Pointwise Estimates for a Class of Singular Integrals and Higher Commutators

§Ⅰ. Pointwise and Weak-type Estimates Denote M(R~k), M(R~k×R~k) as the space of Lebesque measurable functions defined on R~k, R~k×R~k, respectively. Let L: H→M(R~k×R~k) be a linear operator defined on H, which is a linearsubspace of M(R~k). There exist the follwing conditions on L and H: i) If U is a convex open set in R~k, x, y∈U and a∈H, then x·a∈H, and