Rough singular integrals associated with surfaces of Van der Corput type

In this paper, the authors establish the L (p) -mapping properties for a class of singular integrals along surfaces in a"e (n) of the form {I center dot(|u|)u': u a a"e (n) } as well as the related maximal operators provided that the function I center dot satisfies certain oscillatory integral estimates of Van der Corput type, and the integral kernels are given by the radial function {ie86-1} for gamma > 1 and the sphere function {ie86-2} for some beta > 0, which is distinct from H (1)(S (n-1)).