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Weighted Inequalities for Potential Operators on Differential Forms

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Research Article
Part of the following topical collections:
  1. Inequalities in the A-Harmonic Equations and the Related Topics

Abstract

We develop the weak-type and strong-type inequalities for potential operators under two-weight conditions to the versions of differential forms. We also obtain some estimates for potential operators applied to the solutions of the nonhomogeneous A-harmonic equation.

Keywords

Differential Form Potential Operator Standard Estimate Weak Type Norm Inequality 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

1. Introduction

In recent years, differential forms as the extensions of functions have been rapidly developed. Many important results have been obtained and been widely used in PDEs, potential theory, nonlinear elasticity theory, and so forth; see [1, 2, 3]. In many cases, the process to solve a partial differential equation involves various norm estimates for operators. In this paper, we are devoted to develop some two-weight norm inequalities for potential operator Open image in new window to the versions of differential forms.

We first introduce some notations. Throughout this paper we always use Open image in new window to denote an open subset of Open image in new window , Open image in new window . Assume that Open image in new window is a ball and Open image in new window is the ball with the same center as Open image in new window and with Open image in new window . Let Open image in new window , Open image in new window , be the linear space of all Open image in new window -forms Open image in new window with summation over all ordered Open image in new window -tuples Open image in new window , Open image in new window . The Grassman algebra Open image in new window is a graded algebra with respect to the exterior products Open image in new window . Moreover, if the coefficient Open image in new window of Open image in new window -form Open image in new window is differential on Open image in new window , then we call Open image in new window a differential Open image in new window -form on Open image in new window and use Open image in new window to denote the space of all differential Open image in new window -forms on Open image in new window . In fact, a differential Open image in new window -form Open image in new window is a Schwarz distribution on Open image in new window with value in Open image in new window . For any Open image in new window and Open image in new window , the inner product in Open image in new window is defined by Open image in new window with summation over all Open image in new window -tuples Open image in new window and all Open image in new window . As usual, we still use Open image in new window to denote the Hodge star operator. Moreover, the norm of Open image in new window is given by Open image in new window . Also, we use Open image in new window to denote the differential operator and use Open image in new window to denote the Hodge codifferential operator defined by Open image in new window on Open image in new window , Open image in new window .

A weight Open image in new window is a nonnegative locally integrable function on Open image in new window . The Lebesgue measure of a set Open image in new window is denoted by Open image in new window . Open image in new window is a Banach space with norm

Similarly, for a weight Open image in new window , we use Open image in new window to denote the weighted Open image in new window space with norm Open image in new window .

From [1], if Open image in new window is a differential form defined in a bounded, convex domain Open image in new window , then there is a decomposition

where Open image in new window is called a homotopy operator. Furthermore, we can define the Open image in new window -form Open image in new window by

for all Open image in new window , Open image in new window .

For any differential Open image in new window -form Open image in new window , we define the potential operator Open image in new window by

where the kernel Open image in new window is a nonnegative measurable function defined for Open image in new window and the summation is over all ordered Open image in new window -tuples Open image in new window . It is easy to find that the case Open image in new window reduces to the usual potential operator. That is,

where Open image in new window is a function defined on Open image in new window . Associated with Open image in new window , the functional Open image in new window is defined as

where Open image in new window is some sufficiently small constant and Open image in new window is a ball with radius Open image in new window . Throughout this paper, we always suppose that Open image in new window satisfies the following conditions: there exists Open image in new window such that

and there exists Open image in new window such that

On the potential operator Open image in new window and the functional Open image in new window , see [4] for details.

For any locally Open image in new window -integrable form Open image in new window , the Hardy-Littlewood maximal operator Open image in new window is defined by

where Open image in new window is the ball of radius Open image in new window , centered at Open image in new window , Open image in new window .

Consider the nonhomogeneous Open image in new window -harmonic equation for differential forms as follows:

where Open image in new window and Open image in new window are two operators satisfying the conditions

for almost every Open image in new window and all Open image in new window . Here Open image in new window are some constants and Open image in new window is a fixed exponent associated with (1.10). A solution to (1.10) is an element of the Sobolev space Open image in new window such that

for all Open image in new window with compact support. Here Open image in new window are those differential Open image in new window -forms on Open image in new window whose coefficients are in Open image in new window . The notation Open image in new window is self-explanatory.

2. Weak Type Open image in new window Inequalities for Potential Operators

In this section, we establish the weighted weaks type Open image in new window inequalities for potential operators applied to differential forms. To state our results, we need the following definitions and lemmas.

We first need the following generalized Hölder inequality.

Lemma 2.1.

for any Open image in new window .

Definition 2.2.

Proposition 2.3.

Proof.

Choose Open image in new window and Open image in new window . From the Hölder inequality, we have the estimate

we obtain that Open image in new window satisfies (2.3) as required.

In [4], Martell proved the following two-weight weak type norm inequality applied to functions.

Lemma 2.4.

Let Open image in new window and Open image in new window . Assume that Open image in new window is the potential operator defined in (1.5) and that Open image in new window is a functional satisfying (1.7) and (1.8). Let Open image in new window be a pair of weights for which there exists Open image in new window such that
Then the potential operator Open image in new window verifies the following weak type Open image in new window inequality:

where Open image in new window for any set Open image in new window and Open image in new window .

The following definition is introduced in [5].

Definition 2.5.

A kernel Open image in new window on Open image in new window satisfies the standard estimates if there exist Open image in new window , Open image in new window , and constant Open image in new window such that for all distinct points Open image in new window and Open image in new window in Open image in new window , and all Open image in new window with Open image in new window , the kernel Open image in new window satisfies Open image in new window ; Open image in new window ; Open image in new window .

Theorem 2.6.

Let Open image in new window be the potential operator defined in (1.4) with the kernel Open image in new window satisfying the condition Open image in new window of the standard estimates and let Open image in new window be a differential form in a domain Open image in new window . Assume that Open image in new window satisfies (2.3) for some Open image in new window and Open image in new window . Then, there exists a constant Open image in new window , independent of Open image in new window , such that the potential operator Open image in new window satisfies the following weak type Open image in new window inequality:

where Open image in new window for any set Open image in new window and Open image in new window .

Proof.

Since Open image in new window satisfies condition Open image in new window of the standard estimates, for any ball Open image in new window of radius Open image in new window , we have
Here Open image in new window and Open image in new window are two constants independent of Open image in new window . Therefore, Open image in new window and Open image in new window are some constants independent of Open image in new window . Thus, from Open image in new window satisfying (2.3) for some Open image in new window and Open image in new window , it follows that
Set Open image in new window and Open image in new window , where Open image in new window corresponds to all ordered Open image in new window -tuples and Open image in new window . It is easy to find that there must exist some Open image in new window such that Open image in new window whenever Open image in new window . Since the reverse is obvious, we immediately get Open image in new window . Thus, using Lemma 2.4 and the elementary inequality Open image in new window , where Open image in new window is any constant, we have
Combining the above inequality (2.11), the elementary inequality and Lemma 2.4 yield

We complete the proof of Theorem 2.6.

3. The Strong Type Open image in new window Inequalities for Potential Operators

In this section, we give the strong type Open image in new window inequalities for potential operators applied to differential forms. The result in last section shows that Open image in new window -weights are stronger than those of condition (2.3), which is sufficient for the weak Open image in new window inequalities, while the following conclusions show that Open image in new window -condition is sufficient for strong Open image in new window inequalities.

The following weak reverse Hölder inequality appears in [6].

Lemma 3.1.

Let Open image in new window , Open image in new window be a solution of the nonhomogeneous A-harmonic equation in Open image in new window , Open image in new window and Open image in new window . Then there exists a constant Open image in new window , independent of Open image in new window , such that

for all balls Open image in new window with Open image in new window .

The following two-weight inequality appears in [7].

Lemma 3.2.

Let Open image in new window and Open image in new window . Assume that Open image in new window is the potential operator defined in (1.5) and Open image in new window is a functional satisfying (1.7) and (1.8). Let Open image in new window be a pair of weights for which there exists Open image in new window such that
Then, there exists a constant Open image in new window , independent of Open image in new window , such that

Lemma 3.3.

Let Open image in new window , Open image in new window , Open image in new window , be a differential form defined in a domain Open image in new window and Open image in new window be the potential operator defined in (1.4) with the kernel Open image in new window satisfying condition Open image in new window of standard estimates. Assume that Open image in new window for some Open image in new window and Open image in new window . Then, there exists a constant Open image in new window , independent of Open image in new window , such that

Proof.

By the proof of Theorem 2.6, note that (3.2) still holds whenever Open image in new window satisfies the Open image in new window -condition. Therefore, using Lemma 3.2, we have
Also, Lemma 3.2 yields that
for all ordered Open image in new window -tuples Open image in new window . From (3.5) and (3.6), it follows that

We complete the proof of Lemma 3.3.

Lemma 3.3 shows that the two-weight strong Open image in new window inequality still holds for differential forms. Next, we develop the inequality to the parametric version.

Theorem 3.4.

Let Open image in new window , Open image in new window , Open image in new window , be the solution of the nonhomogeneous A-harmonic equation in a domain Open image in new window and let Open image in new window be the potential operator defined in (1.4) with the kernel Open image in new window satisfying condition Open image in new window of standard estimates. Assume that Open image in new window for some Open image in new window and Open image in new window . Then, there exists a constant Open image in new window , independent of Open image in new window , such that

for all balls Open image in new window with Open image in new window . Here Open image in new window and Open image in new window are constants with Open image in new window .

Proof.

for all balls Open image in new window with Open image in new window . Choosing Open image in new window to be a ball and Open image in new window in Lemma 3.3, then there exists a constant Open image in new window , independent of Open image in new window , such that
Choosing Open image in new window and using Lemma 3.1, we obtain
where Open image in new window . Combining (3.9), (3.10), and (3.11), it follows that
Since Open image in new window , using the Hölder inequality with Open image in new window , we obtain
From the condition Open image in new window , we have
Combining (3.12), (3.13), and (3.14) yields

for all balls Open image in new window with Open image in new window . Thus, we complete the proof of Theorem 3.4.

Next, we extend the weighted inequality to the global version, which needs the following lemma about Whitney cover that appears in [6].

Lemma 3.5.

Each open set Open image in new window has a modified Whitney cover of cubes Open image in new window such that

for all Open image in new window and some Open image in new window , where Open image in new window is the characteristic function for a set Open image in new window .

Theorem 3.6.

Let Open image in new window , Open image in new window , Open image in new window , be the solution of the nonhomogeneous A-harmonic equation in a domain Open image in new window and let Open image in new window be the potential operator defined in (1.4) with the kernel Open image in new window satisfying condition Open image in new window of standard estimates. Assume that Open image in new window for some Open image in new window and Open image in new window . Then, there exists a constant Open image in new window , independent of Open image in new window , such that

where Open image in new window is some constant with Open image in new window .

Proof.

From Lemma 3.5, we note that Open image in new window has a modified Whitney cover Open image in new window . Hence, by Theorem 3.4, we have that

This completes the proof of Theorem 3.6.

Remark 3.7.

Note that if we choose the kernel Open image in new window to satisfy the standard estimates, then the potential operators Open image in new window reduce to the Calderón-Zygmund singular integral operators. Hence, Theorems 3.4 and 3.6 as well as Theorem 2.6 in last section still hold for the Calderón-Zygmund singular integral operators applied to differential forms.

4. Applications

In this section, we apply our results to some special operators. We first give the estimate for composite operators. The following lemma appears in [8].

Lemma 4.1.

Let Open image in new window be the Hardy-Littlewood maximal operator defined in (1.9) and let Open image in new window , Open image in new window , Open image in new window , be a differential form in a domain Open image in new window . Then, Open image in new window and

for some constant Open image in new window independent of Open image in new window .

Observing Lemmas 4.1 and 3.3, we immediately have the following estimate for the composition of the Hardy-Littlewood maximal operator Open image in new window and the potential operator Open image in new window .

Theorem 4.2.

Let Open image in new window , Open image in new window , Open image in new window , be a differential form defined in a domain Open image in new window , Open image in new window be the Hardy-Littlewood maximal operator defined in (1.9), Open image in new window , and let Open image in new window be the potential operator with the kernel Open image in new window satisfying condition Open image in new window of standard estimates. Then, there exists a constant Open image in new window , independent of Open image in new window , such that

Next, applying our results to some special kernels, we have the following estimates.

Consider that the function Open image in new window is defined by

where Open image in new window . For any Open image in new window , we write Open image in new window . It is easy to see that Open image in new window and Open image in new window . Such functions are called mollifiers. Choosing the kernel Open image in new window and setting each coefficient of Open image in new window satisfing Open image in new window , we have the following estimate.

Theorem 4.3.

Let Open image in new window , Open image in new window , be a differential form defined in a bounded, convex domain Open image in new window , and let Open image in new window be coefficient of Open image in new window with Open image in new window for all ordered Open image in new window -tuples Open image in new window . Assume that Open image in new window and Open image in new window is the potential operator with Open image in new window for any Open image in new window . Then, there exists a constant Open image in new window , independent of Open image in new window , such that

Proof.

By the decomposition for differential forms, we have
where Open image in new window is the homotopy operator. Also, from [1], we have
for any differential form Open image in new window defined in Open image in new window . Therefore,
Note that
where the notation Open image in new window denotes convolution. Hence, we have
Since Open image in new window , it is easy to find that Open image in new window . Therefore, we have
From (4.7) and (4.10), we obtain

This ends the proof of Theorem 4.3.

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Copyright information

© Hui Bi. 2010

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.

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina
  2. 2.Department of Applied MathematicsHarbin University of Science and TechnologyHarbinChina

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