Rough multiple singular integrals along hypersurfaces

In this paper, the authors study the mapping properties of singular integrals on product domains with kernels in L(log+L)ɛ(Sm-1×Sn-1)(ɛ=1 or 2) supported by hyper-surfaces. The Lp bounds for such singular integral operators as well as the related Marcinkiewicz integral operators are established, provided that the lower dimensional maximal function is bounded on Lq(R3) for all q>1. The condition on the integral kernels is known to be optimal.