  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.
