Weighted Decomposition Estimates for Differential Forms

Let e1, e2, . . . , en denote the standard orthogonal basis of R. Suppose that Λ Λ R is the linear space of all l-vectors, spanned by the exterior product eI ei1∧ei2∧· · ·∧eil corresponding to all ordered l-tuples I i1, i2, . . . , il , 1 ≤ i1 < i2 < · · · < il ≤ n. Throughout this paper, we always assume that Ω is an open subset of R. We use D′ M,Λ to denote the space of all differential l-forms and L M,Λ to denote the l-formsω x ∑ I ωIdxI ∑ ωi1i2···il x dx1∧ dx2 ∧ · · · ∧ dxl on M with the coefficient ωI ∈ L Ω,R for all ordered l-tuples I, where M is a manifold. Thus L M,Λ is a Banach space with the norm