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International Journal of Analysis
Volume 2013 (2013), Article ID 108623, 6 pages
Integral Estimates for the Potential Operator on Differential Forms
Department of Mathematics, Seattle University, Seattle, WA 98122, USA
Received 2 October 2012; Accepted 21 November 2012
Academic Editor: Tohru Ozawa
Copyright © 2013 Casey Johnson and Shusen Ding. 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.
We develop the local inequalities with new weights for the potential operator applied to differential forms. We also prove the global weighted norm inequalities for the potential operator in averaging domains and explore applications of our new results.
This paper deals with the weighted estimates for the potential operator applied to differential forms. Throughout this paper, will denote an open subset of , , and . Let , ,, be the standard unit basis of . For , the linear space of -vectors, spanned by the exterior products , corresponding to all ordered -tuples , , is denoted by . The Grassman algebra is a graded algebra with respect to the exterior products. For and , the inner product in is given by with summation over all -tuples and all integers . We should also notice that , and .
Assume that is a ball and is the ball with the same center as and with . Differential forms are extensions of functions defined in . A function in is called a -form. A differential -form is of the form in . Differential forms have become invaluable tools for many fields of sciences and engineering, including theoretical physics, general relativity, potential theory, and electromagnetism. They can be used to describe various systems of PDEs and to express different geometrical structures on manifolds. Many interesting and useful results about the differential forms have been obtained during recent years; particularly, for the differential forms satisfying some version of -hrmonic equation, see [1–8]. The -dimensional Lebesgue measure of a set is denoted by . We call a weight if and a.e. For , we denote the weighted norm of a measurable function over by if the above integral exists. Here is a real number. It should be noticed that the Hodge star operator can be defined equivalently as follows.
Definition 1. If , is a differential -form, then where , , and
The following -weights were introduced in .
Definition 2. One says that a measurable function defined on a subset satisfies the -condition for some positive constants , writes if a.e., and writes
where the supremum is over all balls . One says that satisfies the -condition if (7) holds for and write .
Notice that there are three parameters in the definition of the -class. We obtain some existing weighted classes if we choose some particular values for these parameters. For example, it is easy to see that the -class reduces to the usual -class if and .
Recently, Bi extended the definition of the potential operator to the case of differential forms; see . For any differential -form , the potential operator is defined by where the kernel is a nonnegative measurable function defined for and the summation is over all ordered -tuples . The case reduces to the usual potential operator, where is a function defined on . A kernel on is said to satisfy the standard estimates if there are , , and constant such that, for all distinct points and in and all with , the kernel satisfies (i) , (ii) , and (iii) . In this paper, we always assume that is the potential operator with the kernel satisfying the above condition (i) in the standard estimates. In , Bi proved the following inequality for the potential operator: where , is a differential form defined in a bounded and convex domain .
Definition 3. One says that a differential form belongs to the -class and writes , , if for any constants , the inequality
holds for any ball with , where and are constants.
From , we know that any solution of -harmonic equations satisfies (11). Hence, the -class is a large class of differential forms. We will use the following Hölder inequality repeatedly in this paper.
Lemma 4. Let both and be measurable functions in and and . Then for any .
2. Local Inequalities
In this section, we will prove some local weighted norm inequalities for the potential operator.
Theorem 5. Let be the potential operator applied to a differential form , where is a domain. Assume that with . Then, there exists a constant , independent of , such that for all balls with , where is a constant.
Proof. Let , then . Using Lemma 4 with yields
Set , then . Since is in the class,
where in a constraint. Since is in the -class again (note that ),
where we have used the following calculation:
Combining (14), (15) and, (16) gives Note that since , it follows that Plugging (19) into (18), we have We should notice that Combining (20) and (21) gives us The proof of Theorem 5 has been completed.
A continuously increasing function with is called an Orlicz function. A convex Orlicz function is often called a Young function. The Orlicz space consists of all measurable functions on such that for some . If is a Young function, then defines a norm in , which is called the Orlicz norm or the Luxemburg norm.
Definition 6 (see ). One says that a Young function lies in the class , , , if (i) and (ii) for all , where is a convex increasing function and is a concave increasing function on .
From , each of , and in the above definition is doubling in the sense that its values at and are uniformly comparable for all and the consequent fact that where and are constants. Also, for all and , the function belongs to for some constant . Here is defined by for and for . Particularly, if , we see that lies in , .
Theorem 7. Let be the potential operator applied to a differential form and a Young function in the class , and where is a bounded domain. Assume that . Then, there exists a constant , independent of , such that for all balls with , where is a constant.
Proof. Using Jensen's inequality for and (24), we have
Since , by (11) we obtain where is a constant. Using (13), (24), and Jensen's inequality, Note that , then . Thus, Using the above inequality and (i) in Definition 6, we find that . Therefore, Combining (28) and (30) yields Finally, substituting (31) into (26), we obtain The proof of Theorem 7 has been completed.
3. Global Inequalities
In 1989, Staples introduced the following -averaging domains in . A proper subdomain is called an -averaging domain, , if there exists a constant such that for all . Here the supremum is over all balls . The -averaging domains were extended into the -averaging domains recently in . We call a proper subdomain an -averaging domain, , if and there exists a constant such that for some ball and all . The -averaging domain was genralized into the following -averaging domain in .
Definition 8. Let be a continuous increasing convex function on with . One calls a proper subdomain an -averaging domain, if and there exists a constant such that
for some ball and all such that , where the measure is defined by is a weight, and are constants with , , and the supremum is over all balls with .
From the above definition, we see that -averaging domains are special -averaging domains when in Definition 8.
Theorem 9. Let be a Young function in the class , , ; and any bounded -averaging domain, and the potential operator applied to a differential form , . Assume that . Then, there exists a constant , independent of , such that where is some fixed ball.
Choosing in Theorem 9, we obtain the following Poincaré inequalities with the -norms.
Corollary 10. Let , , , and the potential operator applied to a differential form , . Assume that . Then, there exists a constant , independent of , such that for any bounded -averaging domain and is some fixed ball.
We have established the local and global weighted estimates for the potential operator applied to the differential forms in the -class. It is well known that any solution to -harmonic equations belongs to the -class. Hence, our inequalities can be used to estimate solutions of -harmonic equations. Next, as applications of the main theorems, we develop some estimates for the Jacobian of a mapping , . We know that the Jacobian of a mapping is an -form, specifically, , where . For example, let be a differential mapping in . Then, where we have used the property if , and if . Clearly, .
Let , be a mapping, whose distributional differential is a locally integrable function on with values in the space of all -matrices. We use to denote the Jacobian determinant of . Assume that is the subdeterminant of Jacobian , which is obtained by deleting the rows and columns, ; that is, which is an subdeterminant of , and . Note that is an -form. Thus, all estimates for differential forms are applicable to the -form . For example, choosing and applying Theorems 7 and 9 to , respectively, we have the following theorems.
Theorem 11. Let be a Young function in the class , . Let be a mapping such that , where is the Jacobian of the mapping and is a bounded domain in . Assume that . Then, there exists a constant , independent of , such that for all balls and some constant .
Theorem 12. Let be a Young function in the class , . Let be a mapping such that , where is the Jacobian of the mapping and is a bounded -averaging domain in . Assume that . Then, there exists a constant , independent of , such that for some ball .
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