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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 158281, 17 pages
doi:10.1155/2010/158281
Research Article
Weighted Rellich Inequality on H-Type Groups and
Nonisotropic Heisenberg Groups
Yongyang Jin1 and Yazhou Han2
1
2
Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310032, China
Department of Mathematics, College of Science, China Jiliang University, Hangzhou 310018, China
Correspondence should be addressed to Yongyang Jin, [email protected]
Received 19 March 2010; Accepted 15 June 2010
Academic Editor: Nikolaos Papageorgiou
Copyright q 2010 Y. Jin and Y. Han. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We prove a sharp weighted Rellich inequality associated with a class of Greiner-type vector
fields on H-type groups. We also obtain some weighted Hardy- and Rellich-type inequalities on
nonisotropic Heisenberg groups. As an application, we get a Rellich-Sobolev-type inequality on
Heisenberg groups.
1. Introduction
The study of partial differential operators constructed from noncommutative vector fields
satisfying the Hörmander condition 1 has had much development. We refer to 2, 3 and the
references therein for a systematic account of the study. Recently there have been considerable
interests in studying the sub-Laplacians as square sums of vector fields that are not invariant
or do not satisfy the Hörmander condition. Among the examples of such sub-Laplacians are
the Grushin operators, Greiner-type operators, and the sub-Laplacian constructed by Kohn
4. Those noninvariant sub-Laplacians also appear naturally in complex analysis. In 5 Beals
n
et al. considered the CR operators {Zj , Zj }j1 on R2n1 as boundary of the complex domain
⎧
⎪
⎨
⎞k ⎫
⎪
n ⎬
2⎠
⎝
zj
,
>
⎪
⎭
j1
⎛
z1 , . . . , zn1 ∈ Cn1 : Im zn1
⎪
⎩
1.1
2
Journal of Inequalities and Applications
where Zj 1/2Xj − iYj ,
Xj ∂
∂
2kyj |z|2k−2 ,
∂xj
∂t
Yj ∂
∂
− 2kxj |z|2k−2 ,
∂yj
∂t
1.2
and k is a positive integer. The space R2n1 has a natural structure of a Heisenberg group, but
the vector fields are not left or right invariant. In 6 Zhang and Niu studied the Greiner vector
fields on R2n1 for general parameter k ≥ 1 and got the corresponding Hardy-type inequality.
Note that for nonintegral k these vector fields do not satisfy the Hörmander condition and
are not smooth.
H-type groups were introduced by Kaplan 7 as direct generalizations of Heisenberg
groups. In 8 we define a class of vector fields X see 2.5 on H-type groups generalizing the
vector fields 1.2 considered in 5, 6 and find the fundamental solution of the corresponding
p-Laplacian with singularity at the identity element. Also we prove a Hardy-type inequality
associated to X.
The goal of the present paper is to continue our study on analysis associated with
Greiner-type vector fields X introduced in 8. We will throughout study the Rellich
inequality which is a generalization of Hardy inequality to higher-order derivatives. They
have various applications in the study of elliptic and parabolic PDEs. The classical Rellich
inequality in Rn states that for n ≥ 5 and φ ∈ C0∞ Rn \ {0},
Δφx2 dx ≥
Rn
nn − 4
4
φx2
2 Rn
|x|4
dx.
1.3
The constant n2 n − 42 /16 is sharp and is never achieved. Davies and Hinz 9 generalized
1.3 to the Lp case and showed that for any p ∈ 1, n/2 there holds
Δφxp dx ≥
Rn
p φxp
n p − 1 n − 2p
dx.
p2
|x|2p
Rn
1.4
See also the works in 10–13.
In a recent paper 14 Yang obtains an L2 version of Rellich inequality associated
with the left-invariant vector fields in the setting of Heisenberg group, there is a similar L2
Rellich inequality on the general carnot group in 15 with different approach. A natural
question is to find an Lp version of Rellich inequality in this general setting. The main
purpose of the present paper is to prove some weighted Lp -Rellich inequalities associated
with Greiner-type vector fields on H-type groups. Our approach depends on the fundamental
solution of the corresponding square operator and the weighted Hardy inequality proved in
our earlier paper 8. We prove also some weighted Lp Hardy and Rellich inequalities on
nonisotropic Heisenberg groups by a different method caused by the absence of the explicit
representation formula for fundamental solution. As an application, we get a Rellich-Sobolevtype inequality on Heisenberg groups.
The plan of the paper is as follows. In Section 2 we introduce a class of Greinertype vector field and prove the corresponding weighted Lp -Rellich inequality on H-type
groups; Section 3 is devoted to the proof of weighted Hardy- and Rellich-type inequalities
Journal of Inequalities and Applications
3
on nonisotropic Heisenberg groups and a Rellich-Sobolev-type inequality on Heisenberg
groups.
2. Rellich Inequality on H-Type Groups
We recall that a simply connected nilpotent group G is of Heisenberg type, or of H-type, if its
Lie algebra n V ⊕ t is of step two, V, V ⊂ t, t, V 0 and if there is an inner product · , ·
on n such that the linear map
J : t −→ EndV ,
2.1
Jt u, v t, u, v u, v ∈ V, z ∈ t,
2.2
Jt2 −|t|2 Id
2.3
defined by the condition
satisfies
for all t ∈ t, where |t|2 t, t.
Groups of H-type were introduced by Kaplan in 7 as direct generalizations of
Heisenberg groups, and they have been studied quite extensively; see 16–19 and the
references therein.
We identify G with its Lie algebra n via the exponential map, exp : V ⊕ t → G. The Lie
group product is given by
u, tv, s 1
u v, t s u, v .
2
2.4
For g ∈ G, we write g zg, tg ∈ V ⊕ t.
In 8 the authors constructed a family of Greiner-type vector fields X {X1 , . . . , Xm }
on G:
1
Xj ∂j k|z|2k−2 ∂z,ej ,
2
j 1, 2 . . . , m,
2.5
where ∂j ∂ej , ∂z,ej are the directional derivatives, {ej }j1,...,m is an orthonormal basis of V ,
and k ≥ 1 is a fixed parameter. If k 1, Xj are the left-invariant vector fields defined by the
orthonormal basis {ej }m
j1 on V. The corresponding degenerate p-sub-Laplacian is
Lp,k u divX |∇X u|p−2 ∇X u ,
2.6
4
Journal of Inequalities and Applications
where
∇X u X1 u, . . . , Xm u,
divX u1 , . . . , um m
Xj uj .
2.7
j1
We denote
ΔX L2,k m
Xi2 .
2.8
i1
There is a one-parameter group of dilations {δλ : λ > 0} on G:
δλ : z, t −→ w, s λz, λ2k t .
2.9
The volume element is transformed by δ via
2.10
dw ds λQ dz dt,
where Q : m2kq with m dim V and q dim t. Q will be called the degree of homogeneity
and is the homogeneous dimension if k 1. We define a corresponding homogeneous norm by
1/4k
d dz, t : |z|4k 16|t|2
.
2.11
Remark 2.1. Note that when p 2 and k 1, Lp,k becomes the sub-Laplacian ΔG on the Htype group G. If p 2, q 1, and k 2, 3, . . ., Lp,k is a Greiner operator see 5, 20. Also
we note that vector fields {Xj } are neither left nor right invariant and they do not satisfy
Hörmander’s condition for k > 1, k /
∈ Z.
The main results in 8 are the following.
Theorem 2.2. Let G be an H-type group, k ≥ 1, and Q m 2kq. Then for 1 < p < ∞,
Γp ⎧
⎪
⎨Cp dp−Q/p−1 ,
p/
Q,
⎪
⎩CQ log 1 ,
d
pQ
2.12
is a fundamental solution of Lp,k with singularity at the identity element 0 ∈ G. Here dz, t is defined
in 2.11,
−1/Q−1
p − 1 −1/p−1
,
CQ − σQ
,
σp
p−Q
q−1/2 qm/2 π
Γ 2k − 1p m /4k
1
σp .
4
Γm/2Γ 2k − 1p Q /4k
Cp 2.13
Journal of Inequalities and Applications
5
Theorem 2.3. Let G be an H-type group, k ≥ 1, and Q m 2kq. Suppose that 1 < p < Q α.
Then the following Hardy-type inequality holds for Φ ∈ C0∞ G \ {0}:
p
d |∇X Φ| ≥
α
G
Qα−p
p
p G
d
α
|z|
d
2k−1p p
Φ
,
d
2.14
where ∇X f X1 f, X2 f, . . . , Xm f is the gradient defined by the vector fields 2.5. Moreover, the
constant Q α − p/pp is sharp.
Based on the above two theorems, we will prove the following Lp version of weighted
Rellich inequality on H-type groups.
Theorem 2.4. Let G be an H-type group, k ≥ 1, and Q m 2kq. Suppose that 1 < p < ∞, 2 −
Q < α < min{p − 1Q − 2, Q − 2}. Then the following Rellich-type inequality holds for Φ ∈
C0∞ G \ {0}:
G
d
α4kp−1
|z|
p
ΔX φ ≥
2−4kp−1 p p
Q α − 2 p − 1 Q − 2 − α
dα−4k |z|4k−2 φ .
p2
G
2.15
Moreover, the constant Q α − 2p − 1Q − 2 − α/p2 p
is sharp.
Remark 2.5. In the abelian case G Rn , the above result recovers the classical Rellich
inequality 1.4 with dx |x|, α 2 − 2p under the condition 2p < n.
Remark 2.6. When we take p 2 and k q 1, our inequality 2.15 is just inequality 1.5 in
14 and inequality 5.2 in 15 for Heisenberg groups.
Now we prove Theorem 2.4.
Proof. We denote u d2−Q , where d is as in 2.11, then
ΔX dα ΔX
d2−Q
α/2−Q ΔX uα/2−Q
α
α
uα/2−Q−1 ΔX u 2−Q
2−Q
α
− 1 uα/2−Q−2 |∇X u|2 ,
2−Q
2.16
for α > 2 − Q; we have
G
p
p
α
α
α
uα/2−Q−1 ΔX uφ − 1 uα/2−Q−2 |∇X u|2 φ
G 2−Q
G 2−Q 2−Q
p
α
α
− 1 uα/2−Q−2 |∇X u|2 φ
G 2−Q 2−Q
p
αα − 2 Q
dα−2 |∇X d|2 φ
p
φ ΔX dα G
2.17
6
Journal of Inequalities and Applications
since C2 · u is the fundamental solution of ΔX at the origin by Theorem 2.2; however the
left-hand side is
G
p
φ ΔX dα −
p
G
G
p−2
pφ φ∇X φ∇X dα d
α
2 p−2
p−2 p − 1 φ ∇X φ φ φΔX φ .
2.18
Thus, by 2.17, 2.18, and the corresponding weighted Hardy inequality Theorem 2.3, we
have
αα−2Q
G
p
dα−2 |∇X d|2 φ −p
G
p−2
dα φ φΔX φ
2
2 p−2 4p p−1
p/2 α
d ∇X φ φ
d
∇X φ
p2
G
G
2
p
4 p−1 Qα−2 2
p−1
2
α p |∇X d|
≥
d φ
dα−2 |∇X d|2 φ ,
Qα−2
2
p
2
p
d
G
G
2.19
p p−1
α
this implies that
−p
G
p
p−1
2
dα−2 |∇X d|2 φ
Q α − 2 − αα − 2 Q
p
G
p
p−1
Q α − 2
dα−2 |∇X d|2 φ
Q α − 2 − α
p
G
p
Q α − 2 p − 1 Q − 2 − α
dα−2 |∇X d|2 φ .
p
G
p−2
d φ φΔX φ ≥
α
2.20
Applying Hölder’s inequality, we get
p
Q α − 2 p − 1 Q − 2 − α
dα−2 |∇X d|2 φ
2
p
G
1/p
21−p
p
p−1/p
2 p
α−2
α ∇X d ΔX φ
≤
d |∇X d| φ
d .
d G
G
2.21
Noticing that
|∇X d| we have thus proved 2.15.
|z|
d
2k−1
,
2.22
Journal of Inequalities and Applications
7
It remains to show the sharpness of the constant Q α − p /pp . Let B be any
constant satisfying the inequality
G
d
α−22p
p
ΔX φ ≥ B
2−2p |∇X d|
G
p
dα−2 |∇X d|2 φ .
2.23
p
We will prove that B ≤ Q α − 2p − 1Q − 2 − α/p2 . The idea is to find functions
{uj }∞
j1 so that the difference between the left- and right-hand sides approximates to 0. Given
any positive integer j it is elementary that there exists ψj in C0∞ 0, ∞ such that supp ψj 2−j−1 , 2, ψj x 1 on 2−j , 1, and |ψj x| ≤ C2j , |ψj x| ≤ C22j on 2−j−1 , 2−j , where C is a
constant independent of j. Let
uj z, t dz, t2−Q−α/p−1/j ψj dz, t.
2.24
Clearly uj ∈ C∞ G \ {0} which is radial. Denoting Cj by 2 − Q − α/p − 1/j, it is easy to see
that
∇X uj ⎧
⎨0,
0 ≤ d < 2−j−1 , or d > 2
⎩C dCj −1 ∇ d, 2−j < d < 1,
j
X
⎧
⎨0,
ΔX uj ⎩C C Q − 2dCj −2 |∇ d|2 ,
j
j
X
0 ≤ d < 2−j−1 , or d > 2
2.25
2−j < d < 1.
Here we used the fact that
2.26
dΔX d Q − 1|∇X d|2 .
The left-hand side of the above inequality 2.23 is
LHS G
2−j <d<1
2−j−1 <d≤2−j
1≤d<2
2−j <d<1
I II.
2.27
The first integration is
2−j <d<1
p
dα−22p |∇X d|2−2p ΔX φ
Qα−2 1
p
j
p p p − 1 Q − 2 − α 1
−
d−Q−p/j |∇X d|2 ,
p
j
2−j <d<1
2.28
which can be evaluated as the last computations in the proof of Theorem 1 in 8, and is
Qα−2 1
p
j
p p
p − 1 Q − 2 − α 1
−
C0 2p − 1j,
p
j
2.29
8
Journal of Inequalities and Applications
where C0 is a positive constant. Similarly,
RHS B
2−j <d<1
III IV.
2.30
The first integration is precisely the same as above and is
B
2−j <d<1
BC0 2p − 1j.
2.31
It is easy to estimate the error terms which they are all bounded:
I, II, III, IV ≤ C.
2.32
The inequality 2.23 now becomes
Qα−2 1
p
j
p p
p − 1 Q − 2 − α 1
−
C0 2p − 1j I II
p
j
2.33
≥ BC0 2p − 1j III IV.
Dividing both sides by j and letting j → ∞ prove our claim.
The following is an immediate consequence of Theorem 2.4, which is an extension of
the uncertainty principle inequality.
Corollary 2.7. Let G be a Heisenberg-type group, k ≥ 1, and Q m 2kq. Suppose that 1 < p <
∞, 1/p 1/q 1, 2 − Q < α < min{p − 1Q − 2, Q − 2}. Then for all φ ∈ C0∞ G \ {0} the
following inequality holds:
G
p
dα2p−1 |∇X d|21−p ΔX φ
1/p G
q
d2−αq/p |∇X d|2q/p φ
2
Q α − 2 p − 1 Q − 2 − α
φ .
≥
2
p
G
1/q
2.34
Remark 2.8. We mention that when p q 2 and k 1, our inequality 2.34 goes back to
inequality 5.7 in 15 in the setting of H-type group.
We end this section with the following Rellich-type inequality on the polarizable group
which can be proved by the same method if we noted Theorem 2.15 and Proposition 2.18 in
21 and the weighted Hardy inequality Theorem 4.1 in 22.
Journal of Inequalities and Applications
9
Theorem 2.9. Let G be a polarizable group with homogeneous dimension Q > 4. Suppose that 1 <
p < ∞, 2 − Q < α < min{p − 1Q − 2, Q − 2}. Then the following inequality holds for all
φ ∈ C0∞ G \ {0}:
p 21−p
2
p
Q α − 2 p − 1 Q − 2 − α
α ∇G N p
ΔG φ ≥
φ .
N N
2
N
N p
G
G
2.35
α ∇G N Here N u1/2−Q is the homogeneous norm associated with the fundamental solution u for the Kohn
p
sub-Laplacian. Moreover, the constant Q α − 2p − 1Q − 2 − α/p2 is sharp.
3. Rellich Inequality on Nonisotropic Heisenberg Groups
Let a a1 , a2 , . . . , an , a1 , a2 , . . . , an > 0. Let H Ha R2n ⊕ R be the corresponding
nonisotropic Heisenberg group with the product
⎞
⎛
n
ζ, t ◦ η, s ⎝ζ η, t s 2 aj ζjn ηj − ζj ηjn ⎠.
3.1
j1
We consider the following nonisotropic Greiner-type vector fields:
⎛
⎞k−1
n
2
∂
∂
,
Xj 2kaj yj ⎝ aj zj ⎠
∂xj
∂t
j1
⎛
Yj ⎞k−1
n
2
∂
− 2kaj xj ⎝ aj zj ⎠
∂yj
j1
j 1, . . . , n,
3.2
∂
,
∂t
where x, y, t ∈ R2n ⊕ R, x x1 , x2 , . . . , xn , y y1 , y2 , . . . , yn . Denote
∇H u X1 u, . . . , Xn u, Y1 u, . . . , Yn u,
divH u1 , . . . , u2n n
Xj uj Yj unj ,
j1
⎞1/4k
⎞2k
n
2
⎟
⎜
,
d ⎝⎝ aj zj ⎠ t2 ⎠
⎛⎛
3.3
j1
n
Q 2 aj 2k.
j1
For further information on the nonisotropic Heisenberg group, see, for example, 23, 24.
10
Journal of Inequalities and Applications
In this section we prove firstly the Lp Hardy-type inequality associated with the
Greiner-type vector fields 3.2 on the nonisotropic Heisenberg group H.
Theorem 3.1. Let H Ha be the anisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Let
α ∈ R, 2 ≤ p < Q α and Φ ∈ C0∞ H \ {0}. Then the following inequality is valid:
dα |∇L Φ|p ≥
H
where d n
j1
⎛
p Qα−p
p
H
1/4k
aj |zj |2 2k t2 ⎞p/2 ⎛
⎞k−1p
n
n
2
2
dα−2kp ⎝ a2j zj ⎠ ⎝ aj zj ⎠
|Φ|p ,
j1
, Q2
n
j1
3.4
j1
aj 2k.
For the proof of the above inequality, we need the following lemma which can be
proved by a similar method in 25.
Lemma 3.2. Let w ≥ 0 be a weight function in Ω ⊂ G and Lp,k,w u divH |∇H u|p−2 w∇H u.
Suppose that for some λ > 0, there exists v ∈ C∞ Ω, v > 0 such that
−Lp,k,w v ≥ λgvp−1
3.5
for some g ≥ 0, in the sense of distribution acting on nonnegative test functions. Then for any u ∈
1,p
HW0 Ω, w, it holds that
Ω
|∇H u|p w ≥ λ
1,p
Ω
g|u|p ,
where HW0 Ω, w denote the closure of C0∞ Ω in the norm 3.6
1/p
Ω
|∇H u|p w
.
We now prove Theorem 3.1.
Proof. Take w dα and v dp−Q−α/p . Noting that
LH,p,w v !
Xi |∇L v|p−2 wXi v Yi |∇L v|p−2 wYi v ,
n
i1
"
Xi d Yi d 2 2k−1
2 k−1
n
aj zj ai xi ai yi t
j1 aj zj
#
d4k−1
#
"
2 2k−1
2 k−1
n
n
ai yi −
ai xi t
j1 aj zj
j1 aj zj
|∇L d|2 n
j1
n
j1
2 n
a2j zj j1
d4k−2
d4k−1
2 2k−2
aj zj ,
3.7
,
,
Journal of Inequalities and Applications
11
we get
α − p −Qα/p p−2
p−2 p − Q − α
−Qα/p
Xi
d
d
Xi d
−LH,p,w v −
|∇L d|
p
p
i1
p−2
p−2 p − Q − α
α Q α − p −Qα/p −Qα/p
Yi d d
d
Yi d
|∇L d|
p
p
n
d Qα−p
p
α Q
3.8
p−1
{I1 I2 I3 I4 },
where
I1 Q α − pQ 2 − p 2k − 1 dQα−pQ−p/p2−p2k−1
p
⎛
⎞p−2/2 ⎛
⎞p−2k−1
n
n
2
⎝ aj zj 2 ⎠
× ⎝ a2 zj ⎠
|∇L d|2 ;
j
j1
j1
⎛
⎞p−4/2 ⎛
⎞p−2k−1
n
n
Qα−pQ/p2−p2k−1 2
2
⎝ a2 zj ⎠
⎝ aj zj ⎠
I2 p − 2 d
j
j1
×
j1
n
a2j xj · Xj d yj · Yj d ;
j1
⎛
⎞p−2/2 ⎛
⎞p−2k−1−1
n
n
2
2
Qα−pQ/p2−p2k−1 2
⎝ a zj ⎠
⎝ aj zj ⎠
I3 2k − 1 p − 2 d
j
j1
×
j1
n
aj xj · Xj d yj · Yj d ;
j1
⎛
⎞p−2/2 ⎛
⎞p−2k−1
n
n
n 2
⎝ aj zj 2 ⎠
Xj2 d Yj2 d .
I4 dQα−pQ/p2−p2k−1 ⎝ a2j zj ⎠
j1
j1
j1
3.9
By direct computations we have
n
aj xj · Xj d yj · Yj d n
j1
d4k−1
j1
n
a2j xj · Xj d yj · Yj d j1
2 2k−1 n 2 2 aj zj j1 aj zj
n
j1
2 2k−1 n 3 2 aj zj j1 aj zj
d4k−1
,
,
12
Journal of Inequalities and Applications
n j1
⎞2k−1 n
⎛
2 n 2 2
n
n
2 j1 aj
j1 aj zj
j1 aj zj
2
2
2
⎠
⎝
Xj d Yj d aj zj
.
d4k−1
j1
3.10
Then
Qα−p
−LH,p,w v p
p−1
⎛
⎞p−4/2 ⎛
⎞k−1p
n
n
2
⎝ aj zj 2 ⎠
dQα/p−1−Q2k−12−p ⎝ a2 zj ⎠
· II,
j
j1
j1
3.11
where
II ⎧
⎪
⎨ Q α
⎪
⎩
p
− Q 2k − 1 2 − p
⎛
⎛
⎞2
⎞⎛
⎞
n
n
n
2
2
2
2
3
⎝ a zj ⎠ p − 2 ⎝ aj zj ⎠⎝ a zj ⎠
j
j
j1
j1
j1
⎛
⎞2 ⎛
⎞
n
n
2
2
2k − 1 p − 2 ⎝ a2j zj ⎠ ⎝ a2j zj ⎠
j1
j1
⎞⎛
⎞⎫⎫
n
n
n
⎬
2
2 ⎬⎪
2 ⎠
⎝
⎝
⎠
⎝
.
2 aj
× 2k − 1
aj zj
aj zj ⎠
⎭⎪
⎩
⎭
j1
j1
j1
⎧
⎨
⎛
⎞
⎛
3.12
Using Cauchy inequality
⎛
⎞⎛
⎞ ⎛
⎞2
n
n
n
2
2
2
3
2
⎝ aj zj ⎠⎝ a zj ⎠ ≥ ⎝ a zj ⎠
j
j
j1
j1
3.13
j1
and that since, by assumption, aj ≤ 1, j 1, . . . , n
n
n
2 2
aj zj ≥
a2j zj ,
j1
3.14
j1
we find
⎞2
⎛
n
Q α − p ⎝
2
II ≥
a2j zj ⎠ ,
p
j1
3.15
Journal of Inequalities and Applications
13
so for p ≥ 2,
−LH,p,w v ≥
Qα−p
p
Qα−p
p
p dp−Q−α/p
p−1
⎛
⎞p/2 ⎛
⎞k−1p
n
n
2
2
dα−2kp ⎝ a2j zj ⎠ ⎝ aj zj ⎠
j1
p
j1
3.16
vp−1 dα−p |∇L d|p .
Hence Theorem 3.1 follows from Lemma 3.2 with λ Q α − p/pp , g dα−p |∇L d|p .
Now it is time to prove the following Rellich inequality on nonisotropic Heisenberg
group.
Theorem 3.3. Let H Ha be a nonisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Suppose
that α ∈ R, 2 − Q < α ≤ 0 and u ∈ C0∞ H \ {0}. Then the following inequality is valid:
H
dα−4k4kp
p
2k−1p−1 |ΔH u|
2
n
2 j1 aj zj
⎛
⎞2k−1
p n
p − 1 Q − 2 − α Q α − 2
2
dα−4k ⎝ a2j zj ⎠
|u|p ,
p2
H
j1
≥
where d n
aj |zj |2 j1
2k
1/4k
t2 , Q2
n
j1
3.17
aj 2k.
Remark 3.4. If p 2, α −2, and k ai 1 i 1, . . . , n, then we get the Rellich inequality
on the Heisenberg group Hn with homogeneous dimension Q 2n 2:
d2
Hn
|z|
2
|ΔHn u| ≥
2
QQ − 4
4
2 Hn
|z|2 2
|u| .
d6
3.18
We also mention that to our knowledge, even in the special case k aj 1, j 1, 2, . . . , n,
our inequality 3.17 is new.
We now give the proof of Theorem 3.3.
Proof. We have
p
|u|p α − 1dα−2 |∇H d|2 dα−1 ΔH d
|u| ΔX d α
H
α
H
3.19
p α−2
αα − 1
|u| d
H
2
p α−1
|∇H d| α
|u| d
H
ΔH d.
14
Journal of Inequalities and Applications
By a direct computation we have
"
Xi d n
j1
2 2k−1
2 k−1
n
aj zj ai xi ai yi t
j1 aj zj
#
,
d4k−1
"
Yi d n
j1
2 2k−1
2 k−1
n
zj aj zj ai yi −
a
ai xi t
j
j1
,
d4k−1
n
j1
|∇H d|2 2 n
2 2k−2
a2j zj j1 aj zj
d4k−2
#
,
⎞2k−1 n
2 n
2k − 1 n a2 zj 2
n
2 j1 aj
2
j1 aj zj
j1 j
ΔH d ⎝ aj zj ⎠
.
d4k−1
j1
⎛
3.20
Noting that ai ≤ 1 then
dΔH d ≥ Q − 1|∇H d|2 ;
3.21
we take α < 0 and thus deduce from 3.19 that
|u|p dα−2 |∇H d|2 .
|u|p ΔX dα ≤ αQ α − 2
H
3.22
H
On the other hand,
|u|p ΔH dα −p
H
|u|p−2 u∇H u∇H dα H
p
d
α
p − 1 |u|p−2 |∇H u|2 |u|p−2 uΔH u .
3.23
H
Combining with 3.22, 3.23, and the L2 weighted Hardy inequality associated with
the Greiner-type vector fields 3.2 on the nonisotropic Heisenberg group 3.4, then we only
need to do the same steps as in Theorem 2.4 and thus finish the proof of Theorem 3.3.
Journal of Inequalities and Applications
15
The following is also an uncertainty principle type inequality which is an immediate
consequence of Theorem 3.3.
Corollary 3.5. Let H Ha be an anisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Then
for u ∈ C0∞ H \ {0}, 1/p 1/q 1 1 < p < ∞, the following inequality holds:
⎛ ⎛
⎜ ⎜
⎝ ⎝
n
H
⎞p−1
d4k
⎟
2k−1 ⎠
2
2 j1 aj zj
⎞1/p ⎛ ⎛
⎟ ⎜ ⎜
|ΔH u|p ⎠ ⎜
⎝ ⎝ n
H
⎞1/p−1
⎞1/q
⎟
|u|q ⎟
⎠
d4k
⎟
2k−1 ⎠
2
2 j1 aj zj
≥
p − 1 Q − 22
|u|2 ,
p2
H
where d n
1/4k
2
j1
3.24
aj |zj |2k t2 , Q2
n
j1
aj 2k.
By a similar method, we can get the following inequality which does not contain the
weight |∇H d|, so we omit the proof.
Theorem 3.6. Let H Ha be a nonisotropic Heisenberg group with aj ≤ 1, j 1, . . . , n. Let
α ∈ R, 2 − Q < α ≤ 0 and u ∈ C0∞ H \ {0}. Then the following inequality is valid:
p p − 1 2 ni1 ai − α 2 ni1 ai α − 2p
|z|α−2p |u|p .
p2
H
α
p
|z| |ΔH u| ≥
H
3.25
With the help of Theorem 3.6, we can also obtain a Rellich-Sobolev-type inequality on
the Heisenberg group.
Corollary 3.7. Let H Hn be the Heisenberg group with homogeneous dimension Q 2n 2.
Suppose that 1 < p < Q − 2/2, 0 ≤ s ≤ p, ps pQ − 2s/Q − 2p. Then there exists a positive
constant C such that for any u ∈ C0∞ H \ {0}, the following inequality holds:
|u|ps
H
1/ps
|ΔH u|p
≤C
|z|2s
1/p
3.26
.
H
Proof. By Hölder inequality we have
|u|ps
H
|z|2s
|u|s
H
|z|
Qp−s/Q−2p
|u|
2s
|u|p
≤
H
|z|2p
s/p |u|pQ/Q−2p
H
p−s/p
.
3.27
16
Journal of Inequalities and Applications
Thanks to the Rellich inequality 3.25 with α 0, ai 1 i 1, 2, . . . , n and the Sobolev
inequality on the nilpotent Lie group Chapter IV Theorem 3.3.1 in 26 we get
|u|ps
H
|z|2s
|H u|p
≤C
s/p H
H
|H u|p
Qp−s/pQ−2p
|H u|p
C
Q−2s/Q−2p
.
H
3.28
Thus we obtain the desired result.
Acknowledgments
The authors would like to thank Professor Genkai Zhang for constant encouragement and
the referee for his/her helpful comments. The research was supported by the Natural Science
Foundation of Zhejiang Province no. Y6090359, Y6090383 and Department of Education of
Zhejiang Province no. Z200803357.
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