → Hm is in the Sobolev space W 1,p (Ω, Hm ) if
u ∈ Lp (Ω, Hm ) and
Z
(1.5)
Ep,Ω (u) =
sup
lim sup f (h)eu, (h)dh < ∞,
f ∈Cc (Ω),0≤f ≤1 →0
Ω
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where
Z
eu, (h) =
|h−q|=
d(u(h), u(q))
p
dσ (q)
.
n−1
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Ep,Ω (u) is called the p-energy of u on Ω.
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Lemma 1.1. If u = (x, y, t) ∈ W 1,p (Ω, Hm ), then
(1.6)
∇t = 2(y∇x − x∇y) in
p
L 2 (Ω).
J. Ineq. Pure and Appl. Math. 6(2) Art. 33, 2005
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The maps satisfying (1.6) are called Legendrian maps.
Lemma 1.2. If u = (z, t) = (x, y, t) ∈ W 1,p (Ω, Hm ), then
Z
Ep,Ω (u) = ωn−1 |∇z|p (q)dq.
Ω
Lemma 1.1 and Lemma 1.2 are due to L. Capogna and Fang-Hua Lin [3].
Lemma 1.3 ([4]). (Cp −inequality) Let p > 0. Then for any ai ∈ R,
!p
n
n
X
X
|ai | ≤ Cp
|ai |p ,
i=1
Lemma 1.4 ([5]). (Poincaré Inequality in the classical case) Let Ω be a bounded
and connected Lipschitz domain in Rm . Let p > 1. Then there exists a constant
C depending only on Ω, m and p, such that for every function u ∈ W 1,p (Ω, R),
we have
Z
Z
p
|u(x) − λu | dx ≤ C
|∇u|p dx,
Ω
where λu =
R
u(x)dx.
Ω
Gao Jia
i=1
where Cp = 1 if 0 < p < 1 and Cp = np−1 if p ≥ 1.
1
|Ω|
Poincaré Inequalities for the
Heisenberg Group Target
Ω
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2.
The Poincaré Type Inequalities for the
Heisenberg Group Target
Theorem 2.1 (Poincaré type inequality). Let Ω be a bounded and connected
Lipschitz domain in Rn . Then there exists a constant C depending only on
Ω, n, m and p, such that for every function u = (x, y, t) = (z, t) ∈ W 1,p (Ω, Hm ),
Z
Z
p
(d(u(q), λu )) dq ≤ CΩ Ep,Ω (u) = CΩ
(2.1)
Ω
Here λu = (λx , λy , λt ) and λf =
|∇z|p (q)dq.
Ω
1
|Ω|
R
Ω
Gao Jia
f (q)dq.
Proof. Obviously, λu ∈ W 1,p (Ω, Hm ). From (1.3), using the Cp −inequality,
we have
(2.2)
Poincaré Inequalities for the
Heisenberg Group Target
(d(u(q), λu ))p
p
= |z(q) − λz |4 + (t(q) − λt + 2(λx y(q) − λy x(q)))2 4
h
≤ Cp |x(q) − λx |p + |y(q) − λy |p
i
p
+ |t(q) − λt + 2(λx y(q) − λy x(q))| 2 ,
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where Cp depends on p. By the Poincaré inequality in the classical case, noting
that
2(λx y(q) − λy x(q)) = 2λx (y(q) − λy ) − 2λy (x(q) − λx ),
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we obtain
Z
(d(u(q), λu ))p dq
Ω
Z p
p
p
≤ Cp
|x(q) − λx | + |y(q) − λy | + |t(q) − λt + 2(λx y(q) − λy x(q))| 2 dq
Ω
Z
Z
Z
p
p
p
≤ C1 |∇x| (q)dq + C2 |∇y| (q)dq + C3 |∇t + 2(λx ∇y − λy ∇x)| 2 dq.
Ω
Ω
Ω
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
(d(u(q), λu ))p dq
Ω
Z
Z
p
≤ C1 |∇x| dq + C2 |∇y|p (q)
Ω
Ω
Z
p
p
+ C3 2 2
|∇y(x − λx ) − ∇x(y − λy )| 2 dq
Ω
Z
Z
p
≤ C1 |∇x| dq + C2 |∇y|p dq
Ω
Ω
Z
Z
p
p
p
p
+ C4
|∇x| 2 |y − λy | 2 dq + |∇y| 2 |x − λx | 2 dq
Ω
Z
≤ C1
p
Ω
Z
|∇x| dq + C2
Ω
p
Z
|∇y| dq + C5
Ω
p
Z
|∇x| dq
Ω
|∇y|p dq
12
Poincaré Inequalities for the
Heisenberg Group Target
Gao Jia
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Ω
J. Ineq. Pure and Appl. Math. 6(2) Art. 33, 2005
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Z
p
≤ C1 |∇x| dq + C2
ZΩ
≤C
|∇z|p (q)dq,
Z
p
Z
|∇y| dq + C6
Ω
p
Z
|∇x| dq +
Ω
p
|∇y| dq
Ω
Ω
where C1 , C2 , C3 , C4 , C5 , C6 and C are dependent on Ω, n, m and p.
Corollary 2.2. If u ∈ W 1,p (B(h0 , r), Hm ), then
Z
(2.3)
(d(u(q), λu ))p dq ≤ Crp Ep,B(h0 ,r) (u)
B(h0 ,r)
Z
p
= Cr
|∇z|p (q)dq.
Poincaré Inequalities for the
Heisenberg Group Target
Gao Jia
B(h0 ,r)
Proof. Observe that
(d(u(q), λu ))p
p
= |z(q) − λz |4 + (t(q) − λt + 2(λx y(q) − λy x(q)))2 4
h
i
p
≤ Cp |x(q) − λx |p + |y(q) − λy |p + |t(q) − λt + 2(λx y(q) − λy x(q))| 2 .
Here Cp depends on p. By the Poincaré inequality in the classical case, noting
that
2(λx y(q) − λy x(q)) = 2λx (y(q) − λy ) − 2λy (x(q) − λx ),
we deduce
Z
(d(u(q), λu ))p dq
Br (h0 )
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Z
p
≤ Cp
p
2
p
(|x − λx | + |y − λy | + |t − λt + 2(λx y − λy x)| )dq
Z
Z
p
p
p
≤ C1 r
|∇x| (q)dq + C2 r
|∇y|p (q)dq
Br (h0 )
Br (h0 )
Z
p
p
+ C3 r 2
|∇t + 2(λx ∇y − λy ∇x)| 2 dq.
Br (h0 )
Br (h0 )
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
(d(u(q), λu ))p dq
Br (h0 )
Z
Z
p
p
p
≤ C1 r
|∇x| (q)dq + C2 r
|∇y|p (q)dq
Br (h0 )
Br (h0 )
Z
p
p
+ C3 r 2
|∇y(q)(x(q) − λx ) − ∇x(q)(y(q) − λy )| 2 dq
Br (h0 )
Z
Z
p
p
p
≤ C1 r
|∇x| (q)dq + C2 r
|∇y|p (q)dq
+ C4 rp
Z
Br (h0 )
≤ Crp
Z
|∇x|p dq
Z
Gao Jia
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Br (h0 )
Br (h0 )
Poincaré Inequalities for the
Heisenberg Group Target
|∇y|p dq
Br (h0 )
|∇z|p (q)dq,
12
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Ω
where C1 , C2 , C3 , C4 and C depend on Ω, n, m and p.
J. Ineq. Pure and Appl. Math. 6(2) Art. 33, 2005
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References
[1] D. CHRISTODOULOU, On the geometry and dynamics of crystalline continua, Ann. Inst. H. Poincaré, 13 (1998), 335–358.
[2] A. VENERUSO, Hömander multipliers on the Heisenberg group, Monatsh.
Math., 130 (2000), 231–252.
[3] L. CAPOGNA AND FANG-HUA LIN, Legendrian energy minimizers, Part
I: Heisenberg group target, Calc. Var., 12 (2001), 145–171.
[4] B.F. BECKENBACH
1961.
AND
R. BELLMAN, Inequalities, Springer-Verlag,
[5] W.P. ZIEMER, Weakly Differentiable Function, Springer-Verlag, New
York, 1989.
[6] Z.Q. WANG AND M. WLLEM, Singular minimization problems, J. Diff.
Equa., 161 (2000), 307–370.
Poincaré Inequalities for the
Heisenberg Group Target
Gao Jia
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