Imbedding theorems

Abstract

In recent years various versions of imbedding theorems for differential forms have been established. The imbedding theorems for functions can be found in almost every book on partial differential equations, see Sections 7.7 and 7.8 in [274], for example. For different versions of imbedding theorems, see [20, 275–280, 32, 268, 81]. Many results for Sobolev functions have been extended to differential forms in R ⁿ. The imbedding theorems play crucial role in generalizing the theory of Sobolev functions into the theory of differential forms. The objective of this chapter is to discuss several other versions of imbedding theorems for differential forms. We also explore some imbedding theorems related to operators, such as the homotopy operator T and Green’s operator G. We first study the imbedding theorems for quasiconformal mappings in Section 5.2. Then, we establish an imbedding theorem for differential forms satisfying the nonhomogeneous A-harmonic equation in Section 5.3. In Section 5.4, we present the A _r (Ω)-weighted imbedding theorems related to the gradient operator and the homotopy operator. In Section 5.5, we explore some A ^r(λ, Ω)-weighted imbedding theorems and $A_{r}^{\lambda}$(Ω)-weighted imbedding theorems. In Section 5.6, we develop some L ^s-estimates and imbedding theorems for the compositions of operators. Finally, in Section 5.7, we study the two-weight imbedding inequalities.