Weighted Decomposition Estimates for Differential Forms
© Y. Wang and G. Li. 2010
Received: 28 October 2009
Accepted: 24 March 2010
Published: 4 May 2010
After introducing the definition of -weights, we establish the -weighted decomposition estimates and -weighted Caccioppoli-type estimates for -harmonic tensors. Furthermore, by Whitney covering lemma, we obtain the global results in domain . These results can be used to study the integrability of differential forms and to estimate the integrals for differential forms.
The solutions to (1.5) are called -harmonic tensors. See  for the recent research on -harmonic equation.
Caccioppoli-type estimates have been widely studied and frequently used in analysis and related fields, including partial differential equations and the theory of elasticity. These inequalities provide upper bounds for the -norm of if is a function or if is a form with the -norm of the differential form . Different versions of the Caccioppoli-type inequality have been established in [2, 3].
In , Nolder obtains the following local Caccioppoli-type estimate.
The next lemma is the generalized Hölder inequality which will be widely used in this paper.
The following weak reverse Hölder inequality plays an important role in founding the integral estimate of the nonhomogeneous and homogeneous -harmonic tensor; see .
The following Whitney covering lemma appears in .
The proof of Theorem 2.6 is completed.
Since there are fore real parameters, , in Theorem 2.6, we can obtain some desired versions of weighted Caccioppoli-type estimates by different choices of them. Let in Theorem 2.6, we obtain the following corollaries.
3. The Local Weighted Estimates for the Decomposition
The Hodge decomposition theorem has been playing an important part in partial differential equation, the operator theory, and so on. In the recent years, there are some interesting conclusions on the Hodge decomposition of differential forms; see [5, 6]. In , there is the following decomposition theorem for differential form .
The next lemma states the reverse Hölder inequality for -weight; see .
In this section, we will extend (3.2) to the weighted form.
4. The Global Weighted Estimates
The proof of Theorem 4.1 is completed.
In Theorems 3.4 and 4.1, there is a real parameter , which makes the results more flexible. By choosing different value of the parameter , we get the estimates in different forms. For example, if we take , we have the following corollary.
This work was supported by the National Natural Science Foundation of China (Grant no. 10771044) and the Natural Science Foundation of Heilongjiang Province (Grant no. 200605).
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