We prove A r -weighted imbedding theorems for differential forms. These results can be used to study the weighted norms of the homotopy operator T from the Banach space L p D ⧠l to the Sobolev space W 1 p D ⧠l-1 l = 1 2 n, and to establish the basic weighted L p -estimates for differential forms.  2001 Academic Press 1. INTRODUCTION In recent years many results about Sobolev functions have been extended to differential forms in R n . The imbedding theorems play crucial roles in generalizing these results to differential forms. The objective of this paper is to prove the weighted versions of imbedding theorems for differential forms and establish weighted norm estimates for the homotopy operator T . Let e 1 e 2
e n be the standard unit basis of R n , n β₯ 2, and let β§ l = β§ l (R n ) be the linear space of l-vectors, spanned by the exterior products e I = e i 1 β§ e i 2 β§ β’ β’ β’ β§ e i l , corresponding to all ordered l-tuples
The Grassman algebra β§ = ββ§ l is a graded algebra with respect to the exterior products. For Ξ± = Ξ± I e I β β§ and Ξ² = Ξ² I e I β β§, the inner product in β§ is given by Ξ± Ξ² = Ξ± I Ξ² I with summation over all l-tuples I = i 1 i 2 i l and all integers l = 0 1 n. The Hodge star operator : β§ β β§ is defined by the rule 1 = e 1 β§ e 2 β§ β’ β’ β’ β§ e n and Ξ± β§ Ξ² = Ξ² β§ Ξ± = Ξ± Ξ² 1 for all Ξ± Ξ² β β§. The norm of Ξ± β β§ is given by the formula Ξ± 2 = Ξ± Ξ± = Ξ± β§ Ξ± β β§ 0 = R. The Hodge star is an isometric isomorphism on β§ with β§ l β β§ n-l and -1 l n-l β§ l β β§ l .