Ostrowski Type Inequalities in the Grushin Plane

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.

1. Introduction

The classical Ostrowski inequality [1] is as follows:

(1.1)

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

(1.2)

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

(1.3)

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

(1.4)

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

(1.5)

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

(1.6)

where

(1.7)

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 ,

(1.8)

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

(2.1)

The associated Hamiltonian function is of the form

(2.2)

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

(2.3)

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

(2.4)

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

(2.5)

and hence the correct normalization is .

Set

(2.6)

and define by , where

(2.7)

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 ,

(2.8)

where

(2.9)

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

(2.10)

Therefore,

(2.11)

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]:

(2.12)

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 :

(2.13)

3. The Proofs

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

Lemma 3.1.

Let and . There holds

(3.1)

Proof.

Let be a point on the sphere, that is, , where . We consider for the following difference using the fundamental theorem of calculus:

(3.2)

where . Using (2.7), we have

(3.3)

Combining (3.2) and (3.3) and rewriting the expression into a solid integral using the polar coordinates, we get

(3.4)

To finish our proof, it is enough to show that

(3.5)

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

Lemma 3.2.

There hold, for ,

(3.6)

Proof.

Recall that if , then

(3.7)

where The simple calculation shows

(3.8)

Therefore, if , then

(3.9)

On the other hand,

(3.10)

Therefore, we obtain, by (2.11),

(3.11)

This completes the proof of Lemma 3.2.

Proof of Theorem 1.1.

We have, by Lemma 3.1, since ,

(3.12)

where

(3.13)

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

(3.14)

and the right-hand side of (1.8) is

(3.15)

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 ,

(3.16)

Here we use the fact Therefore,

(3.17)

By dominated convergence, letting , we have

(3.18)

By Hölder's inequality,

(3.19)

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