Abstract and Applied Analysis

Abstract and Applied Analysis / 2014 / Article
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Lp-Theory of Differential Forms and Related Operators with Applications

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Research Article | Open Access

Volume 2014 |Article ID 941658 | 7 pages | https://doi.org/10.1155/2014/941658

Some Weighted Norm Estimates for the Composition of the Homotopy and Green’s Operator

Academic Editor: Yuming Xing
Received12 Nov 2013
Accepted23 Jan 2014
Published02 Mar 2014

Abstract

We establish the -weighted integral inequality for the composition of the Homotopy and Green’s operator on a bounded convex domain and also motivated it to the global domain by the Whitney cover. At the same time, we also obtain some -type norm inequalities. Finally, as applications of above results, we obtain the upper bound for the norms of or in terms of norms of or .

1. Introduction

Our purpose is to study the theory of the composition of the Homotopy and Green’s operator acting on differential forms on a bounded convex domain. Both operators play an important role in many fields, including harmonic analysis, potential theory, and partial equations (see [16]). In the present paper, we will obtain some -type norm inequalities for the composition of the Homotopy and Green’s operator and also prove the -weighted integral inequality on a bounded convex domain. These results will provide effective tools for studying behavior of solutions of -harmonic equations and related differential systems on manifolds.

We start this paper by introducing some notations and definitions. Let be a Riemannian, compact, oriented, and -smooth manifold without boundary on and let be an open subset of . Also, we use to denote Green’s operator throughout this paper. Furthermore, we use to denote a ball and to denote the ball with the same center as and with . We do not distinguish balls from cubs in this paper.

We assume that   is the linear space of all -forms with summation over all ordered -tuples ,  . If the coefficient of -form is differential on , then we call a differential -form on . A differential -form on is a de Rham current (see [7]) on with values in . Let be the th exterior power of the cotangent bundle and be the space of smooth -forms on . As usual, we use to denote the space of all differential -forms and to denote the -form with the norm on . Thus is a Banach space. As usual, we still use to denote the Hodge star operator. Also, we use to denote the differential operator and use to denote the Hodge codifferential operator which is defined by on . The -dimensional Lebesgue measure of a set is denoted by . We call a weight if and . For , we denote the weighted -norm of a measurable function over by where is a real number.

Let be a bounded, convex domain. Iwaniec and Lutoborski in [1] first introduced a linear operator satisfying that and the decomposition . Then by averaging over all points in , they constructed a Homotopy operator satisfying that , where is normalized by . The -form is defined by , if , and if , then

2. Boundedness of the Composition of the Homotopy and Green’s Operator in Space

In this section, we will prove the -weighted norm inequality for the composition of the Homotopy and Green’s operator on a bounded convex domain. Then using the Whitney cover, we develop the local result to the global domain. In [8], Gol’dshtein and Troyanov proved the following lemma.

Lemma 1. Let be a bounded convex domain. The operator maps continuously to in the following cases:

From [3], we have the following lemma about -estimates for Green’s operator.

Lemma 2. Let and . Then there exists a constant , independent of , such that

Definition 3. We say that a weight satisfies the condition for and write , if a.e. and

For weight, we also need the following result which appears in [9].

Lemma 4. If , then there exist constants and , independent of , such that for all balls .

Theorem 5. Let be a bounded convex domain, , and let be the Homotopy operator, . Then there exists a constant , independent of , such that for any ball , , and .

Proof. Since , by Lemma 4, there exist constants and , independent of , such that for any ball .
Choosing , then by Hölder inequality with , we have Thus, substituting (11) into (12), we obtain Taking , it is easy to see that and . Hence communicating Lemmas 1 and 2, we have Combining (13) and (14), we have Using Hölder inequality with , we have Note ; then, Thus, observing (15) and (16), we immediately obtain that Here is a constant independent of . Thus we complete the proof of Theorem 5.

Furthermore, if is an -harmonic tensor on , and , , then there exists a constant , independent of , such that for all balls or cubs with (for more details about -harmonic tensors, see [10]). By the property of -harmonic tensor, using the same method developed in the proof of Theorem 5, we can easily extend into the following -weighted version.

Corollary 6. Let be a bounded convex domain, , be an -harmonic tensor, and be the Homotopy operator, . Then there exists a constant , independent of , such that for any ball , , and , , .

In order to obtain the boundedness of the composition , we need the following modified Whitney cover in [10] and see [11] for more details about Whitney cover.

Lemma 7. Each open subset has a modified Whitney cover of cubs satisfying and , for all and some , where is the characteristic function for the set .

Theorem 8. Let be a bounded convex domain, . Then the composite operator is bounded, . Here and .

Proof. From Lemma 7, we know that there exists a sequence of cubs such that and for all , where is some constant. Hence, for , we have where and is independent of and each . Thus, we complete the proof of Theorem 8.

3. Norm Estimates with Power-Type Weights

Let be a bounded domain and be a nonempty of . If we use to denote the distance of the point from the set , then for is called power-type weight. In this section, we will establish some strong -type norm inequalities with power-type weights for the composition of the Homotopy and Green’s operator acting on differential form. In the following proof, we will use the following Lemma which appears in [8].

Lemma 9. The operator is bounded provided that

Theorem 10. Let be a bounded convex domain, , , , and let be the Homotopy operator, . Then there exists a constant , independent of , such that for any .

Proof. From (4), we have the following decomposition: for any differential form , .
Note that is an element of , . From (4) and Lemmas 1 and 9, we have Here is a constant independent of . Applying (24) and (5), we have Applying Lemma 2 into (26), we obtain Thus Here is independent of . Thus, we complete the proof of Theorem 10.

Next, we consider the following norm comparison equipped with power-type weights.

Theorem 11. Let be a bounded convex domain, , , , let be the Homotopy operator, , and that continuous functions and defined in satisfy ; . Then there exists a constant , independent of , such that for any , , .

Proof. From Theorem 10, we know that there exists a constant , independent of , such that Fixing , then there exists such that for all with . Let and . Then for all , we have Therefore, by the continuity of , we know that there exists , such that for all . Thus we have Here . Communicating (30) and (33), we have Note that . Then there exists such that for all with . Let and . Then for all , we have Therefore, by the continuity of , we know that there exists , such that for all . Therefore, we obtain Here . By (34) and (37), we have Here is independent of . Thus, we complete the proof of Theorem 11.

In Theorem 11, if we choose and , , , we can easily obtain the following corollary.

Corollary 12. Let be a bounded convex domain, , , , and let be the Homotopy operator, . Then there exists a constant , independent of , such that Here , .

Note that, in the proof of Theorem 11, if we let the composite operator act on the solution of nonhomogeneous -harmonic equation, then we can drop . Next, we state the result as follows.

Corollary 13. Let be a bounded convex domain, , , , let be the Homotopy operator, and is a solution of nonhomogeneous -harmonic equation, . If continuous functions and defined in satisfy that , and . Then there exists a constant , independent of , such that for all balls with . Here is some constant.

It is easy to find that the above corollary does not hold for balls with but holds for those balls with . Next, we introduce the following singular integral inequality.

Theorem 14. Let be a bounded convex domain, , , , let be the Homotopy operator, and is a solution of nonhomogeneous -harmonic equation, . If continuous functions and defined in and is an increasing function, then there exists a constant , independent of , such that for all balls with and . Here is some constant.

Proof. Let . From , it is easy to see that . Using the Hölder inequality, we have Note that . Therefore, there exists a positive number such that for all . Furthermore, by the continuity of function in , has a positive lower bound in . Thus, from Theorem 10 and (42), we have where is a constant. Let and . Since is the solution of nonhomogenous -harmonic equation. By (19), we know where is a constant. It is easy to find that . Using the Hölder inequality, we have The continuity and monotonicity of function imply that Hence, combining (41)–(47), we have Here is dependent of and but independent of . Thus, we complete the proof of Theorem 11.

4. Application

In this section, we will use the estimates in Section 3 to obtain the upper bound for the norms of or in terms of norms of or .

Example 15. For , let be a -form defined in by It is easy to find that If we choose the usual -type norm inequality to estimate and take , where is a ball, then by Theorem 10, we have However, if we choose the -type norm inequality to estimate and take , , then , satisfy the condition . Hence by using Theorem 10, we obtain Compare (51) and (52), we can easily find that if we choose different -type norm inequality to estimate the oscillation , we also obtain the different upper bound.

Example 16. In , consider that It is easy to check that is harmonic in the upper half plane. Note that Therefore, we have which implies that is a closed form and hence is a solution of nonhomogenous -harmonic equation. It is easy to see that Let denote a bound convex domain in the upper half plane and let be a closed ball without the points and . If and satisfy that , then both and have the upper bounds in . Thus, for the term it is usually not easy to be estimated due to the complexity of the compositions and the function . However, by Theorem 14, (57) can be controlled by the term Thus, we obtain an upper bound of (57).

Conflict of Interests

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

Acknowledgments

The first author was supported by the foundation at the Jiangxi University of Science and Technology (no. jxxj12073) and by the Youth Foundation of Jiangxi Provincial Education Department of China (no. GJJ13376).

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Copyright © 2014 Huacan Li and Qunfang Li. 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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