New Weighted Integral Inequalities for Differential Forms in Some Domains

Differential forms are interesting and important generalizations of real functions and distributions. Many interesting results and applications of differential forms have recently been found in some fields, such as tensor analysis, potential theory, partial differential equations and quasiregular mappings, see [B], [C], [D1], [HKM], [I], [IL] and [IM]. In many cases, we need to know the integrability of differential forms and estimate the integrals for differential forms. In this paper we prove local weighted Poincaré-type inequalities for differential forms in any kind of domains and global weighted Poincaré-type inequalities for differential forms in John domains and Ls(μ)averaging domains, where μ is a measure defined by dμ = w(x)dx and w ∈ Ar . These integral inequalities can be used to study the integrability of differential forms and estimate the integrals for differential forms. As we know, A-harmonic tensors are the special differential forms which are solutions to the A-harmonic equation for differential forms: d?A(x, du) = 0, where A : Ω× ∧l(Rn) → ∧l(Rn) is an operator satisfying some conditions, see [I], [IL] and [N]. So that all of the results about differential forms in this paper remain true for A-harmonic tensors. Therefore, our new results concerning differential forms are of interest in some fields, such as those mentioned above. Throughout this paper, we always assume Ω is a connected open subset of Rn. Let e1, e2, . . . , en denote the standard unit basis of Rn. For l = 0, 1, . . . , n, the linear space of l-vectors, spanned by the exterior products eI = ei1∧ei2∧· · · eil , corresponding to all ordered l-tuples I = (i1, i2, . . . , il), 1 ≤ i1 < i2 < · · · < il ≤ n, is denoted by ∧l = ∧l(Rn). The Grassman algebra ∧ = ⊕∧l is a graded algebra with respect to the exterior products. For α = ∑ αeI ∈ ∧ and β = ∑ βeI ∈ ∧, the inner product in ∧ is given