- Research Article
- Open Access
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.
- Differential Form
- Potential Operator
- Standard Estimate
- Weak Type
- Norm Inequality
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 , .
From , if is a differential form defined in a bounded, convex domain , then there is a decomposition
On the potential operator and the functional , see  for details.
We first need the following generalized Hölder inequality.
In , Martell proved the following two-weight weak type norm inequality applied to functions.
The following definition is introduced in .
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 .
The following two-weight inequality appears in .
We complete the proof of Lemma 3.3.
Next, we extend the weighted inequality to the global version, which needs the following lemma about Whitney cover that appears in .
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 .
Next, applying our results to some special kernels, we have the following estimates.
This ends the proof of Theorem 4.3.
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