# Weighted Inequalities for Potential Operators with Lipschitz and BMO Norms

- Yuxia Tong
^{1}Email author and - Jiantao Gu
^{1}

**2011**:659597

https://doi.org/10.1155/2011/659597

© Y. Tong and J. Gu. 2011

**Received: **1 January 2011

**Accepted: **7 March 2011

**Published: **15 March 2011

## Abstract

## 1. Introduction

In many situations, the process to study solutions of PDEs involves estimating the various norms of the operators. Hence, we are motivated to establish some Lipschitz norm inequalities and BMO norm inequalities for potential operator to the versions of differential forms.

We keep using the traditional notation.

where is a real number. We denote the exterior derivative by : for . Its formal adjoint operator : is given by on .

for some . When is a 0-form, (1.6) reduces to the classical definition of .

for some , where the Radon measure is defined by is a weight, and is a real number. Again, we use to replace whenever it is clear that the integral is weighted.

On the potential operator and the functional , see [2] for details.

Lots of results have been obtained in recent years about different versions of the -harmonic equation, see [3–5].

## 2. The Estimate for Potential Operators with Lipschitz Norm and BMO Norm

In this section, we give the estimate for potential operators with Lipschitz norm and BMO norm applied to differential forms. The following strong type inequality for potential operators appears in [6].

Lemma 2.1 (see [6]).

We will establish the following estimate for potential operators.

Theorem 2.2.

Proof.

since and , where is a constant and .

We have completed the proof of Theorem 2.2.

## 3. The Weight

In this section, we introduce the weight appeared in [7].

Definition 3.1.

The following results show that the weights have the properties similar to those of the weights.

Theorem 3.2.

Proof.

for all balls since . Therefore, , and hence .

Theorem 3.3.

where is a ball in and is a measurable subset of .

Proof.

The desired result is obtained.

which is called the strong doubling property of weights; see [8].

## 4. The Weighted Inequality for Potential Operators

In this section, we are devoted to develop some two-weight norm inequalities for potential operator to the versions of differential forms. We need the following lemmas.

Lemma 4.1 (see [9]).

Lemma 4.2.

Lemma 4.3 (see [10]).

Theorem 4.4.

Proof.

for all balls with and . We have completed the proof of Theorem 4.4.

## Declarations

### Acknowledgments

The authors are supported by NSF of Hebei Province (A2010000910) and Scientific Research Fund of Zhejiang Provincial Education Department (Y201016044).

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.