Research Article | Open Access

# Norm Comparison Estimates for the Composite Operator

**Academic Editor:**Shusen Ding

#### 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. *
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.

#### 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.

#### References

- C. Scott, “${L}^{p}$-theory of differential forms on manifolds,”
*Transactions of the American Mathematical Society*, vol. 347, no. 6, pp. 2075–2096, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Ling and G. Bao, “Some local Poincaré inequalities for the composition of the sharp maximal operator and the Green's operator,”
*Computers & Mathematics with Applications*, vol. 63, no. 3, pp. 720–727, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Dai, Y. Xing, S. Ding, and Y. Wang, “Inequalities for the composition of Green's operator and the potential operator,”
*Journal of Inequalities and Applications*, vol. 2012, article 271, 13 pages, 2012. View at: Publisher Site | Google Scholar - C. J. Neugebauer, “Inserting ${A}_{p}$-weights,”
*Proceedings of the American Mathematical Society*, vol. 87, no. 4, pp. 644–648, 1983. View at: Publisher Site | Google Scholar | MathSciNet - R. P. Agarwal, S. Ding, and C. Nolder,
*Inequalities for Differential Forms*, Springer, New York, NY, USA, 2009. View at: Publisher Site | MathSciNet - G. Bao, “${A}_{r}(\lambda )$-weighted integral inequalities for $A$-harmonic tensors,”
*Journal of Mathematical Analysis and Applications*, vol. 247, no. 2, pp. 466–477, 2000. View at: Publisher Site | Google Scholar | MathSciNet - Y. Xing, “A new weight class and Poincaré inequalities with the Radon measure,”
*Journal of Inequalities and Applications*, vol. 2012, article 32, 11 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Bi, “Weighted inequalities for potential operators on differential forms,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 713625, 13 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Xing and S. Ding, “Norms of the composition of the maximal and projection operators,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 72, no. 12, pp. 4614–4624, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Iwaniec and A. Lutoborski, “Integral estimates for null Lagrangians,”
*Archive for Rational Mechanics and Analysis*, vol. 125, no. 1, pp. 25–79, 1993. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Ding, “Two-weight Caccioppoli inequalities for solutions of nonhomogeneous $A$-harmonic equations on Riemannian manifolds,”
*Proceedings of the American Mathematical Society*, vol. 132, no. 8, pp. 2367–2375, 2004. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

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.