In this first chapter, we discuss various versions of the Hardy–Littlewood inequality for differential forms, including the local cases, the global cases, one weight cases, and two-weight cases. We know that differential forms are generalizations of the functions, which have been widely used in many fields, including potential theory, partial differential equations, quasiconformal mappings, nonlinear analysis, electromagnetism, and control theory; see [1–19], for example. During recent years new interest has developed in the study of the L p theory of differential forms on manifolds [20, 21]. For p = 2, the L p theory has been well studied. However, in the case of p ≠ 2, the L p theory is yet to be fully developed. The development of the L p theory of differential forms makes it possible to transport all notations of differential calculus in R n to the field of differential forms. The outline of this chapter is first to provide background materials, such as the definitions of differential forms and A-harmonic equations, some classes of weight functions and domains, and then, introduce different versions of Hardy–Littlewood inequalities on various domains with some specific weights or norms.
KeywordsQuasiconformal Mapping Weight Class Carnot Group Average Domain Weighted Inequality
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