1. Introduction

Let be a bounded, convex domain and a ball in , . We use to denote the ball with the same center as and with , . We do not distinguish the balls from cubes in this paper. We use to denote the n-dimensional Lebesgue measure of the set . We say that is a weight if and , a.e.

Differential forms are extensions of functions in . For example, the function is called a -form. Moreover, if is differentiable, then it is called a differential -form. The -form in can be written as . If the coefficient functions , , are differentiable, then is called a differential -form. Similarly, a differential -form is generated by , , that is,

(1.1)

where is the Wedge Product, , . Let

(1.2)

be the set of all -forms in ,

(1.3)

the space of all differential -forms on , and

(1.4)

the -forms on satisfying for all ordered -tuples , . We denote the exterior derivative by

(1.5)

for , and define the Hodge star operator

(1.6)

as follows. If , is a differential -form, then

(1.7)

where , , and . The Hodge codifferential operator

(1.8)

is given by on , . We write

(1.9)

The differential forms can be used to describe various systems of PDEs and to express different geometric structures on manifolds. For instance, some kinds of differential forms are often utilized in studying deformations of elastic bodies, the related extrema for variational integrals, and certain geometric invariance. Differential forms have become invaluable tools for many fields of sciences and engineering; see [1, 2] for more details.

In this paper, we will focus on a class of differential forms satisfying the well-known nonhomogeneous -harmonic equation

(1.10)

where and satisfy the conditions

(1.11)

for almost every and all . Here are constants and is a fixed exponent associated with (1.10). If the operator , (1.10) becomes , which is called the (homogeneous) -harmonic equation. A solution to (1.10) is an element of the Sobolev space such that for all with compact support. Let be defined by with . Then, satisfies the required conditions and becomes the -harmonic equation

(1.12)

for differential forms. If is a function (-form), (1.12) reduces to the usual -harmonic equation for functions. A remarkable progress has been made recently in the study of different versions of the harmonic equations; see [3] for more details.

Let be the space of smooth -forms on and

(1.13)

The harmonic -fields are defined by

(1.14)

The orthogonal complement of in is defined by

(1.15)

Then, the Green's operator is defined as

(1.16)

by assigning to be the unique element of satisfying Poisson's equation , where is the harmonic projection operator that maps onto so that is the harmonic part of . See [4] for more properties of these operators.

For any locally -integrable form , the Hardy-Littlewood maximal operator is defined by

(1.17)

where is the ball of radius , centered at , . We write if . Similarly, for a locally -integrable form , we define the sharp maximal operator by

(1.18)

where the l-form is defined by

(1.19)

for all , and is the homotopy operator which can be found in [3]. Also, from [5], we know that both and are -integrable 0-form.

Differential forms, the Green's operator, and maximal operators are widely used not only in analysis and partial differential equations, but also in physics; see [2–4, 6–9]. Also, in real applications, we often need to estimate the integrals with singular factors. For example, when calculating an electric field, we will deal with the integral , where is a charge density and is the integral variable. The integral is singular if . When we consider the integral of the vector field , we have to deal with the singular integral if the potential function contains a singular factor, such as the potential energy in physics. It is clear that the singular integrals are more interesting to us because of their wide applications in different fields of mathematics and physics. In recent paper [10], Ding and Liu investigated singular integrals for the composition of the homotopy operator and the projection operator and established some inequalities for these composite operators with singular factors. In paper [11], they keep working on the same topic and derive global estimates for the singular integrals of these composite operators in -John domains. The purpose of this paper is to estimate the Poincaré type inequalities for the composition of the maximal operator and the Green's operator over the -John domain.

2. Definitions and Lemmas

We first introduce the following definition and lemmas that will be used in this paper.

Definition 2.1.

A proper subdomain is called a -John domain, , if there exists a point which can be joined with any other point by a continuous curve so that

(2.1)

for each . Here is the Euclidean distance between and .

Lemma 2.2 (see [12]).

Let be a strictly increasing convex function on with and a domain in . Assume that is a function in such that and for any constant , where is a Radon measure defined by for a weight . Then, one has

(2.2)

for any positive constant , where .

Lemma 2.3 (see [13]).

Each has a modified Whitney cover of cubes such that

(2.3)

and some , and if , then there exists a cube (this cube need not be a member of ) in such that . Moreover, if is -John, then there is a distinguished cube which can be connected with every cube by a chain of cubes from and such that , , for some .

Lemma 2.4 (see [14]).

Let be a smooth differential form satisfying (1.10) in a domain , , and . Then, there exists a constant , independent of , such that

(2.4)

for all balls B with , where is a constant.

Lemma 2.5 (see [5]).

Let be the Hardy-Littlewood maximal operator defined in (1.17), the Green's operator, and , , , a smooth differential form in a bounded domain . Then,

(2.5)

for some constant , independent of u.

Lemma 2.6 (see [5]).

Let , , , be a smooth differential form in a bounded domain , the sharp maximal operator defined in (1.18), and the Green's operator. Then,

(2.6)

for some constant , independent of u.

Lemma 2.7.

Let , , be a smooth differential form satisfying the -harmonic equation (1.10) in convex domain , the Green's operator, and the Hardy-Littlewood maximal operator defined in (1.17) with . Then, there exists a constant , independent of , such that

(2.7)

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

Proof.

Let be small enough such that and any ball with , center and radius . Taking we see that . Note that ; using Hlder's inequality, we obtain

(2.8)

where . Since , using Lemma 2.5, we get

(2.9)

Let , then . Using Lemma 2.4, we have

(2.10)

where is a constant and By Hlder's inequality with again, we find

(2.11)

Note that for all , it follows that

(2.12)

Hence, we have

(2.13)

Now, by the elementary integral calculation, we obtain

(2.14)

Substituting (2.9)–(2.14) into (2.8), we obtain

(2.15)

We have completed the proof.

Similarly, by Lemma 2.6, we can prove the following lemma.

Lemma 2.8.

Let , , , be a smooth differential form satisfying the -harmonic equation (1.10) in convex domain , the sharp maximal operator defined in (1.18), and G Green's operator. Then, there exists a constant , independent of , such that

(2.16)

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

3. Main Results

Theorem 3.1.

Let , , be a smooth differential form satisfying the -harmonic equation (1.10), Green's operator, and the Hardy-Littlewood maximal operator defined in (1.17) with . Then, there exists a constant , independent of , such that

(3.1)

for any bounded and convex -John domain , where

(3.2)

and are constants, the fixed cube , the cubes , the constant appeared in Lemma 2.3, and is the center of .

Proof.

First, we use Lemma 2.3 for the bounded and convex -John domain . There is a modified Whitney cover of cubes for such that , and for some . Moreover, there is a distinguished cube which can be connected with every cube by a chain of cubes from such that , , for some . Then, by the elementary inequality , , we have

(3.3)

The first sum in (3.3) can be estimated by using Lemma 2.2 with , , and Lemma 2.7:

(3.4)

where and are the Radon measures defined by and , respectively.

To estimate the second sum in (3.3), we need to use the property of -John domain. Fix a cube and let be the chain in Lemma 2.3. Then we have

(3.5)

The chain also has property that for each , , . Thus, there exists a cube such that and , , so,

(3.6)

Note that

(3.7)

where is a positive constant. By (3.6), (3.7), and Lemma 2.7, we have

(3.8)

Then, by (3.5), (3.8), and the elementary inequality , we finally obtain

(3.9)

Substituting (3.4) and (3.9) in (3.3), we have completed the proof of Theorem 3.1.

Using the proof method for Theorem 3.1 and Lemma 2.8, we get the following theorem.

Theorem 3.2.

Let , , be a smooth differential form satisfying the -harmonic equation (1.10), Green's operator, and the sharp maximal operator defined in (1.18). Then, there exists a constant , independent of , such that

(3.10)

for any bounded and convex -John domain , where

(3.11)

and are constants, the fixed cube , the cubes , the constant appeared in Lemma 2.3, and is the center of .