- Research Article
- Open Access

# Global Estimates for Singular Integrals of the Composition of the Maximal Operator and the Green's Operator

- Yi Ling
^{1}Email author and - HansonM Umoh
^{1}Email author

**2010**:723234

https://doi.org/10.1155/2010/723234

© Y. Ling and H. M. Umoh. 2010

**Received:**31 December 2009**Accepted:**12 March 2010**Published:**6 April 2010

## Abstract

We establish the Poincaré type inequalities for the composition of the maximal operator and the Green's operator in John domains.

## Keywords

- Maximal Operator
- Differential Form
- Exterior Derivative
- Elementary Inequality
- John Domain

## 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.

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.

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.

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 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.

for each . Here is the Euclidean distance between and .

Lemma 2.2 (see [12]).

for any positive constant , where .

Lemma 2.3 (see [13]).

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]).

for all balls B with , where is a constant.

Lemma 2.5 (see [5]).

for some constant , independent of u.

Lemma 2.6 (see [5]).

for some constant , independent of u.

Lemma 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.

We have completed the proof.

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

Lemma 2.8.

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.

Proof.

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

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.

## Authors’ Affiliations

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