Skip to content


  • Research Article
  • Open Access

Ostrowski Type Inequalities in the Grushin Plane

Journal of Inequalities and Applications20102010:987484

  • Received: 7 January 2010
  • Accepted: 14 March 2010
  • Published:


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.


  • Initial Date
  • Hamiltonian System
  • Simple Calculation
  • Fundamental Solution
  • Heisenberg Group

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 [25]). 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. [710]). 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.

Let . Then for , there holds
denote the volume of the and is the gradient operator defined by . The constants in (1.6) are the best possible, equality that can be attained for nontrivial radial functions at any .

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 . There holds, for ,

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 .



and define by , where


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




since is a diffeomorphism.

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.

Let and . There holds


Let be a point on the sphere, that is, , where . We consider for the following difference using the fundamental theorem of calculus:
where . Using (2.7), we have
Combining (3.2) and (3.3) and rewriting the expression into a solid integral using the polar coordinates, we get
To finish our proof, it is enough to show that

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

Lemma 3.2.

There hold, for ,


Recall that if , then
where The simple calculation shows
Therefore, if , then
On the other hand,
Therefore, we obtain, by (2.11),

This completes the proof of Lemma 3.2.

Proof of Theorem 1.1.

We have, by Lemma 3.1, since ,
To see that the estimate in (1.8) is sharp, we consider the function and that is fixed in . Notice that we have . We look at inequality (1.6) evaluating the function at . Since , the left-hand side of (1.6) is
and the right-hand side of (1.8) is

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 . Then . In fact, has the same support as . Putting in Lemma 3.1 and letting and , we get, since ,
Here we use the fact Therefore,
By dominated convergence, letting , we have
By Hölder's inequality,

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



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

Authors’ Affiliations

School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China


  1. Ostrowski A: Über die absolutabweichung einer differentiierbaren funktion von ihrem Integralmittelwert. Commentarii Mathematici Helvetici 1938, 10(1):226–227.MathSciNetView ArticleMATHGoogle Scholar
  2. 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
  3. Anastassiou GA: Quantitative Approximations. Chapman & Hall, Boca Raton, Fla, USA; 2001.MATHGoogle Scholar
  4. Anastassiou GA, Goldstein JA: Ostrowski type inequalities over euclidean domains. Atti della Accademia Nazionale dei Lincei 2007, 18(3):305–310.MathSciNetMATHGoogle Scholar
  5. 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
  6. 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
  7. 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
  8. 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
  9. Monti R, Morbidelli D: The isoperimetric inequality in the grushin plane. The Journal of Geometric Analysis 2004, 14(2):355–368.MathSciNetView ArticleMATHGoogle Scholar
  10. Paulat M: Heat kernel estimates for the grusin operator.
  11. Chow WL: Über systeme von linearen partiellen differentialgleichungen erster ordnung. Mathematische Annalen 1939, 117: 98–105.MathSciNetMATHGoogle Scholar
  12. 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
  13. 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
  14. 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
  15. 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