Weighted Inequalities for Potential Operators on Differential Forms

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 [1–3]. 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 to the versions of differential forms.

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

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

(1.1)

Similarly, for a weight , we use to denote the weighted space with norm .

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

(1.2)

where is called a homotopy operator. Furthermore, we can define the -form by

(1.3)

for all , .

For any differential -form , we define the potential operator by

(1.4)

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

(1.5)

where is a function defined on . Associated with , the functional is defined as

(1.6)

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

(1.7)

and there exists such that

(1.8)

On the potential operator and the functional , see [4] for details.

For any locally -integrable form , the Hardy-Littlewood maximal operator is defined by

(1.9)

where is the ball of radius , centered at , .

Consider the nonhomogeneous -harmonic equation for differential forms as follows:

(1.10)

where and are two operators satisfying the conditions

(1.11)

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

(1.12)

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

In this section, we establish the weighted weaks type 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.

Let , , and . If and are two measurable functions on , then

(2.1)

for any .

Definition 2.2.

A pair of weights satisfies the -condition in a set ; write for some and with if

(2.2)

Proposition 2.3.

If for some and with , then satisfies the following condition:

(2.3)

Proof.

Choose and . From the Hölder inequality, we have the estimate

(2.4)

Since

(2.5)

we obtain that satisfies (2.3) as required.

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

Lemma 2.4.

Let and . Assume that is the potential operator defined in (1.5) and that is a functional satisfying (1.7) and (1.8). Let be a pair of weights for which there exists such that

(2.6)

Then the potential operator verifies the following weak type inequality:

(2.7)

where for any set and .

The following definition is introduced in [5].

Definition 2.5.

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

Theorem 2.6.

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

(2.8)

where for any set and .

Proof.

Since satisfies condition of the standard estimates, for any ball of radius , we have

(2.9)

Here and are two constants independent of . Therefore, and are some constants independent of . Thus, from satisfying (2.3) for some and , it follows that

(2.10)

Set and , where corresponds to all ordered -tuples and . It is easy to find that there must exist some such that whenever . Since the reverse is obvious, we immediately get . Thus, using Lemma 2.4 and the elementary inequality , where is any constant, we have

(2.11)

Combining the above inequality (2.11), the elementary inequality and Lemma 2.4 yield

(2.12)

We complete the proof of Theorem 2.6.

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

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

Lemma 3.1.

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

(3.1)

for all balls with .

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

Lemma 3.2.

Let and . Assume that is the potential operator defined in (1.5) and is a functional satisfying (1.7) and (1.8). Let be a pair of weights for which there exists such that

(3.2)

Then, there exists a constant , independent of , such that

(3.3)

Lemma 3.3.

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

(3.4)

Proof.

By the proof of Theorem 2.6, note that (3.2) still holds whenever satisfies the -condition. Therefore, using Lemma 3.2, we have

(3.5)

Also, Lemma 3.2 yields that

(3.6)

for all ordered -tuples . From (3.5) and (3.6), it follows that

(3.7)

We complete the proof of Lemma 3.3.

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

Theorem 3.4.

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

(3.8)

for all balls with . Here and are constants with .

Proof.

Take . By , where and the Hölder inequality, we have

(3.9)

for all balls with . Choosing to be a ball and in Lemma 3.3, then there exists a constant , independent of , such that

(3.10)

Choosing and using Lemma 3.1, we obtain

(3.11)

where . Combining (3.9), (3.10), and (3.11), it follows that

(3.12)

Since , using the Hölder inequality with , we obtain

(3.13)

From the condition , we have

(3.14)

Combining (3.12), (3.13), and (3.14) yields

(3.15)

for all balls with . 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 has a modified Whitney cover of cubes such that

(3.16)

(3.17)

for all and some , where is the characteristic function for a set .

Theorem 3.6.

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

(3.18)

where is some constant with .

Proof.

From Lemma 3.5, we note that has a modified Whitney cover . Hence, by Theorem 3.4, we have that

(3.19)

This completes the proof of Theorem 3.6.

Remark 3.7.

Note that if we choose the kernel to satisfy the standard estimates, then the potential operators 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.

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

(4.1)

for some constant independent of .

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

Theorem 4.2.

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

(4.2)

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

Consider that the function is defined by

(4.3)

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

Theorem 4.3.

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

(4.4)

Proof.

By the decomposition for differential forms, we have

(4.5)

where is the homotopy operator. Also, from [1], we have

(4.6)

for any differential form defined in . Therefore,

(4.7)

Note that

(4.8)

where the notation denotes convolution. Hence, we have

(4.9)

Since , it is easy to find that . Therefore, we have

(4.10)

From (4.7) and (4.10), we obtain

(4.11)

This ends the proof of Theorem 4.3.