Weighted Inequalities for Potential Operators on Differential Forms
© Hui Bi. 2010
Received: 23 December 2009
Accepted: 10 February 2010
Published: 14 February 2010
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.
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
Similarly, for a weight , we use to denote the weighted space with norm .
From , if is a differential form defined in a bounded, convex domain , then there is a decomposition
where is called a homotopy operator. Furthermore, we can define the -form by
for all , .
For any differential -form , we define the potential operator by
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,
where is a function defined on . Associated with , the functional is defined as
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
and there exists such that
On the potential operator and the functional , see  for details.
For any locally -integrable form , the Hardy-Littlewood maximal operator is defined by
where is the ball of radius , centered at , .
Consider the nonhomogeneous -harmonic equation for differential forms as follows:
where and are two operators satisfying the conditions
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
for all with compact support. Here are those differential -forms on whose coefficients are in . The notation is self-explanatory.
2. Weak Type Inequalities for Potential Operators
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.
for any .
we obtain that satisfies (2.3) as required.
In , Martell proved the following two-weight weak type norm inequality applied to functions.
where for any set and .
The following definition is introduced in .
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 ; ; .
where for any set and .
We complete the proof of Theorem 2.6.
3. The Strong Type Inequalities for Potential Operators
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 .
for all balls with .
The following two-weight inequality appears in .
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.
for all balls with . Here and are constants with .
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 .
for all and some , where is the characteristic function for a set .
where is some constant with .
This completes the proof of Theorem 3.6.
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 .
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 .
Next, applying our results to some special kernels, we have the following estimates.
Consider that the function is defined by
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.
This ends the proof of Theorem 4.3.
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