Journal of Function Spaces

Volume 2014 (2014), Article ID 943986, 7 pages

http://dx.doi.org/10.1155/2014/943986

Research Article

Norm Comparison Estimates for the Composite Operator

Xuexin Li, Yong Wang, and Yuming Xing

Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China

Received 12 November 2013; Revised 6 February 2014; Accepted 20 February 2014; Published 30 March 2014

Academic Editor: Shusen Ding

Copyright © 2014 Xuexin Li et al. 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.

Abstract.

This paper obtains the Lipschitz and BMO norm estimates for the composite operator applied to differential forms. Here, is the Hardy-Littlewood maximal operator, and is the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.

1. Introduction

Since the differential form has been proposed by Elie Cartan, its development is obvious. Differential form theory has been applied to many fields, such as partial differential equations, nonlinear analysis, and control theory. Particularly, it is an important part in differential forms to research norm inequalities for an operator or composite operator. norm has been studied very thoroughly [1–3], so we try to establish the Lipschitz norm and BMO norm inequalities for the composition of the Hardy-Littlewood maximal operator and potential operator. It is useful to explore the properties of the weighted norms; see [4–7]. So, we also research the weighted Lipschitz norm and the weighted BMO norm inequalities.

As the traditional notations, we write for a bounded convex domain in , , endowed with the usual Lebesgue measure denoted by . and are concentric balls, with . denotes a weight defined by , and a.e.. The , denoted by , is a , spanned by exterior products , for all ordered -tuples , . The -form is called a differential -form, if is differential. We use to denote the differential -form and to denote the -form on satisfying . In particular, we know that the -form is a function. We define the exterior derivative by

We define by

where is a permutation of and is the signature of the permutation. Now, we can define the Hodge codifferential like this

The differential form we research here satisfies the nonhomogeneous -harmonic equation

where and satisfy the conditions

for almost every and all . Here are constants and is a fixed exponent associated with (4).

2. Lipschitz and BMO Norm Inequalities

In this section, we introduce some definitions. Then, we give main lemmas used in theorems. Finally, we give the norm comparison estimates for the composite operator.

For , we write , if

for some .

For , we say , if

for some .

Hardy-Littlewood maximal operator is defined by

where , is a locally -integrable form and is a ball with radius .

Potential operator is first extended to differential form by Bi in [8], which is defined as follows:

where the kernel is a nonnegative measurable function defined for , is a differential -form, and the summation is over all ordered -tuples .

We need the following two lemmas for the Hardy-Littlewood maximal operator and potential operator to prove Theorem 7.

Lemma 1 (see [9]). Let , be a differential form in a domain and be the Hardy-Littlewood maximal operator, . Then, and there exists a constant , independent of , such that

Lemma 2 (see [8]). Let , be a differential form in a domain and let be the potential operator. Then, there exists a constant , independent of , such that

where .

The following lemma was established by Iwaniec and Lutoborski in [10].

Lemma 3. Let , be a differential form in a domain . Then, there exists a constant , independent of , such that

where is a closed form.

We will use the following generalized Hölder inequality repeatedly in this paper.

Lemma 4. Let , and . If and are measurable functions on , then

for any .

We also need the Caccioppoli inequality for differential form.

Lemma 5 (see [11]). Let , be a solution of the nonhomogeneous A-harmonic equation (4) in . Then, there exists a constant , independent of , such that

for all balls with and any closed form , where , and .

The following Poincaré inequality appears in [5].

Lemma 6. Let , be a solution of the nonhomogeneous A-harmonic equation (4) in . Then, there exists a constant , independent of , such that

for any ball in .

First, we establish a Poincaré-type inequality for the composite operator .

Theorem 7. Let , be a differential form in a domain . Then, there exists a constant , independent of , such that

Proof. From Lemma 3 and Minkowski inequality, we have

Replacing with and combining Lemmas 1 and 2, we obtain

Theorem 7 has been completed.

Then, we estimate the Lipschitz norm of in terms of -norm.

Theorem 8. Let , be a solution of the nonhomogeneous A-harmonic equation (4) in . Then, there exists a constant , independent of , such that

where is a constant with .

Proof. Setting in Lemma 6 we have

for all balls with . Using Hölder inequality, we have

From the definition of Lipschitz norm and (21), it follows that

Theorem 8 has been completed.

Now, we establish the norm comparison theorems of the composite operator between Lipschitz norm and BMO norm.

Theorem 9. Let , be a differential form in . Then, there exists a constant , independent of , such that

where is a constant with .

Proof. From the definition of BMO norm, we have

Theorem 9 has been completed.

Theorem 10. Let , be a solution of the nonhomogeneous A-harmonic equation (4) in . Then, there exists a constant , independent of , such that

where is a constant with .

Proof. From (21), we obtain

Based on Lemma 5, we get

From the weak reverse Hölder inequality, it follows that

where . So, we get

Let ; we obtain

Theorem 10 has been completed.

Combining Theorems 8 and 9, we obtain the following result easily.

Corollary 11. Let , be a solution of the nonhomogeneous A-harmonic equation (4) in . Then, there exists a constant , independent of , such that

3. The Weighted Lipschitz and BMO Norm Inequalities

In this section, we obtain some weighted inequalities for the composition operator .

The weight function we use here is weight. The -class is a new weight class which was first proposed by Xing in [7]. It contains the well-known -weight as a proper subset.

Definition 12. One says that a measurable function defined on a subset satisfies the -condition for some positive constants , if a.e., and

Let . We say , if

for some , where is bounded in , is a weight, and is Radon measure defined by .

Let . We say , if

for some , where is bounded in , is a weight, and is Radon measure defined by .

Now, we estimate the weighted Lipschitz and BMO norm for the composition operator .

Theorem 13. Let , be a solution of the nonhomogeneous A-harmonic equation (4) in . Radon measure is defined by , and for some , where , , , and . Then, there exists a constant , independent of , such that

Proof. Using Hölder inequality with , we have

Using Hölder inequality with , we obtain

Since , we get

From (36), (37), and (38), we know that

So, we obtain

Based on the definition of the weighted Lipschitz norm and (40), we have

Theorem 13 has been completed.

Theorem 14. Let , be a differential form in . Radon measure is defined by , and . Then, there exists a constant , independent of , such that

where are some positive constants.

Proof. From the definition of the weighted BMO norm, we have

Theorem 14 has been completed.

Combining Theorems 13 and 14, we obtain the following corollary.

Corollary 15. Let , be a solution of the nonhomogeneous A-harmonic equation (4) in . Radon measure is defined by , and for some , where , , , and . Then, there exists a constant , independent of , such that

4. Applications

In this section, we use the theorems we obtain to estimate the norms of Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.

Example 16. Let be a map from to . is the Jacobian determinant. Now choosing the subdeterminant of the Jacobian determinant,

We know that is a -form. Let . If , , from Theorems 9 and 14, we obtain the following results:

where is a constant, , is the Hardy-Littlewood maximal operator, is the potential operator, and is a weight satisfying , , , and .

Example 17. Let be a -quasiregular mapping, . Then,

or

is a generalized solution of the following equation:

Here, can be expressed as

in the formula above are some functions, which can be expressed in terms of the differential matrix , and satisfy

where . If we assume that is the Hardy-Littlewood maximal operator, is the potential operator, , and is a weight satisfying , , , and , according to Theorem 13 and Corollary 15, we obtain the following inequalities:

where is a constant.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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