Some Estimates of Integrals with a Composition Operator

Abstract

We give some estimates of integrals with a composition operator, namely, composition of homotopy, differential, and Green's operators , with the Lipschitz and norms. We also have estimates of those integrals with a singular factor.

1. Introduction

The purpose of this paper is to establish the Poincaré-type inequalities for the composition of the homotopy operator , differential operator , and Green's operator under Lipschitz and norms. One of the reasons that we consider this composition operator is due to the Hodge theorem. It is well known that Hodge decomposition theorem plays important role in studying harmonic analysis and differential forms; see [1–3]. It gives a relationship of the three key operators in harmonic analysis, namely, Green's operator , the Laplacian operator , and the harmonic projection operator . This relationship offers us a tool to apply the composition of the three operators under the consideration to certain harmonic forms and to obtain some estimates for certain integrals which are useful in studying the properties of the solutions of PDEs. We also consider the integrals of this composition operator with a singular factor because of their broad applications in solving differential and integral equations; see [4].

We first give some notations and definitions which are commonly used in many books and papers; for example, see [1, 4–12]. We use to denote a Riemannian, compact, oriented, and smooth manifold without boundary on . Let be the th exterior power of the cotangent bundle, and let be the space of smooth -forms on and has generalized gradient. The harmonic -fields are defined by , for some The orthogonal complement of in is defined by for all Then, Green's operator is defined as by assigning as the unique element of satisfying Poisson's equation , where is the harmonic projection operator that maps onto so that is the harmonic part of . In this paper, we also assume that is a bounded and convex domain in . The -dimensional Lebesgue measure of a set is denoted by . The operator with the case was first introduced by Cartan in [3]. Then, it was extended to the following version in [13]. To each there corresponds a linear operator defined by and the decomposition A homotopy operator is defined by averaging over all points :

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where is normalized so that . We are particularly interested in a class of differential forms which are solutions of the well-known nonhomogeneous -harmonic equation:

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where satisfy the conditions: , and for almost every and all . Here and are constants, and is a fixed exponent associated with the equation. A significant progress has been made recently in the study of different versions of the harmonic equations; see [1, 4–12].

A function is said to be in if there is a constant such that for all balls with , where is a constant. norm of -forms is defined as the following. Let , . We say if

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for some . Similar way to define the Lipschitz norm for , , we say , , if

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for some .

We will use the following results.

Lemma 1.1 (see [7]).

If , , , then for any bounded ball ,

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One also has the Poincaré type inequality:

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Lemma 1.2 (see [5]).

Let , , , be a solution of the A-harmonic equation in a bounded, convex domain , and let be the homotopy operator defined in (1.1). Then, there exists a constant , independent of , such that

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where is a constant with .

Lemma 1.3 (see [4]).

Let , , , be a solution of the nonhomogeneous -harmonic equation (1.2) in a bounded domain , let be the projection operator and let be the homotopy operator. Then, there exists a constant , independent of , such that

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for all balls with and any real numbers and with , where and is the center of ball and is a constant.

2. The Estimates for Lipschitz and BMO Norms

We first give an estimate of the composition operator with the Lipschitz norm .

Theorem 2.1.

Let , , , be a solution of the -harmonic equation (1.2) in a bounded, convex domain , and let be the homotopy operator defined in (1.1) and Green's operator. Then, there exists a constant , independent of , such that

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where is a constant with .

Proof.

From Lemma 1.1, we have

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for all balls . By Hölder inequality with , we have

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By the definition of Lipschitz norm and noticing that , we have

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Theorem 2.1 is proved.

We learned from [5] that the norm and the Lipschitz norm are related in the following inequality.

Lemma 2.2 (see [5]).

If a differential form is , , , in a bounded domain , then and

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where is a constant.

Applying to (2.5), then using Theorem 2.1, we have the following.

Theorem 2.3.

Let , , , be a solution of the A-harmonic equation (1.2) in a bounded, convex domain , and let be the homotopy operator defined in (1.1), and let be the Green's operator. Then, there exists a constant , independent of , such that

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3. The Lipschitz and BMO Norms with a Singular Factor

We considered the integrals with singular factors in [4]. Here, we will give estimates to Poincaré type inequalities with singular factors in the Lipschitz and norms. If we use the formula (1.7) in Lemma 1.1 and follow the same proof of Lemma in [4], we obtain the following theorem.

Theorem 3.1.

Let , , , be a solution of the nonhomogeneous -harmonic equation (1.2) in a bounded domain , let be Green's operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that

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for all balls with and any real numbers and with , where and is the center of ball and is a constant.

We extend Theorem 3.1 to the Lipschitz norm with a singular factor and have the following result.

Theorem 3.2.

Let , , , be a solution of the non-homogeneous -harmonic equation in a bounded and convex domain , let be Green's operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that

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for all balls with , , where and , and , are real numbers with . Here is the center of the ball .

Proof.

Equation (3.2) is equivalent to

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By using Theorem 3.1, we have

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where . Notice that as . Thus,

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We have completed the proof of Theorem 3.2.

We also obtain a similar version of the Poincaré type inequality with a singular factor for the norm.

Theorem 3.3.

Let , , , be a solution of the non-homogeneous -harmonic equation in a bounded and convex domain , let be Green's operator, and let be the homotopy operator. Then, there exists a constant , independent of , such that

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for all balls with , , where and , and , are real numbers with . Here is the center of the ball .

We omit the proof since it is the same as the proof of Theorem 3.2.

4. The Weighted Inequalities

In this section, we introduce weighted versions of the Poincaré type inequality with the Lipschitz and norms.

Definition 4.1.

We say that a weight belongs to the class, and write , if a.e., and

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for any ball .

Definition 4.2.

We say , for , , if

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for some , where the measure is defined by , is a weight, and is a real number. Similarly, for , , we write if

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Lemma 4.3 (see [7]).

Let , , , be a smooth differential form satisfying equation (1.2) in a bounded domain , and let be the homotopy operator defined in (1.1). Assume that and for some . Then, there exists a constant , independent of , such that

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for all balls with and any real number with .

We extend the Lemma 4.3 to the version with the Lipschitz norm as the following.

Theorem 4.4.

Let , , , be a solution of (1.2) in a bounded domain, convex , and let be the homotopy operator defined in (1.1), where the measure is defined by and for some with for any . Then, there exists a constant , independent of , such that

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where and are constants with and .

Proof.

First, by using the Hölder inequality and inequality (4.4), we see that

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Since , we have . Then,

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due to and . Theorem 4.4 is proved.

Similarly, we have the weighted version for the norm.

Theorem 4.5.

Let , , , be a solution of (1.2) in a bounded domain, convex , and let be the homotopy operator defined in (1.1), where the measure is defined by and for some with for any . Then, there exists a constant , independent of , such that

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where is a constant with .

Proof.

We only need to prove that

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As a matter of fact,

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5. Applications

Example 5.1 .

We consider the homogeneous case of (1.2) as and , . Let be a -form. Then, the operator satisfies the required conditions of (1.2) and (1.2) is reduced to the -harmonic equation:

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For example, , as and as is a solution of -harmonic equation (5.1). Then, also satisfies the results proved in the Theorems 2.1–4.5. Let us consider a special case. Set , and let be the unit sphere in . In particular, one could think of as square root of an attraction force between two objects of masses and , respectively. Then, , where is the gravitational constant. It would be very complicated to estimate the or directly. To estimate their upper bounds by estimating is much easier. As a matter of fact, by using the spherical coordinates, we have

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Example 5.2 (see [5]).

Let be a -quasiregular mapping, ; that is, if are in the Sobolev class , for , and the norm of the corresponding Jacobi matrix satisfies , where is the Jacobian determinant of the , then, each of the functions , or , is a generalized solution of the quasilinear elliptic equation:

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in , where and are some functions that satisfy for some constants . Then, all of functions defined here also satisfy the results in Theorems 2.1–4.5.