# Ostrowski Type Inequalities in the Grushin Plane

- Heng-Xing Liu
^{1}and - Jing-Wen Luan
^{1}Email author

**2010**:987484

https://doi.org/10.1155/2010/987484

© H.-X. Liu and J.-W. Luan 2010

**Received: **7 January 2010

**Accepted: **14 March 2010

**Published: **17 March 2010

## Abstract

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.

## Keywords

## 1. Introduction

The classical Ostrowski inequality [1] is as follows:

for , and it is a sharp inequality. Inequality (1.1) was extended from intervals to rectangles and general domains in (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

and satisfies . By Chow's conditions, the Carnot-Carathéodory distance between any two points is finite (cf. [11]). We denote , where is the origin. Define on the dilation as

For simplicity, we will write it as . It is not difficult to check that and are homogeneous of degree one with respect to the dilation. The Jacobian determinant of is , where is the homogeneous dimension. The Carnot-Carathéodory distance satisfies

Let be the Carnot-Carathéodory ball centered at the origin and of radius . Let be the corresponding unit sphere. Let be the surface measure on . Given any , set , and . For , let

be the averages of over the unit sphere, where denote the volume of the . Then, we can state our result as follows.

Theorem 1.1.

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.

## 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

The associated Hamiltonian function is of the form

It is known that geodesics in the Grushin plane are solutions of the Hamiltonian system (cf. [8])

Taking the initial date and , one can find the solutions (cf. [8])

where the time is exactly the Carnot-Carathéodory distance. Letting , we get the Euclidean geodesics

and hence the correct normalization is .

Set

with . If , the range of is ; if , the range of is . Furthermore, if one fixes , (2.7) with and parameterize .

On the other hand, the Carnot-Carathéodory distance satisfies (cf. [10, Theorem ]), for ,

is a diffeomorphism of the interval onto (cf. [14]), and is the inverse function of . From (2.7), we have

Therefore,

We finally recall the polar coordinates in the Grushin plane associated with . The following coarea formula has been proved in [15]:

where and is the perimeter-measure. Notice that (cf. [15]); and (cf. [9, Proposition ]); we have the following polar coordinates in the Grushin plane, for all :

## 3. The Proofs

To prove the main result, we first need the following representation formula.

Lemma 3.1.

Proof.

in . This is just the following Lemma 3.2. The proof of Lemma 3.1 is now complete.

Lemma 3.2.

Proof.

This completes the proof of Lemma 3.2.

Proof of Theorem 1.1.

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.

Canceling and raising both sides to the power , we get (1.8).

## Declarations

### Acknowledgment

This work was supported by Natural Science Foundation of China no. 10871149 and no. 10671009.

## Authors’ Affiliations

## References

- Ostrowski A: Über die absolutabweichung einer differentiierbaren funktion von ihrem Integralmittelwert.
*Commentarii Mathematici Helvetici*1938, 10(1):226–227.MathSciNetView ArticleMATHGoogle Scholar - Anastassiou GA: Ostrowski type inequalities.
*Proceedings of the American Mathematical Society*1995, 123(12):3775–3781. 10.1090/S0002-9939-1995-1283537-3MathSciNetView ArticleMATHGoogle Scholar - Anastassiou GA:
*Quantitative Approximations*. Chapman & Hall, Boca Raton, Fla, USA; 2001.MATHGoogle Scholar - Anastassiou GA, Goldstein JA: Ostrowski type inequalities over euclidean domains.
*Atti della Accademia Nazionale dei Lincei*2007, 18(3):305–310.MathSciNetMATHGoogle Scholar - Anastassiou GA, Goldstein JA: Higher order ostrowski type inequalities over euclidean domains.
*Journal of Mathematical Analysis and Applications*2008, 337(2):962–968. 10.1016/j.jmaa.2007.04.033MathSciNetView ArticleMATHGoogle Scholar - Lian B-S, Yang Q-H: Ostrowski type inequalities on H-type groups.
*Journal of Mathematical Analysis and Applications*2010, 365(1):158–166. 10.1016/j.jmaa.2009.10.030MathSciNetView ArticleMATHGoogle Scholar - Calin O, Chang D-C, Greiner P, Kannai Y: On the geometry induced by a grusin operator. In
*Complex Analysis and Dynamical Systems II, Contemporary Mathematics*.*Volume 382*. American Mathematical Society, Providence, RI, USA; 2005:89–111.View ArticleGoogle Scholar - Greiner PC, Holcman D, Kannai Y: Wave kernels related to second-order operators.
*Duke Mathematical Journal*2002, 114(2):329–386. 10.1215/S0012-7094-02-11426-4MathSciNetView ArticleMATHGoogle Scholar - Monti R, Morbidelli D: The isoperimetric inequality in the grushin plane.
*The Journal of Geometric Analysis*2004, 14(2):355–368.MathSciNetView ArticleMATHGoogle Scholar - Paulat M: Heat kernel estimates for the grusin operator. http://arxiv.org/abs/0707.4576
- Chow WL: Über systeme von linearen partiellen differentialgleichungen erster ordnung.
*Mathematische Annalen*1939, 117: 98–105.MathSciNetMATHGoogle Scholar - D'Ambrosio L: Hardy inequalities related to grushin type operators.
*Proceedings of the American Mathematical Society*2004, 132(3):725–734. 10.1090/S0002-9939-03-07232-0MathSciNetView ArticleMATHGoogle Scholar - Monti R: Some properties of carnot-varathéodory balls in the Heisenberg group.
*Atti della Accademia Nazionale dei Lincei*2000, 11(3):155–167.MathSciNetMATHGoogle Scholar - Beals R, Gaveau B, Greiner PC: Hamilton-Jacobi theory and the heat kernel on heisenberg groups.
*Journal de Mathématiques Pures et Appliquées*2000, 79(7):633–689.MathSciNetView ArticleMATHGoogle Scholar - Monti R, Serra Cassano F: Surface measures in carnot-carathéodory spaces.
*Calculus of Variations and Partial Differential Equations*2001, 13(3):339–376. 10.1007/s005260000076MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.