Journal of Inequalities and Applications Volume 2010, Article ID 987484,9pages 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
Copyrightq2010 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. Yang2010we proved an Ostrowski inequality associated with Carnot-Carath´eodory 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-Carath´eodory distance in the Grushin plane.
1. Introduction
The classical Ostrowski inequality1is as follows:
1 b−a
b
a
fydy−fx≤
1 4
x−ab/22 b−a2
b−af∞ 1.1
forf ∈C1a, b,x∈a, b,and it is a sharp inequality. Inequality1.1was extended from
intervals to rectangles and general domains inRnsee2–5. Recently, it has been proved by
the same authors6that there exists an Ostrowski inequality on the 3-dimension Heisenberg group associated with horizontal gradient and Carnot-Carath´eodory 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 operatorcf.7–10. Recall that in the Grushin plane, the sub-Riemannian metric is given by
the vectors
X1 ∂
∂x, X2x ∂
and satisfies X1, X2 ∂/∂y. By Chow’s conditions, the Carnot-Carath´eodory distance
dccu, vbetween any two pointsu, v ∈ R2 is finitecf.11. We denotedccu dcco, u,
whereo 0,0is the origin. Define onR2the dilationδ
λas
δλuδλ
x, y: λx, λ2y, ux, y∈R2. 1.3
For simplicity, we will write it asλu λx, λ2y. It is not difficult to check thatX
1andX2are
homogeneous of degree one with respect to the dilation. The Jacobian determinant ofδλis
λQ, whereQ12 3 is the homogeneous dimension. The Carnot-Carath´eodory distance
dccsatisfies
dcc
λx, yλdcc
x, y, λ >0. 1.4
LetBR be the Carnot-Carath´eodory ball centered at the originoand of radiusR >0.
LetΣ ∂B1be the corresponding unit sphere. Letdσbe the surface measure onΣ. Given any
o /u x, y∈R2, setx∗x/d
ccu,y∗y/d2ccu,andu∗ x∗, y∗. Forf∈CBRo, let
fr 1 |Σ|
Σfru
∗dσ, 0< r ≤R, 1.5
be the averages offover the unit sphere, where|Σ|denote the volume of theΣ. Then, we can state our result as follows.
Theorem 1.1. Letf ∈C1B
R. Then foru x, y ru∗, there holds
fu−
1 |BR|
BR
fvdv≤ Nf
3 4R−r
r4
2RQ
∇Lf
∞, 1.6
where
Nf := sup
u∈BR
fu−frf−f
∞, 1.7
|BR|denote the volume of the BR,and ∇L is the gradient operator defined by∇L X1, X2. The
constants in1.6are the best possible, equality that can be attained for nontrivial radial functions at anyr∈0, R.
We also obtain the following Hardy type inequalities in the Grushin plane. We refer to 12the Hardy inequalities associated with nonisotropic gauge induced by the fundamental solution.
Theorem 1.2. Letf ∈C∞0 R2. There holds, for1< p <3,
R2
∇Lfup
du≥
3−p p
p
R2
fp
dpccu
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
∂x2 x 2 ∂2
∂y2. 2.1
The associated Hamiltonian functionHx, y, ξ, θis of the form
Hx, y, ξ, η 1 2 ξ
2x2η2. 2.2
It is known that geodesics in the Grushin plane are solutions of the Hamiltonian systemcf. 8
˙
xs ∂H ∂ξ ξs,
˙
ξs −∂H ∂x −xη
2s,
˙
ys ∂H ∂η x
2η,
˙
ηs −∂H
∂y 0, that is, ηs η0.
2.3
Taking the initial datex0, y0 0,0andξ0, η0 A, φ, one can find the solutions cf.8
xs Asinφs φ ,
ys A22φs−sin 2φs
4φ ,
2.4
where the time sis exactly the Carnot-Carath´eodory distance. Letting φ → 0, we get the Euclidean geodesics
xs, ys As,0 2.5
and hence the correct normalization isA21.
Set
and defineΦ:Ω → R2byΦφ, ρ xφ, ρ, yφ, ρ, where
xφ, ρAsinφρ φ ,
yφ, ρ 2φρ−sin 2φρ 4φ2
2.7
withA2 1. IfA1, the range ofΦis0,∞×R; ifA−1, the range ofΦis−∞,0×R.
Furthermore, if one fixesρ >0,2.7withA±1 and−π/ρ≤φ≤π/ρparameterize∂Bρ.
On the other hand, the Carnot-Carath´eodory distancedcc satisfiescf. 10, Theorem
2.6, forx /0,
dcc
x, y θ
sinθ|x|, 2.8
whereθμ−12y/x2
μθ θ
sin2θ −cotθ
2θ−sin 2θ
2sin2θ :−π, π−→R 2.9
is a diffeomorphism of the interval−π, πontoRcf.14, andμ−1 is the inverse function
ofμ. From2.7, we have
μθ 2y x2
2φρ−sin 2φρ 2 sin2φρ μ
φρ. 2.10
Therefore,
θφρ 2.11
sinceμis a diffeomorphism.
We finally recall the polar coordinates in the Grushin plane associated withdcc. The
following coarea formula has been proved in15:
R2
fu|∇Ldccu|du
∞
−∞
{dccuλ}
fudPEλdλ, 2.12
whereEλ{u∈R2:dccu> λ}andPEλis the perimeter-measure. Notice that|∇Ldccu|
1 a.e.cf.15; andPEλ λ2PE1 cf.9, Proposition 2.2; we have the following polar
coordinates in the Grushin plane, for allf ∈L1R2:
R2
fudu
∞
0
Σfλu
3. The Proofs
To prove the main result, we first need the following representation formula.
Lemma 3.1. LetR2> R1>0andf ∈C1BR2\BR1. There holds
ΣfR2u
∗dσ−
ΣfR1u
∗dσ
BR2\BR1
∇Lf,∇Ldcc
· 1 d2
cc
du. 3.1
Proof. Letu∗be a point on the sphere, that is,u∗ x∗, y∗, wheredccx∗, y∗ 1. We consider
for 0< R1< R2the following difference using the fundamental theorem of calculus:
ΣfR2ξ
∗dσ−
ΣfR1ξ
∗dσ
Σ
R2
R1 d dρf
ρξ∗dρdσ
Σ
R2
R1
∂fu ∂x ·
∂x ∂ρ
∂fu ∂y ·
∂y ∂ρ
dρdσ,
3.2
whereu x, y ρu∗. Using2.7, we have
∂fu ∂x ·
∂x ∂ρ
∂fu ∂y ·
∂y
∂ρ Acosφρ ∂fu
∂x
sin2φρ φ
∂fu ∂y
Acosφρ∂fu
∂x Asinφρ·x ∂fu
∂y
AcosφρX1fu AsinφρX2fu.
3.3
Combining3.2and3.3and rewriting the expression into a solid integral using the polar coordinates, we get
ΣfR2ξ
∗dσ−
ΣfR1ξ
∗dσ
BR2\BR1
AcosφρX1fu AsinφρX2fu
d2 cc
du. 3.4
To finish our proof, it is enough to show that
X1dccu Acosφρ, X2dccu Asinφρ 3.5
inR\ {0} ×R. This is just the following Lemma3.2. The proof of Lemma3.1is now complete.
Lemma 3.2. There hold, forx /0,
Proof. Recall that ifx /0, then
dccu dcc
x, y θ
sinθ|x|, 3.7
whereθμ−12y/x2.The simple calculation shows
μθ 2 sinθ−2θcosθ sin3θ ,
∂θ ∂x
1 μθ·
−4y x3 ,
∂θ ∂y
1 μθ·
−2 x2.
3.8
Therefore, ifx /0, then
X1dccu ∂dccu
∂x
∂ ∂x
θ sinθ|x|
|x x|·
θ
sinθ|x| ·
sinθ−θcosθ sin2θ ·
∂θ ∂x
|x x|·
θ
sinθ − |x|sinθ· −2y
x3
x |x|·
θ sinθ −
|x|
x ·sinθ·μθ
A θ
sinθ−Asinθ·
θ
sin2θ −cotθ
,
Acosθ.
3.9
On the other hand,
X2dccu x∂dccu
∂y x
∂ ∂y
θ sinθ|x|
x|x| ·sinθ−θcosθ sin2θ ·
∂θ ∂y
|x|
x ·sinθAsinθ.
3.10
Therefore, we obtain, by2.11,
X1dccu Acosφρ, X2dccu Asinφρ. 3.11
This completes the proof of Lemma3.2.
Proof of Theorem1.1. We have, by Lemma3.1, since|BR|
R
0
Σs2dσds|Σ|R3/3,
fu−
1 |BR|
BR
fvdv
≤fu−fr|Σ|1
Σfru
∗dσ− 3 |Σ|R3
R
0
Σfsu
∗s2dσds
Nc
f ∗,
where
∗ |Σ|1
Σfru
∗dσ− 3 |Σ|R3
R
0
Σfsu
∗s2dσds
|Σ|3 R3
R
0
Σfru
∗dσ−
Σfsu
∗dσs2ds
≤ 3 |Σ|R3
R
0
Σfru
∗dσ−
Σfsξ
∗dσs2ds
≤ 3 R3 ·
R
0
|r−s|s2ds·∇
Lf∞
3 4R−r
r4
2R3
∇Lf
∞.
3.13
To see that the estimate in1.8is sharp, we consider the functionfu fdccu
|r−dccu|and thatris fixed in0, R. Notice that|∇Lfu|1 a.e.; we have|∇Lfu|∞1. We look at inequality1.6evaluating the function atr. Sincefr 0, the left-hand side of 1.6is
L.H.S.1.6 3 R3 ·
R
0
|r−s|s2ds 3 4R−r
r4
2R3 3.14
and the right-hand side of1.8is
R.H.S.1.8 3 4R−r
r4
2R3. 3.15
Thus the equality holds in1.6. This completes the proof of the sharpness of inequality1.6.
The proof of the theorem is now complete.
Proof of Theorem1.2. Let >0. Then 0≤f : |f|22p/2−p ∈C∞0R2. In fact,fhas the
same support asf. Puttingfdcc3−puin Lemma3.1and lettingR2 → ∞andR1 → 0, we
get, sincedcco 0,
R2
∇Lf,∇Ldcc ·
1
dccp−1
3−p R2
f
dpcc
Here we use the fact|∇Ldccu|1 a.e.Therefore,
3−p R2
f
dpcc
−p
R2
f2
2p−2/2f∇
Lf,∇Ldcc
· 1 dpcc−1
≤p
R2
f2
2p−2/2f·∇
Lf
dpcc−1
≤p
R2
f2
2p−1/2·∇
Lf
dpcc−1
.
3.17
By dominated convergence, letting → 0, we have
3−p R2
fp
dccp
≤p
R2
fp−1·∇
Lf
dccp−1
. 3.18
By H ¨older’s inequality,
3−p R2
fp
dccp
≤p
R2
fp
dccp
p−1/p
R2
∇Lfp1/p
. 3.19
Canceling and raising both sides to the powerp, we get1.8.
Acknowledgment
This work was supported by Natural Science Foundation of China no. 10871149 and no. 10671009.
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