Open Access

Norm Comparison Inequalities for the Composite Operator

Journal of Inequalities and Applications20092009:212915

DOI: 10.1155/2009/212915

Received: 2 August 2008

Accepted: 15 January 2009

Published: 22 January 2009

Abstract

We establish norm comparison inequalities with the Lipschitz norm and the BMO norm for the composition of the homotopy operator and the projection operator applied to differential forms satisfying the A-harmonic equation. Based on these results, we obtain the two-weight estimates for Lipschitz and BMO norms of the composite operator in terms of the -norm.

1. Introduction

The purpose of this paper is to establish the Lipschitz norm and BMO norm inequalities for the composition of the homotopy operator and the projection operator applied to differential forms in , . The harmonic projection operator , one of the key operators in the harmonic analysis, plays an important role in the Hodge decomposition theory of differential forms. In the meanwhile, the homotopy operator is also widely used in the decomposition and the -theory of differential forms. In many situations, we need to estimate the various norms of the operators and their compositions.

We always assume that is a bounded, convex domain and is a ball in , , throughout this paper. Let be the ball with the same center as and with , . We do not distinguish the balls from cubes in this paper. For any subset , we use to denote the Lebesgue measure of . We call a weight if and a.e. Differential forms are extensions of functions in . For example, the function is called a 0-form. Moreover, if is differentiable, then it is called a differential 0-form. A differential -form is generated by , , that is, where , , and are differentiable functions. Let be the set of all -forms in , be the space of all differential -forms on and be the -forms on satisfying for all ordered -tuples , . We denote the exterior derivative by for . The Hodge codifferential operator is given by on , . We write and , where is a weight. Let be the th exterior power of the cotangent bundle and be the space of smooth -forms on . We set has generalized gradient . The harmonic -fields are defined by for some The orthogonal complement of in is defined by for all The harmonic projection operator is the operator involved in the Poisson's equation where is the Green's operator. See [14] for more propeties of the projection operator and Green's operator.

The differential equation is called the -harmonic equation and the nonlinear elliptic partial differential equation
(1.1)
is called the nonhomogeneous -harmonic equation for differential forms, where and satisfy the conditions:
(1.2)
for almost every and all . Here are constants and is a fixed exponent associated with (1.1). A solution to (1.1) is an element of the Sobolev space such that for all with compact support. Let be defined by with . Then satisfies required conditions and becomes the -harmonic equation for differential forms. If is a function (a 0-form), the above equation reduces to the usual -harmonic equation
(1.3)

for functions. Some results have been obtained in recent years about different versons of the -harmonic equation, see [29].

Let , . We write , , if
(1.4)
for some . The factor here is for convenience and in fact the norm is independent of this expansion factor, see [8]. Further, we write for those forms whose coefficients are in the usual Lipschitz space with exponent and write for this norm. Similarly, for , , we write if
(1.5)

for some . Again, the factor here is for convenience and the norm is independent of the expansion factor , see [8]. When is a -form, (1.5) reduces to the classical definition of BMO .

The following operator with the case was first introduced by Cartan in [10]. Then, it was extended to the following version in [6]. For each point , there is a linear operator defined by and the decomposition A homotopy operator is defined by , averaging over all points in , where is normalized by and the decomposition
(1.6)
holds for any differential form . The -form is defined by
(1.7)
for all , . From [6], we know that for any differential form , , , we have
(1.8)
(1.9)

2. Lipschitz Norm Estimates

The following Hölder inequality will be used in the proofs of main theorems.

Lemma 2.1.

Let , and . If and are measurable functions on , then for any .

Lemma 2.2 (see [1]).

Let and , . Then, there exists a positive constant , independent of , such that
(2.1)

We first prove the following Poincaré-type inequality for the composition of the homotopy operator and the projection operator.

Theorem 2.3.

Let , , , be a smooth differential form in a bounded, convex domain , be the projection operator and be the homotopy operator. Then, there exists a constant , independent of , such that
(2.2)

for all balls with .

Proof.

Let be the projection operator and be the homotopy operator. For any differential form , we know that
(2.3)
Replacing by in (2.3) yields
(2.4)
Since and , by Lemma 2.2, we have
(2.5)
Using (1.9), (2.4), and (2.5), we find that
(2.6)

The proof of Theorem 2.3 has been completed.

Using Theorem 2.3, we estimate the following Lipschitz norm of the composite operator .

Theorem 2.4.

Let , , , be a smooth differential form in a bounded, convex domain , be the projection operator and be the homotopy operator. Then, there exists a constant , independent of , such that
(2.7)

where is a constant with .

Proof.

From Theorem 2.3, we have
(2.8)
for all balls with . Using the Hölder inequality with , we find that
(2.9)
where we have used . Now, from the definition of Lipschitz norm, (2.9) and we obtain
(2.10)

The proof of Theorem 2.4 has been completed.

In order to prove Theorem 2.6, we extend [11, Lemma 8.2.2] into the following version for differential forms.

Lemma 2.5.

Let be a strictly increasing convex function on with , and be a bounded domain in . Assume that is a smooth differential form in such that for any real number and , where is a Radon measure defined by for a weight . Then, for any positive constant , we have
(2.11)

where is a positive constant.

Proof.

Let Note that . Then, there exists a constant such that that is
(2.12)
Since is an increasing convex function, we obtain
(2.13)

The proof of Lemma 2.5 is completed.

Theorem 2.6.

Let , , be a smooth differential form satisfying the nonhomogeneous -harmonic equation in a bounded, convex domain and the Lebesgue for any ball . Assume that is the projection operator and is the homotopy operator. Then, there exists a constant , independent of , such that
(2.14)

where is a constant with .

Proof.

Using Lemma 2.5 with and the weight over the ball , we have
(2.15)
From Theorem 2.3 and (2.15), we obtain
(2.16)
From the definition of the Lipschitz norm, the Hölder inequality with and (2.16), for any ball with , we find that
(2.17)
Next, from the weak reverse Hölder inequality for solutions of the nonhomogeneous -harmonic equation, we have
(2.18)
for some constant . Combination of (2.17) and (2.18) gives
(2.19)
Hence, we obtain
(2.20)
Thus, taking the supremum on both sides of (2.20) over all balls with and using the definitions of the Lipschitz and BMO norms, we find that
(2.21)
that is,
(2.22)

The proof of Theorem 2.6 has been completed.

Note that inequality (2.14) implies that the norm of can be controlled by the norm when is a 1-form.

Theorem 2.7.

Let , , , be a smooth differential form in a bounded, convex domain , be the projection operator and be the homotopy operator. Then, there exists a constant , independent of , such that
(2.23)

Proof.

From the definitions of the Lipschitz and BMO norms, we obtain
(2.24)
that is
(2.25)

where and are constants with . We have completed the proof of Theorem 2.7.

3. BMO Norm Estimates

We have developed some estimates for the Lipschitz norm in last section. Now, we estimates the BMO norm . We first prove the following inequality between the BMO norm and the Lipschitz norm for the composite operator.

Theorem 3.1.

Let , , , be a smooth differential form in a bounded, convex domain , be the projection operator and be the homotopy operator. Then, there exists a constant , independent of , such that
(3.1)

Proof.

From Theorems 2.4 and 2.7, we have
(3.2)
(3.3)
respectively. Combination of (3.2) and (3.3) yields
(3.4)

The proof of Theorem 3.1 has been completed.

Based on the above results, we discuss the weighted Lipschitz and BMO norms. For , , we write , , if
(3.5)
for some , where is a bounded domain, the Radon measure is defined by , is a weight and is a real number. For convenience, we will write the following simple notation for . Similarly, for , , we will write if
(3.6)

for some , where the Radon measure is defined by , is a weight and is a real number. Again, the factor in the definitions of the norms and is for convenience and in fact these norms are independent of this expansion factor. We also write to replace when it is clear that the integral is weighted.

Definition 3.2.

We say a pair of weights satisfies the -condition in a set , write , for some and with if
(3.7)

Lemma (see [8]).

Let be a smooth differential form satisfying the nonhomogeneous -harmonic equation in , and . Then there exists a constant , independent of , such that
(3.8)

for all balls or cubes with .

Using the reverse Hölder inequality (Lemma 3.3) and Theorem 2.3, one obtains the following weighted version:
(3.9)

for all balls with , where , and , , , and are constants with , , , and .

Theorem 3.4.

Let , , , be a solution of the nonhomogeneous -harmonic equation in a bounded, convex domain , be the projection operator and be the homotopy operator, where the measure and are defined by , and for some and with for any . Then, there exists a constant , independent of , such that
(3.10)

where and are constants with and .

Proof.

Since , we have
(3.11)
for any ball . Using (3.9) and the Hölder inequality with , we find that
(3.12)
Notice that and , from (3.5), (3.11), and (3.12), we have
(3.13)

We have completed the proof of Theorem 3.4.

We now estimate the norm in terms of the norm.

Theorem 3.5.

Let , , , be a solution of the nonhomogeneous -harmonic equation in a bounded domain be the projection operator and be the homotopy operator, where the measure and are defined by , and for some and with for any . Then, there exists a constant , independent of , such that
(3.14)

where is a constant with .

Proof.

From the definitions of the weighted Lipschitz and the weighted BMO norms, we have
(3.15)
where is a positive constant. Replacing by in (3.15), we obtain
(3.16)
where is a constant with . Now, from Theorem 3.4, we find that
(3.17)
Substituting (3.17) into (3.16), we obtain
(3.18)

The proof of Theorem 3.5 has been completed.

Theorem 3.6.

Let and , , , be a smooth differential form in a bounded and convex domain . Then, there exists a constant , independent of , such that
(3.19)

Proof.

From the decomposition (1.6), we have
(3.20)
Using (1.5)), (3.20) and the Hölder inequality, it follows that
(3.21)

This ends the proof of Theorem 3.6.

Declarations

Acknowledgments

The authors thank the referee and the editor, Professor András Rontó, for their precious and thoughtful suggestions on this paper. The first author was supported by Science Research Foundation in Harbin Institute of Technology (HITC200709) and Development Program for Outstanding Young Teachers in HIT.

Authors’ Affiliations

(1)
Department of Mathematics, Harbin Institute of Technology
(2)
Department of Mathematics, Seattle University

References

  1. Scott C: theory of differential forms on manifolds. Transactions of the American Mathematical Society 1995,347(6):2075–2096. 10.2307/2154923MathSciNetMATHGoogle Scholar
  2. Wang Y, Wu C: Poincaré inequalities for Green's operator on solutions of the nonhomogeneous -harmonic equation. Computers & Mathematics with Applications 2004,47(10–11):1545–1554. 10.1016/j.camwa.2004.06.006MathSciNetView ArticleMATHGoogle Scholar
  3. Warner FW: Foundations of Differentiable Manifolds and Lie Groups, Graduate Texts in Mathematics. Volume 94. Springer, New York, NY, USA; 1983:ix+272.View ArticleGoogle Scholar
  4. Xing Y, Wu C: Global weighted inequalities for operators and harmonic forms on manifolds. Journal of Mathematical Analysis and Applications 2004,294(1):294–309. 10.1016/j.jmaa.2004.02.018MathSciNetView ArticleMATHGoogle Scholar
  5. Ding S: Weighted Hardy-Littlewood inequality for -harmonic tensors. Proceedings of the American Mathematical Society 1997,125(6):1727–1735. 10.1090/S0002-9939-97-03762-3MathSciNetView ArticleMATHGoogle Scholar
  6. Iwaniec T, Lutoborski A: Integral estimates for null Lagrangians. Archive for Rational Mechanics and Analysis 1993,125(1):25–79. 10.1007/BF00411477MathSciNetView ArticleMATHGoogle Scholar
  7. Liu B: -weighted imbedding inequalities for -harmonic tensors. Journal of Mathematical Analysis and Applications 2002,273(2):667–676. 10.1016/S0022-247X(02)00331-1MathSciNetView ArticleMATHGoogle Scholar
  8. Nolder CA: Hardy-Littlewood theorems for -harmonic tensors. Illinois Journal of Mathematics 1999,43(4):613–632.MathSciNetMATHGoogle Scholar
  9. Xing Y: Two-weight imbedding inequalities for solutions to the -harmonic equation. Journal of Mathematical Analysis and Applications 2005,307(2):555–564. 10.1016/j.jmaa.2005.03.019MathSciNetView ArticleMATHGoogle Scholar
  10. Cartan H: Differential Forms. Houghton Mifflin, Boston, Mass, USA; 1970.MATHGoogle Scholar
  11. Agarwal RP, Ding S, Nolder CA: Inequalities for Differential Forms. Springer, New York, NY, USA; 2009.View ArticleMATHGoogle Scholar

Copyright

© Y. Xing and S. Ding. 2009

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.