- Research Article
- Open Access
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.
- Initial Date
- Hamiltonian System
- Simple Calculation
- Fundamental Solution
- Heisenberg Group
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
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.
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.
In this section, we will follow  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. )
Taking the initial date and , one can find the solutions (cf. )
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. ), 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 :
To prove the main result, we first need the following representation formula.
in . This is just the following Lemma 3.2. The proof of Lemma 3.1 is now complete.
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).
This work was supported by Natural Science Foundation of China no. 10871149 and no. 10671009.
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