Integral Estimates for the Potential Operator on Differential Forms

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. 1. Introduction 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 [8]. 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 [2]. For any differential -form , the potential