The Higher Integrability of Commutators of Calderón-Zygmund Singular Integral Operators on Differential Forms

Niu, Jinling; Xing, Yuming

doi:https://doi.org/10.1155/2018/1949747

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Research Article | Open Access

Volume 2018 | Article ID 1949747 | https://doi.org/10.1155/2018/1949747

The Higher Integrability of Commutators of Calderón-Zygmund Singular Integral Operators on Differential Forms

Jinling Niu¹and Yuming Xing¹

Academic Editor: Gestur Ólafsson

Received18 Aug 2018

Accepted30 Sept 2018

Published06 Nov 2018

Abstract

In this article, the higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms is derived. Also, the higher order Poincaré-type inequalities for the commutators acting on the solutions of Dirac-harmonic equations are obtained. Finally, some applications of the main results are demonstrated by examples.

1. Introduction

Differential forms as the key tools are widely used in many fields including quasiconformal analysis, nonlinear elasticity, and differential geometry, due to their advantage of being coordinate system independent; see [1–5]. The integrability of various operators and the upper bound estimates for the norms of operators are very important and core topics while studying the -theory of differential forms and investigating the qualitative and quantitative properties of the solutions of partial differential equations. In last few decades, a lot of related research has been done and many results on estimates for the -norms of various operators applied to the differential form in terms of the -norms of have been obtained; see [6–12]. In this paper, we define the commutators of Calderón-Zygmund singular integral operators on differential forms and give the strong type estimates for the commutators, which allows one to estimate in terms of the norm with . Meanwhile, we make a contribution to the estimates of the commutators in terms of the norm , where , that is the higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms. Then, we will establish the higher order Poincaré-type inequalities for the commutators applied to the solutions of Dirac-harmonic equations. The higher integrability and higher order inequalities in this paper can be used to study the regularity properties of the related operators. More results on the problem of higher order estimates and their applications in potential theory, quantum mechanics, and partial differential equations can be found in [13–17].

This paper is organised as follows. Section 2 contains, in addition to definitions and other preliminary material, the main lemmas. In Section 3, Theorems 12 and 13 show the local higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms. Based on these local results, the global higher integrability is presented in Theorems 14 and 15 by the well-known covering lemma. Especially, when the differential forms satisfy the Dirac-harmonic equations (in [18]), we establish the higher order Poincaré-type inequalities for the commutators in Section 4. The local higher order Poincaré-type inequalities are given in Theorems 16 and 17 and the global higher order Poincaré-type inequalities are obtained in Theorems 18 and 19. Finally, we demonstrate some applications of the main results by examples in Section 5. These results obtained in this paper will provide a further insight into the -theory and regularity theory of the related operators and differential forms.

2. Preliminary

Before specifying the main results precisely, we introduce some notation. Let be a bounded domain, , and be the balls with the same center, and for any real number . We denote by the -dimensional Lebesgue measure of a set . The set of all -forms, denoted by , is a -vector, spanned by exterior products , for all ordered -tuples , . The -form is called a differential -form if its coefficients are differentiable functions. We shall denote by the differential -forms space on and denote by the space of differential -forms on satisfying and with norm . In particular, we know that a 0-form is a function. A differential -form is called a closed form if in . Similarly, a differential -form is called a coclosed form if . From the Poincaré lemma, , we know that is a closed form. The operator is the Hodge-star operator which is an isometric isomorphism and the linear operator , , is called the exterior differential. The Hodge codifferential operator , the formal adjoint of , is defined by ; see [19] for more introduction. The following Dirac-harmonic equation for differential forms was initially introduced by S. Ding and B. Liu in [18]:for almost every , where is the Dirac operator, is a domain, and satisfies the following conditions:for almost every and all . Here, is a constant and is a fixed exponent associated with (1).

We should point out that the Dirac-harmonic equation is a kind of general equation which includes many existing harmonic equations as special cases, such as the -harmonic equation; see [18, 20, 21] for more information.

The Calderón-Zygmund singular integral operator on differential forms is defined bywhere is defined on , has mean 0, and is sufficiently smooth.

If , the commutator of Calderón-Zygmund singular integral operator on differential forms is of the form

When taking as a 0-form, the commutator in (4) reduces to the corresponding operator on function space as follows:

For the degenerated operator and the related applications in partial differential equations, see [22–24].

In order to prove our conclusions, we need several lemmas. The following -boundedness result for commutator on function spaces was proved in [25].

Lemma 1. Let be a weight satisfying condition: for all cubes , if there is a constant such thatwhere and . is any Calderón-Zygmund singular integral operator. Then, given any function , satisfies the following inequality:

The following lemma was given by S. Ding and B. Liu in [18].

Lemma 2. Let be a solution of -harmonic equation (1) in ; and are constants. Then, there exists a constant , independent of , such that for all cubes or balls with .

In [26], T. Iwaniec and A. Lutoborski gave the following three lemmas which will be used repeatedly in this paper.

Lemma 3. Let , , , be a differential form and be the homotopy operator. Then, we have the following decomposition:and the inequalityholds for any bounded domain , where is a constant, independent of .

From [26], the -form is defined by if and if , .

Lemma 4. Let , . Then, and where is a constant, independent of and .

Lemma 5. Let and . Then is in and where , , and is a constant independent of .

The following lemma appears in [27].

Lemma 6. Let defined on be a strictly increasing convex function, , and be a domain. Assume that satisfies for any real number , and where is a Radon measure defined by with a weight , then for any , we obtain where and are constants.

Choose , , and and let be a ball in Lemma 6; we find that the norms and are comparable; that is,for any ball with .

The covering lemma below belongs to [19].

Lemma 7. Each domain has a modified Whitney cover of cubes such that 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 .

3. Higher Integrability

In this section, we show the higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms. We first concentrate on the local higher integrability of the commutators. We need the following lemma appearing in [28].

Lemma 8. Letwhere , are points in n-dimensional Euclidean space and is a homogeneous function of degree with mean value zero on , and let . If and are sufficiently smooth and is bounded, then,for , where is a constant, independent of .

Lemma 9. Let , , , be the Calderón-Zygmund singular integral operator on differential forms, and be sufficiently smooth and bounded. Then, there exists a constant , independent of , such that

Proof. For any differential -form , by the definition of commutator of Calderón-Zygmund singular integral operator on differential forms, we haveLet and then according to the definition of the exterior differential operator, we obtainUsing the elementary inequality , for constants , we deduceBy Lemma 8, we obtainSubstituting (22) into (21) and applying the fundamental inequality , yields thatUsing the inequality again, we easily have that is,Substituting (25) into (23) gives This completes the proof of Lemma 9.

In Lemma 9, let in bounded domain and in ; we can easily obtain the following lemma.

Lemma 10. Let , , , be the Calderón-Zygmund singular integral operator on differential forms, and be sufficiently smooth and bounded. Then, there exists a constant , independent of , such that where is any bounded domain.

Using Lemma 1 and the analogous method developed in Lemma 9, we have the following estimate for .

Lemma 11. Let , , , and be a Calderón-Zygmund singular integral operator on differential forms. Then, given any function , satisfies the strong inequality for any bounded domain , where is a constant independent of .

We now present the local higher integrability of commutators of Calderón-Zygmund singular integral operators on differential forms.

Theorem 12. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If , , , then for any . Moreover, there exists a constant , independent of , such thatfor all balls with for some .

Proof. For any ball with for some , in the case that the measure . We have obviously that almost everywhere in , which implies is a closed form and a solution of the Dirac-harmonic equation (1) in . Applying Lemmas 4 and 2 to with any , we havefor all balls B with for some . We note that is bounded; thus . By Lemma 11, it follows thatOn the other hand, if the measure . Applying Lemma 5 to , we get Then by Lemma 10, it follows thatNoticing that the measure , we could use Lemma 6. Taking in Lemma 6, we havefor any differential form and any ball with . Replacing by in (34), we obtainCombining (33) and (35) yields thatIn view of the monotonic property of the -space with , we haveSubstituting (36) and (37) givesTherefore, inequality (29) follows in both cases, which indicates that if , then . We have completed the proof of Theorem 12.

Moreover, inequality (38) can be written as the following integral average inequality:

Clearly, with close to , the integral exponent on the left hand side could be much larger than the integral exponent on the right hand side since the condition . Hence the higher integrability of operator for the case that is obtained.

Next, we consider the higher integrability of for the case .

Theorem 13. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If , , , then for any . Moreover, there exists a constant , independent of , such thatfor all balls with for some .

Proof. It is immediate by the same method developed in the proof of Theorem 12 that (40) holds for . Let us consider the remain case that . Select and . Then it is easy to check that since . Notice that . Using Lemma 5 to and combining Lemma 10 and the monotonic property of the -space, we haveNotice that the measure . Therefore, we can select in Lemma 6. Then for any differential form , we haveReplacing by in (43), we getTaking into account the fact that , we see from the monotonic property of the -space, (42), and (44) thatwhich means that if , then . We have completed the proof of Theorem 13.

Now, we are ready to assert the global higher integrability of the commutator of Calderón-Zygmund singular integral operator on differential forms.

Theorem 14. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If , , , then for any . Moreover, there exists a constant , independent of , such thatfor any bounded domain .

Proof. Notice that since . By Lemma 7 and Theorem 12, we havewhich finishes the proof of Theorem 14.

Using the similar method as we did in Theorem 14 and combining with Theorem 13, we can deduce the following global result for the case .

Theorem 15. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If , , , then for any . Moreover, there exists a constant C, independent of u, such thatfor any bounded domain .

4. Higher Order Poincaré-Type Inequalities

In this section, we shall state the higher order Poincaré-type inequalities for commutator of Calderón-Zygmund singular integral operator acting on the solutions of the Dirac-harmonic equations.

Theorem 16. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If is a solution of the Dirac-harmonic equation (1), , . Then, for any , there exists a constant , independent of , such thatfor all balls with for some .

Proof. For any , using the -decomposition formula (9) to , we obtainNoticing that for any differential form , by (50), (10) and Lemma 10, we getSince is a solution of the Dirac-harmonic equation (1), by Lemma 2, we havewhere is a constant. Substituting (52) into (51) gives thatMoreover, for any , applying the monotonic properties of -space, it follows thatCombined with (53), the above inequality becomeswhich finishes the proof of Theorem 16.

Next, we will prove the higher order Poincaré-type inequality still holds for the case that .

Theorem 17. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If is a solution of the Dirac-harmonic equation (1), , . Then, for any , there exists a constant , independent of , such thatfor all balls with for some .

Proof. Select and . Then, we easily obtainthat is, , since . We can also find . Using the same technique as in the proof of Theorem 16, we haveNoticing that and combining the monotonic property of the -space, we obtainCombining (58) and (59), we finally haveThe proof of Theorem 17 is completed.

Based on Theorems 16 and 17, we can obtain the following global higher order Poincaré-type inequalities for the commutator using the analogous method developed in Theorem 14.

Theorem 18. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If is a solution of the Dirac-harmonic equation (1), , . Then, for any , there exists a constant , independent of , such thatfor any bounded domain .

Theorem 19. Let be the Calderón-Zygmund singular integral operator on differential forms and be sufficiently smooth and bounded. If is a solution of the Dirac-harmonic equation (1), , . Then, for any , there exists a constant , independent of , such thatfor any bounded domain .

5. Applications

In this section, we demonstrate some applications of our main results established in the previous sections.

Example 20. Let and be any constants and . Consider the 1-form defined in such that . It is easy to check that . Hence, is a solution of the Dirac-harmonic equation (1) for any operators satisfying (2). Also, it can be calculated thatfor any . Thus,Applying Theorem 14 for , we have for any . Analogously, we have for any when by Theorem 15. In the meantime, we have the higher estimate for ; that is,In addition, we can evaluate the following integrals by Theorem 18 and Theorem 19 that

We should notice that the above example can be extended to the case of as follows.

Example 21. We can check that the 1-form defined in is a solution of the Dirac-harmonic equation (1) for any operators satisfying (2). Hence, Theorems 14, 15, 18, and 19 are also applicable to .

Remark. It is worth pointing out that the global results obtained in this paper can be extended to larger classes of domains, such as -averaging domains and -averaging domains; see [1, 27]. Also, the techniques developed in this paper provide an effective method to study the higher integrability of bilinear commutators of singular integrals on differential forms, which are defined in [25]. We leave the statements and proofs to the interested readers.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant no. 11601105).

References

R. P. Agarwal, S. Ding, and C. Nolder, Inequalities for Differential Forms, Springer, New York, NY, USA, 2009.
View at: MathSciNet
Y. Xing and C. Wu, “Global weighted inequalities for operators and harmonic forms on manifolds,” Journal of Mathematical Analysis and Applications, vol. 294, no. 1, pp. 294–309, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
J. B. Perot and C. J. Zusi, “Differential forms for scientists and engineers,” Journal of Computational Physics, vol. 257, no. part B, pp. 1373–1393, 2014.
View at: Publisher Site | Google Scholar | MathSciNet
C. Scott, “L^p-theory of differential forms on manifolds 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 | MathSciNet
S. Ding, “Two-weight Caccioppoli inequalities for solutions of nonhomogeneous -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
S. Ding and B. Liu, “A singular integral of the composite operator,” Applied Mathematics Letters, vol. 22, no. 8, pp. 1271–1275, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
S. Ding and B. Liu, “Global estimates for singular integrals of the composite operator,” Illinois Journal of Mathematics, vol. 53, no. 4, pp. 1173–1185, 2009.
View at: Google Scholar | MathSciNet
S. Ding and B. Liu, “Global integrability of the Jacobian of a composite mapping,” Journal of Inequalities and Applications, Art. ID 89134, 9 pages, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
H. Bi and S. Ding, “Some strong (p,q)-type inequalities for the homotopy operator,” Computers and Mathematics with Applications. An International Journal, vol. 62, no. 4, pp. 1780–1789, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
H. Bi, “Weighted inequalities for potential operators on differential forms,” Journal of Inequalities and Applications, Art. ID 713625, 13 pages, 2010.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Ling and B. Gejun, “Some local Poincaré inequalities for the composition of the sharp maximal operator and the Green's operator,” Computers and Mathematics with Applications, vol. 63, no. 3, pp. 720–727, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
S. Ding, “Integral estimates for the Laplace-Beltrami and Green's operators applied to differential forms on manifolds,” Journal for Analysis and its Applications, vol. 22, no. 4, pp. 939–957, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
G. Lu and R. L. Wheeden, “High order representation formulas and embedding theorems on stratified groups and generalizations,” Studia Mathematica, vol. 142, no. 2, pp. 101–133, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
G. Lu, “Polynomials, higher order Sobolev extension theorems and interpolation inequalities on weighted Folland-STEin spaces on stratified groups,” Acta Mathematica Sinica, vol. 16, no. 3, pp. 405–444, 2000.
View at: Publisher Site | Google Scholar | MathSciNet
E. W. Stredulinsky, “Higher integrability from reverse Hölder inequalities,” Indiana University Mathematics Journal, vol. 29, no. 3, pp. 407–413, 1980.
View at: Publisher Site | Google Scholar | MathSciNet
F. W. Gehring, “The L^p-integrability of the partial derivatives of a quasiconformal mapping,” Acta Mathematica, vol. 130, pp. 265–277, 1973.
View at: Publisher Site | Google Scholar | MathSciNet
W. S. Cohn, G. Lu, and S. Lu, “Higher order Poincaré inequalities associated with linear operators on stratified groups and applications,” Mathematische Zeitschrift, vol. 244, no. 2, pp. 309–335, 2003.
View at: Publisher Site | Google Scholar | MathSciNet
S. Ding and B. Liu, “Dirac-harmonic equations for differential forms,” Nonlinear Analysis. Theory, Methods and Applications. An International Multidisciplinary Journal, vol. 122, pp. 43–57, 2015.
View at: Publisher Site | Google Scholar | MathSciNet
C. A. Nolder, “Hardy-Littlewood theorems for -harmonic tensors,” Illinois Journal of Mathematics, vol. 43, no. 4, pp. 613–632, 1999.
View at: Google Scholar | MathSciNet
A. Carbonaro, A. McIntosh, and A. J. Morris, “Local Hardy spaces of differential forms on Riemannian manifolds,” The Journal of Geometric Analysis, vol. 23, no. 1, pp. 106–169, 2013.
View at: Publisher Site | Google Scholar | MathSciNet
K. Kodaira, “Harmonic fields in Riemannian manifolds (generalized potential theory),” Annals of Mathematics: Second Series, vol. 50, pp. 587–665, 1949.
View at: Publisher Site | Google Scholar | MathSciNet
R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Annals of Mathematics, vol. 103, no. 3, pp. 611–635, 1976.
View at: Publisher Site | Google Scholar | MathSciNet
L. Chaffee and D. Cruz-Uribe, “Necessary conditions for the boundedness of linear and bilinear commutators on Banach function spaces,” Mathematical Inequalities and Applications, vol. 21, no. 1, pp. 1–16, 2018.
View at: Publisher Site | Google Scholar | MathSciNet
C. Segovia and J. L. Torrea, “Weighted inequalities for commutators of fractional and singular integrals,” Publicacions Matemàtiques, vol. 35, no. 1, pp. 209–235, 1991.
View at: Publisher Site | Google Scholar | MathSciNet
C. Pérez, “Sharp estimates for commutators of singular integrals via iterations of the Hardy-Littlewood maximal function,” Journal of Fourier Analysis and Applications, vol. 3, no. 6, pp. 743–756, 1997.
View at: Publisher Site | Google Scholar | 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 | MathSciNet
S. Ding, “L^φ(μ)-averaging domains and the quasi-hyperbolic metric,” Computers and Mathematics with Applications. An International Journal, vol. 47, no. 10-11, pp. 1611–1618, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
A. P. Calderón, “Commutators of singular integral operators,” Proceedings of the National Acadamy of Sciences of the United States of America, vol. 53, pp. 1092–1099, 1965.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2018 Jinling Niu and Yuming Xing. 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.

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