Weighted norm inequalities for rough singular integrals

Weighted norm estimates and representation formulae are proved for nonhomogeneous singular integrals with no regularity condition on the kernel and only an L log L integrability condition: the kernel of the operator is the product of a homogeneous function Ω in L log L and a radial function that satisfies certain integrability conditions.; The representation formulae involve averages over a star-shaped set naturally associated with the kernel. The proof of the norm estimates is based on the representation formulae, some new variations of the Hardy-Littlewood maximal function, and weighted Littlewood-Paley theory.; Weighted norm estimates are also proved for oscillatory singular integrals, where the oscillating term is the exponential of an arbitrary imaginary polynomial. Again the homogeneous part of the kernel is in L log L with no regularity condition but now the radial part is assumed to have bounded variation.; The conditions on the weight functions are similar to Muckenhoupt's A_p condition, but with rectangles related to the kernel of the operator instead of cubes. In general this is a more restrictive condition than the A_p condition, since the eccentricities of these rectangles vary unboundedly when Ω is essentially unbounded.