Ostrowski Type Inequalities in the Grushin Plane
© H.-X. Liu and J.-W. Luan 2010
Received: 7 January 2010
Accepted: 14 March 2010
Published: 17 March 2010
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.
The classical Ostrowski inequality  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  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. ). 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
We also obtain the following Hardy type inequalities in the Grushin plane. We refer to  the Hardy inequalities associated with nonisotropic gauge induced by the fundamental solution.
2. Geodesics in the Grushin Plane
In this section, we will follow  to give a parametrization of Grushin plane using the geodesics. Recall that the Grushin operator is given by
It is known that geodesics in the Grushin plane are solutions of the Hamiltonian system (cf. )
Taking the initial date and , one can find the solutions (cf. )
On the other hand, the Carnot-Carathéodory distance satisfies (cf. [10, Theorem ]), for ,
is a diffeomorphism of the interval onto (cf. ), and is the inverse function of . From (2.7), we have
We finally recall the polar coordinates in the Grushin plane associated with . The following coarea formula has been proved in :
3. The Proofs
To prove the main result, we first need the following representation formula.
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.
This work was supported by Natural Science Foundation of China no. 10871149 and no. 10671009.
- 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
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.