Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 987484, 9 pages
doi:10.1155/2010/987484
Research Article
Ostrowski Type Inequalities in the Grushin Plane
Heng-Xing Liu and Jing-Wen Luan
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
Correspondence should be addressed to Jing-Wen Luan, jwluan@whu.edu.cn
Received 7 January 2010; Accepted 14 March 2010
Academic Editor: Kunquan Q. Lan
Copyright q 2010 H.-X. Liu and J.-W. Luan. 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.
Motivated by the work of B.-S. Lian and Q.-H. Yang 2010 we proved an Ostrowski inequality
associated with Carnot-Carathéodory distance in the Grushin plane. The procedure is based on
a representation formula. Using the same representation formula, we prove some Hardy type
inequalities associated with Carnot-Carathéodory distance in the Grushin plane.
1. Introduction
The classical Ostrowski inequality 1 is as follows:
1 b 1 x − a b/22
f y dy − fx ≤
b − af ∞
2
b − a a
4
b − a
1.1
for f ∈ C1 a, b, x ∈ a, b, and it is a sharp inequality. Inequality 1.1 was extended from
intervals to rectangles and general domains in Rn see 2–5. Recently, it has been proved by
the same authors 6 that there exists an Ostrowski inequality on the 3-dimension Heisenberg
group associated with horizontal gradient and Carnot-Carathéodory distance, and it is also a
sharp inequality.
The aim of this note is to establish some Ostrowski type inequality in the Grushin
plane, known as the simplest example of sub-Riemannian metric associated with Grushin
operator cf. 7–10. Recall that in the Grushin plane, the sub-Riemannian metric is given by
the vectors
X1 ∂
,
∂x
X2 x
∂
∂y
1.2
2
Journal of Inequalities and Applications
and satisfies X1 , X2 ∂/∂y. By Chow’s conditions, the Carnot-Carathéodory distance
dcc u, v between any two points u, v ∈ R2 is finite cf. 11. We denote dcc u dcc o, u,
where o 0, 0 is the origin. Define on R2 the dilation δλ as
δλ u δλ x, y : λx, λ2 y ,
u x, y ∈ R2 .
1.3
For simplicity, we will write it as λu λx, λ2 y. It is not difficult to check that X1 and X2 are
homogeneous of degree one with respect to the dilation. The Jacobian determinant of δλ is
λQ , where Q 1 2 3 is the homogeneous dimension. The Carnot-Carathéodory distance
dcc satisfies
dcc λ x, y λdcc x, y ,
1.4
λ > 0.
Let BR be the Carnot-Carathéodory ball centered at the origin o and of radius R > 0.
Let Σ ∂B1 be the corresponding unit sphere. Let dσ be the surface measure on Σ. Given any
2
o/
u x, y ∈ R2 , set x∗ x/dcc u, y∗ y/dcc
u, and u∗ x∗ , y∗ . For f ∈ CBR o, let
1
fr
|Σ|
Σ
fru∗ dσ,
0 < r ≤ R,
1.5
be the averages of f over the unit sphere, where |Σ| denote the volume of the Σ. Then, we can
state our result as follows.
Theorem 1.1. Let f ∈ C1 BR . Then for u x, y ru∗ , there holds
3
1
r4
∇L f ,
R−r
fvdv ≤ N f fu −
∞
Q
4
|BR | BR
2R
1.6
where
N f := supfu − fr
f − f ,
∞
u∈BR
1.7
|BR | denote the volume of the BR , and ∇L is the gradient operator defined by ∇L X1 , X2 . The
constants in 1.6 are the best possible, equality that can be attained for nontrivial radial functions at
any r ∈ 0, R.
We also obtain the following Hardy type inequalities in the Grushin plane. We refer to
12 the Hardy inequalities associated with nonisotropic gauge induced by the fundamental
solution.
Theorem 1.2. Let f ∈ C0∞ R2 . There holds, for 1 < p < 3,
R2
∇L fup du ≥
3−p
p
p
f p R2
p
dcc u
du.
1.8
Journal of Inequalities and Applications
3
2. Geodesics in the Grushin Plane
In this section, we will follow 13 to give a parametrization of Grushin plane using the
geodesics. Recall that the Grushin operator is given by
ΔL ∂2
∂2
x2 2 .
2
∂x
∂y
2.1
The associated Hamiltonian function Hx, y, ξ, θ is of the form
1 2
ξ x2 η2 .
H x, y, ξ, η 2
2.2
It is known that geodesics in the Grushin plane are solutions of the Hamiltonian system cf.
8
ẋs ∂H
ξs,
∂ξ
ξ̇s −
∂H
−xη2 s,
∂x
∂H
x2 η,
ẏs ∂η
η̇s −
∂H
0,
∂y
2.3
that is, ηs η0.
Taking the initial date x0, y0 0, 0 and ξ0, η0 A, φ, one can find the solutions
cf. 8
xs A
ys A
sin φs
,
φ
2 2φs
− sin 2φs
,
4φ
2.4
where the time s is exactly the Carnot-Carathéodory distance. Letting φ → 0, we get the
Euclidean geodesics
xs, ys As, 0
2.5
and hence the correct normalization is A2 1.
Set
Ω
φ, ρ ∈ R2 : −π ≤ φρ ≤ π, ρ ≥ 0
2.6
4
Journal of Inequalities and Applications
and define Φ : Ω → R2 by Φφ, ρ xφ, ρ, yφ, ρ, where
sin φρ
x φ, ρ A
,
φ
2φρ − sin 2φρ
y φ, ρ 4φ2
2.7
with A2 1. If A 1, the range of Φ is 0, ∞ × R; if A −1, the range of Φ is −∞, 0 × R.
Furthermore, if one fixes ρ > 0, 2.7 with A ±1 and −π/ρ ≤ φ ≤ π/ρ parameterize ∂Bρ .
On the other hand, the Carnot-Carathéodory distance dcc satisfies cf. 10, Theorem
2.6, for x / 0,
dcc x, y θ
|x|,
sin θ
2.8
where θ μ−1 2y/x2 μθ θ
2
sin θ
2θ − sin 2θ
− cot θ 2sin2 θ
: −π, π −→ R
2.9
is a diffeomorphism of the interval −π, π onto R cf. 14, and μ−1 is the inverse function
of μ. From 2.7, we have
μθ 2y 2φρ − sin 2φρ
μ φρ .
2
2
x
2 sin φρ
2.10
Therefore,
θ φρ
2.11
since μ is a diffeomorphism.
We finally recall the polar coordinates in the Grushin plane associated with dcc . The
following coarea formula has been proved in 15:
R2
fu|∇L dcc u|du ∞ −∞
{dcc uλ}
fudP Eλ dλ,
2.12
where Eλ {u ∈ R2 : dcc u > λ} and P Eλ is the perimeter-measure. Notice that |∇L dcc u| 1 a.e. cf. 15; and P Eλ λ2 P E1 cf. 9, Proposition 2.2; we have the following polar
coordinates in the Grushin plane, for all f ∈ L1 R2 :
R2
fudu ∞ 0
Σ
fλu∗ λ2 dσdλ.
2.13
Journal of Inequalities and Applications
5
3. The Proofs
To prove the main result, we first need the following representation formula.
Lemma 3.1. Let R2 > R1 > 0 and f ∈ C1 BR2 \ BR1 . There holds
∗
Σ
fR2 u dσ −
∗
Σ
fR1 u dσ BR2 \BR1
1
∇L f, ∇L dcc · 2 du.
dcc
3.1
Proof. Let u∗ be a point on the sphere, that is, u∗ x∗ , y∗ , where dcc x∗ , y∗ 1. We consider
for 0 < R1 < R2 the following difference using the fundamental theorem of calculus:
∗
Σ
fR2 ξ dσ −
∗
Σ
fR1 ξ dσ R2
Σ
d ∗
f ρξ dρdσ
R1 dρ
R2 Σ
R1
∂fu ∂x ∂fu ∂y
·
·
dρdσ,
∂x
∂ρ
∂y
∂ρ
3.2
where u x, y ρu∗ . Using 2.7, we have
∂fu sin2 φρ ∂fu
∂fu ∂x ∂fu ∂y
·
·
A cos φρ
∂x
∂ρ
∂y
∂ρ
∂x
φ
∂y
A cos φρ
3.3
∂fu
∂fu
A sin φρ · x
∂x
∂y
A cos φρX1 fu A sin φρX2 fu.
Combining 3.2 and 3.3 and rewriting the expression into a solid integral using the polar
coordinates, we get
∗
Σ
fR2 ξ dσ −
∗
Σ
fR1 ξ dσ A cos φρX1 fu A sin φρX2 fu
BR2 \BR1
2
dcc
du.
3.4
To finish our proof, it is enough to show that
X1 dcc u A cos φρ,
X2 dcc u A sin φρ
3.5
in R \ {0} × R. This is just the following Lemma 3.2. The proof of Lemma 3.1 is now complete.
Lemma 3.2. There hold, for x /
0,
X1 dcc u A cos φρ,
X2 dcc u A sin φρ.
3.6
6
Journal of Inequalities and Applications
Proof. Recall that if x /
0, then
dcc u dcc x, y θ
|x|,
sin θ
3.7
where θ μ−1 2y/x2 . The simple calculation shows
μ θ 2 sin θ − 2θ cos θ
sin3 θ
,
−4y
1
∂θ
·
,
∂x μ θ x3
1
−2
∂θ
·
.
∂y μ θ x2
3.8
Therefore, if x /
0, then
X1 dcc u ∂
∂dcc u
∂x
∂x
θ
|x|
sin θ
sin θ − θ cos θ ∂θ
θ
x
|x| ·
·
·
∂x
|x| sin θ
sin2 θ
−2y
x
θ
θ
x
|x|
− |x| sin θ · 3 −
· sin θ · μθ
·
·
x
|x| sin θ
|x| sin θ
x
θ
θ
A
− A sin θ ·
−
cot
θ
,
sin θ
sin2 θ
3.9
A cos θ.
On the other hand,
X2 dcc u x
∂
∂dcc u
x
∂y
∂y
θ
|x|
sin θ
x|x| ·
sin θ − θ cos θ ∂θ
·
∂y
sin2 θ
|x|
· sin θ A sin θ.
x
3.10
Therefore, we obtain, by 2.11,
X1 dcc u A cos φρ,
X2 dcc u A sin φρ.
3.11
This completes the proof of Lemma 3.2.
Proof of Theorem 1.1. We have, by Lemma 3.1, since |BR | R
0 Σ
s2 dσds |Σ|R3 /3,
1
fvdv
fu −
|BR | BR
R
1 3
∗ 2
≤ fu − fr
fru∗ dσ −
fsu
dσds
s
|Σ| Σ
|Σ|R3 0 Σ
Nc f ∗,
3.12
Journal of Inequalities and Applications
7
where
R
1
3
∗
∗ 2
fru dσ −
fsu
dσds
∗ s
3
|Σ| Σ
|Σ|R 0 Σ
3 R
∗
∗
2
fru dσ − fsu dσ s ds
3
|Σ|R 0
Σ
Σ
3
≤
|Σ|R3
3
≤ 3·
R
R ∗
fru∗ dσ −
fsξ dσ s2 ds
0
R
0
Σ
3.13
Σ
|r − s|s2 ds · ∇L f ∞
r4
3
R−r
4
2R3
∇L f .
∞
To see that the estimate in 1.8 is sharp, we consider the function fu fdcc u |r − dcc u| and that r is fixed in 0, R. Notice that |∇L fu| 1 a.e.; we have |∇L fu|∞ 1.
We look at inequality 1.6 evaluating the function at r. Since fr 0, the left-hand side of
1.6 is
L.H.S. 1.6 3
·
R3
R
|r − s|s2 ds 0
r4
3
R−r
4
2R3
3.14
and the right-hand side of 1.8 is
R.H.S. 1.8 r4
3
R−r
.
4
2R3
3.15
Thus the equality holds in 1.6. This completes the proof of the sharpness of inequality 1.6.
The proof of the theorem is now complete.
Proof of Theorem 1.2. Let > 0. Then 0 ≤ f : |f|2 2 p/2 − p ∈ C0∞ R2 . In fact, f has the
3−p
same support as f. Putting f dcc u in Lemma 3.1 and letting R2 → ∞ and R1 → 0, we
get, since dcc o 0,
R2
∇L f , ∇L dcc ·
1
p−1
dcc
3−p
f
R2
p
dcc
0.
3.16
8
Journal of Inequalities and Applications
Here we use the fact |∇L dcc u| 1 a.e. Therefore,
3−p
f
p
R2 dcc
−p
p−2/2 2
f 2
f ∇L f, ∇L dcc ·
R2
p−2/2 f · ∇L f f 2 2
≤p
p−1
R2
dcc
p−1/2 f 2 2
· ∇L f ≤p
1
p−1
dcc
p−1
R2
dcc
3.17
.
By dominated convergence, letting → 0, we have
p
f 3−p
R2
p
dcc
p−1 f · ∇L f ≤p
R2
p−1
dcc
.
3.18
By Hölder’s inequality,
3−p
p
f R2
p
dcc
≤p
p p−1/p f 1/p
∇L f p
.
p
R2 dcc
R2
3.19
Canceling and raising both sides to the power p, we get 1.8.
Acknowledgment
This work was supported by Natural Science Foundation of China no. 10871149 and no.
10671009.
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