Global Estimates for Singular Integrals of the Composition of the Maximal Operator and the Green's Operator

Let Ω be a bounded, convex domain and B a ball in R, n ≥ 2. We use σB to denote 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. We use |E| to denote the n-dimensional Lebesgue measure of the set E ⊆ R. We say that w is a weight if w ∈ Lloc R and w > 0, a.e. Differential forms are extensions of functions in R. For example, the function u x1, x2, . . . , xn is called a 0-form. Moreover, if u x1, x2, . . . , xn is differentiable, then it is called a differential 0-form. The 1-form u x in R can be written as u x ∑n i 1 ui x1, x2, . . . , xn dxi. If the coefficient functions ui x1, x2, . . . , xn , i 1, 2, . . . , n, are differentiable, then u x is called a differential 1-form. Similarly, a differential k-form u x is generated by {dxi1 ∧ dxi2 ∧ · · · ∧ dxik}, k 1, 2, . . . , n, that is,