Weighted Norm Estimates and Representation Formulas for Rough Singular Integrals

Weighted norm estimates and representation formulas are proved for non-homogeneous singular integrals with no regularity condition on the kernel and only an L log L integrability condition. The representation formulas involve averages over a star-shaped set naturally associated with the kernel. The proof of the norm estimates is based on the representation formulas, some new variations of the Hardy-Littlewood maximal function, and weighted Littlewood-Paley theory.