Norm Comparison Inequalities for the Composite Operator

The purpose of this paper is to establish the Lipschitz norm and BMO norm inequalities for the composition of the homotopy operator T and the projection operator H applied to differential forms in R, n ≥ 2. The harmonic projection operator H, one of the key operators in the harmonic analysis, plays an important role in the Hodge decomposition theory of differential forms. In the meanwhile, the homotopy operator T is also widely used in the decomposition and the L-theory of differential forms. In many situations, we need to estimate the various norms of the operators and their compositions. We always assume that M is a bounded, convex domain and B is a ball in R, n ≥ 2, throughout this paper. Let σB be the ball with the same center as B and with diam σB σ diam B , σ > 0. We do not distinguish the balls from cubes in this paper. For any subset E ⊂ R, we use |E| to denote the Lebesgue measure of E. We call w a weight if w ∈ Lloc R and w > 0 a.e. Differential forms are extensions of functions in R. For example, the function